m. r. spiegel, vector analysis, schaum's series
TRANSCRIPT
8/19/2019 M. R. Spiegel, Vector Analysis, Schaum's Series
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8/19/2019 M. R. Spiegel, Vector Analysis, Schaum's Series
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C o p y r i g h t Q 1 9 5 9 b y M c G r a w - H i l l , I n c .
A l l R i g h t s R e s e r v e d . P r i n t e d i n t h e
U n i t e d S t a t e s o f A m e r i c a . N o p a r t o f t h i s p u b l i c a t i o n m a y b e r e p r o d u c e d ,
s t o r e d i n a r e t r i e v a l s y s t e m , o r t r a n s m i t t e d , i n a n y f o r m o r b y a n y m e a n s ,
e l e c t r o n i c , m e c h a n i c a l . p h o t o c o p y i n g . r e c o r d i n g . o r o t h e r w i s e . w i t h o u t t h e p r i o r
w r i t t e n p e r m i s s i o n o f t h e p u b l i s h e r .
I S B N 0 7 - 0 6 0 2 2 8 - X
2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 3 0 S H S H 8 7 6
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P r e f a c e
V e c t o r a n a l y s i s , w h i c h h a d i t s b e g i n n i n g s i n t h e m i d d l e o f t h e 1 9 t h c e n t u r y , h a s i n r e c e n t
y e a r s b e c o m e a n e s s e n t i a l p a r t o f t h e m a t h e m a t i c a l b a c k g r o u n d r e q u i r e d o f e n g i n e e r s , p h y -
s i c i s t s , m a t h e m a t i c i a n s a n d o t h e r s c i e n t i s t s . T h i s r e q u i r e m e n t i s f a r f r o m a c c i d e n t a l , f o r n o t
o n l y d o e s v e c t o r a n a l y s i s p r o v i d e a c o n c i s e n o t a t i o n f o r p r e s e n t i n g e q u a t i o n s a r i s i n g f r o m
m a t h e m a t i c a l f o r m u l a t i o n s o f p h y s i c a l a n d g e o m e t r i c a l p r o b l e m s b u t i t i s a l s o a n a t u r a l a i d
i n f o r m i n g m e n t a l p i c t u r e s o f p h y s i c a l a n d g e o m e t r i c a l i d e a s . I n s h o r t , i t m i g h t v e r y w e l l b e
c o n s i d e r e d a m o s t r e w a r d i n g l a n g u a g e a n d m o d e o f t h o u g h t f o r t h e p h y s i c a l s c i e n c e s .
T h i s b o o k i s d e s i g n e d t o b e u s e d e i t h e r a s a t e x t b o o k f o r a f o r m a l c o u r s e i n v e c t o r
a n a l y s i s o r a s a v e r y u s e f u l s u p p l e m e n t t o a l l c u r r e n t s t a n d a r d t e x t s . I t s h o u l d a l s o b e o f
c o n s i d e r a b l e v a l u e t o t h o s e t a k i n g c o u r s e s i n p h y s i c s , m e c h a n i c s , e l e c t r o m a g n e t i c t h e o r y ,
a e r o d y n a m i c s o r a n y o f t h e n u m e r o u s o t h e r f i e l d s i n w h i c h v e c t o r m e t h o d s a r e e m p l o y e d .
E a c h c h a p t e r b e g i n s w i t h a c l e a r s t a t e m e n t o f p e r t i n e n t d e f i n i t i o n s , p r i n c i p l e s a n d
t h e o r e m s t o g e t h e r w i t h i l l u s t r a t i v e a n d o t h e r d e s c r i p t i v e m a t e r i a l . T h i s i s f o l l o w e d b y
g r a d e d s e t s o f s o l v e d a n d s u p p l e m e n t a r y p r o b l e m s . T h e s o l v e d p r o b l e m s s e r v e t o i l l u s t r a t e
a n d a m p l i f y t h e t h e o r y , b r i n g i n t o s h a r p f o c u s t h o s e f i n e p o i n t s w i t h o u t w h i c h t h e s t u d e n t
c o n t i n u a l l y f e e l s h i m s e l f o n u n s a f e g r o u n d , a n d p r o v i d e t h e r e p e t i t i o n o f b a s i c p r i n c i p l e s
s o v i t a l t o e f f e c t i v e t e a c h i n g . N u m e r o u s p r o o f s o f t h e o r e m s a n d d e r i v a t i o n s o f f o r m u l a s
a r e i n c l u d e d a m o n g t h e s o l v e d p r o b l e m s . T h e l a r g e n u m b e r o f s u p p l e m e n t a r y p r o b l e m s
w i t h a n s w e r s s e r v e a s a c o m p l e t e r e v i e w o f t h e m a t e r i a l o f e a c h c h a p t e r .
T o p i c s c o v e r e d i n c l u d e t h e a l g e b r a a n d t h e d i f f e r e n t i a l a n d i n t e g r a l c a l c u l u s o f v e c -
t o r s , S t o k e s ' t h e o r e m , t h e d i v e r g e n c e t h e o r e m a n d o t h e r i n t e g r a l t h e o r e m s t o g e t h e r w i t h
m a n y a p p l i c a t i o n s d r a w n f r o m v a r i o u s f i e l d s . A d d e d f e a t u r e s a r e t h e c h a p t e r s o n c u r v i l i n -
e a r c o o r d i n a t e s a n d t e n s o r a n a l y s i s w h i c h s h o u l d p r o v e e x t r e m e l y u s e f u l i n t h e s t u d y o f
a d v a n c e d e n g i n e e r i n g , p h y s i c s a n d m a t h e m a t i c s .
C o n s i d e r a b l y m o r e m a t e r i a l h a s b e e n i n c l u d e d h e r e t h a n c a n b e c o v e r e d i n m o s t f i r s t
c o u r s e s . T h i s h a s b e e n d o n e t o m a k e t h e b o o k m o r e f l e x i b l e , t o p r o v i d e a m o r e u s e f u l b o o k
o f r e f e r e n c e , a n d t o s t i m u l a t e f u r t h e r i n t e r e s t i n t h e t o p i c s .
T h e a u t h o r g r a t e f u l l y a c k n o w l e d g e s h i s i n d e b t e d n e s s t o M r . H e n r y H a y d e n f o r t y p o -
g r a p h i c a l l a y o u t a n d a r t w o r k f o r t h e f i g u r e s . T h e r e a l i s m o f t h e s e f i g u r e s a d d s g r e a t l y t o
t h e e f f e c t i v e n e s s o f p r e s e n t a t i o n i n a s u b j e c t w h e r e s p a t i a l v i s u a l i z a t i o n s p l a y s u c h
a n i m -
p o r t a n t r o l e .
M . R . S P i E G E L
R e n s s e l a e r P o l y t e c h n i c I n s t i t u t e
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C o n t e n t s
C H A P T E R
P A G E
1 . V E C T O R S A N D S C A L A R S - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1
V e c t o r s . S c a l a r s . V e c t o r a l g e b r a . L a w s o f v e c t o r a l g e b r a . U n i t v e c t o r s . R e c t a n g u l a r u n i t
v e c t o r s . C o m p o n e n t s o f a v e c t o r . S c a l a r f i e l d s . V e c t o r f i e l d s .
2 . T H E D O T A N D C R O S S P R O D U C T - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1 6
D o t o r s c a l a r p r o d u c t s . C r o s s o r v e c t o r p r o d u c t s . T r i p l e p r o d u c t s . R e c i p r o c a l s e t s o f
v e c t o r s .
3 . V E C T O R D I F F E R E N T I A T I O N - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
3 5
O r d i n a r y d e r i v a t i v e s o f v e c t o r s . S p a c e c u r v e s . C o n t i n u i t y a n d d i f f e r e n t i a b i l i t y . D i f f e r e n -
t i a t i o n f o r m u l a s . P a r t i a l d e r i v a t i v e s o f v e c t o r s
D i f f e r e n t i a l s o f v e c t o r s . D i f f e r e n t i a l
g e o m e t r y . M e c h a n i c s .
4 . G R A D I E N T , D I V E R G E N C E A N D C U R L - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
5 7
T h e v e c t o r d i f f e r e n t i a l o p e r a t o r d e l . G r a d i e n t . D i v e r g e n c e . C u r l . F o r m u l a s i n v o l v i n g d e l .
I n v a r i a n c e .
5 . V E C T O R I N T E G R A T I O N - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
8 2
O r d i n a r y i n t e g r a l s o f v e c t o r s . L i n e i n t e g r a l s . S u r f a c e i n t e g r a l s . V o l u m e i n t e g r a l s .
6 . T H E D I V E R G E N C E T H E O R E M , S T O K E S ' T H E O R E M ,
A N D R E L A T E D I N T E G R A L T H E O R E M S - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1 0 6
T h e d i v e r g e n c e t h e o r e m o f G a u s s . S t o k e s ' t h e o r e m . G r e e n ' s t h e o r e m i n t h e p l a n e . R e -
l a t e d i n t e g r a l t h e o r e m s . I n t e g r a l o p e r a t o r f o r m f o r d e l .
7 . C U R V I L I N E A R C O O R D I N A T E S - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1 3 5
T r a n s f o r m a t i o n o f c o o r d i n a t e s . O r t h o g o n a l c u r v i l i n e a r c o o r d i n a t e s . U n i t v e c t o r s i n
c u r v i l i n e a r s y s t e m s . A r c l e n g t h a n d v o l u m e e l e m e n t s . G r a d i e n t , d i v e r g e n c e a n d c u r l .
S p e c i a l o r t h o g o n a l c o o r d i n a t e s y s t e m s . C y l i n d r i c a l c o o r d i n a t e s . S p h e r i c a l c o o r d i n a t e s .
P a r a b o l i c c y l i n d r i c a l c o o r d i n a t e s . P a r a b o l o i d a l c o o r d i n a t e s . E l l i p t i c c y l i n d r i c a l c o o r d i n a t e s .
P r o l a t e s p h e r o i d a l c o o r d i n a t e s . O b l a t e s p h e r o i d a l c o o r d i n a t e s . E l l i p s o i d a l c o o r d i n a t e s .
B i p o l a r c o o r d i n a t e s .
8 . T E N S O R A N A L Y S I S
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1 6 6
P h y s i c a l l a w s . S p a c e s o f N d i m e n s i o n s . C o o r d i n a t e t r a n s f o r m a t i o n s . T h e s u m m a t i o n
c o n v e n t i o n . C o n t r a v a r i a n t a n d c o v a r i a n t v e c t o r s . C o n t r a v a r i a n t , c o v a r i a n t a n d m i x e d
t e n s o r s . T h e K r o n e c k e r d e l t a . T e n s o r s o f r a n k g r e a t e r t h a n t w o . S c a l a r s o r i n v a r i a n t s .
T e n s o r f i e l d s . S y m m e t r i c a n d s k e w - s y m m e t r i c t e n s o r s . F u n d a m e n t a l o p e r a t i o n s w i t h
t e n s o r s . M a t r i c e s . M a t r i x a l g e b r a . T h e l i n e e l e m e n t a n d m e t r i c t e n s o r . C o n j u g a t e o r
r e c i p r o c a l t e n s o r s . A s s o c i a t e d t e n s o r s . L e n g t h o f a v e c t o r . A n g l e b e t w e e n v e c t o r s . P h y s i c a l
c o m p o n e n t s . C h r i s t o f f e l ' s s y m b o l s . T r a n s f o r m a t i o n l a w s o f C h r i s t o f f e l ' s s y m b o l s . G e o -
d e s i c s . C o v a r i a n t d e r i v a t i v e s . P e r m u t a t i o n s y m b o l s a n d t e n s o r s . T e n s o r f o r m o f g r a d i e n t ,
d i v e r g e n c e a n d c u r l . T h e i n t r i n s i c o r a b s o l u t e d e r i v a t i v e . R e l a t i v e a n d a b s o l u t e t e n s o r s .
I N D E X - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
2 1 8
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A V E C T O R i s a q u a n t i t y h a v i n g b o t h m a g i i i t u d a n d d i r e c t i o n s u c h a s d i s p l a c e m e n t , _ v e l o c i t y , f o r c e
a n d a c c e l e r a t i o n .
G r a p h i c a l l y a v e c t o r i s r e p r e s e n t e d b y a n a r r o w O P ( F i g . l ) d e -
f i n i n g t h e d i r e c t i o n , t h e m a g n i t u d e o f t h e v e c t o r b e i n g i n d i c a t e d b y
t h e l e n g t h o f t h e a r r o w . T h e t a i l e n d 0 o f t h e a r r o w i s c a l l e d t h e
o r i g i n o r i n i t i a l p o i n t o f t h e v e c t o r , a n d t h e h e a d P i s c a l l e d t h e
t e r m i n a l p o i n t o r t e r m i n u s .
A n a l y t i c a l l y a v e c t o r i s r e p r e s e n t e d b y a l e t t e r w i t h a n a r r o w
o v e r i t , a s A i n F i g . 1 , a n d i t s m a g n i t u d e i s d e n o t e d b y
I A I o r A . I n
p r i n t e d w o r k s , b o l d f a c e d t y p e , s u c h a s A , i s u s e d t o i n d i c a t e t h e
v e c t o r A w h i l e J A I o r A i n d i c a t e s i t s m a g n i t u d e . W e s h a l l u s e t h i s
b o l d f a c e d n o t a t i o n i n t h i s b o o k . T h e v e c t o r O P i s a l s o i n d i c a t e d a s
O P o r O P ; i n s u c h c a s e w e s h a l l d e n o t e i t s m a g n i t u d e b y O F , O P I ,
o r o f .
F i g . 1
A S C A L A R i s a q u a n t i t y h a v i n g m a g n i t u d e b u t ( n
d i r e c t i o n , e . g .
m
a
I S h , t f e , t e m e r
a n d
a n y r e a l n u m b e r . S c a l a r s a r e i n d i c a t e d b y l e t t e r s i n o r d i n a r y t y p e a s i n e l e m e n t a r y a l g e -
b r a . O p e r a t i o n s w i t h s c a l a r s f o l l o w t h e s a m e r u l e s a s i n e l e m e n t a r y a l g e b r a .
V E C T O R A L G E B R A . T h e o p e r a t i o n s o f a d d i t i o n , s u b t r a c t i o n a n d m u l t i p l i c a t i o n f a m i l i a r i n t h e a l g e -
b r a o f n u m b e r s o r s c a l a r s a r e , w i t h s u i t a b l e d e f i n i t i o n , c a p a b l e o f e x t e n s i o n
t o a n a l g e b r a o f v e c t o r s . T h e f o l l o w i n g d e f i n i t i o n s a r e f u n d a m e n t a l .
1 . T w o v e c t o r s A a n d B a r e e q u a l i f t h e y h a v e t h e s a m e m a g n i t u d e a n d d i r e c t i o n r e g a r d l e s s o f
t h e p o s i t i o n o f t h e i r i n i t i a l p o i n t s . T h u s A = B i n F i g . 2 .
2 . A v e c t o r h a v i n g d i r e c t i o n o p p o s i t e t o t h a t o f v e c t o r A b u t h a v i n g t h e s a m e m a g n i t u d e i s d e -
n o t e d b y - A ( F i g . 3 ) .
F i g . 2 F i g . 3
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2
V E C T O R S a n d S C A L A R S
3 .
T h e s u m o r r e s u l t a n t o f v e c t o r s A a n d B i s a
v e c t o r C f o r m e d b y p l a c i n g t h e i n i t i a l p o i n t o f B
o n t h e t e r m i n a l p o i n t o f A a n d t h e n j o i n i n g t h e
i n i t i a l p o i n t o f A t o t h e t e r m i n a l p o i n t o f B
( F i g . 4 ) . T h i s s u m i s w r i t t e n A + B , i . e . C = A + B .
T h e d e f i n i t i o n h e r e i s e q u i v a l e n t t o t h e p a r -
a l l e l o g r a m l a w f o r v e c t o r a d d i t i o n ( s e e P r o b . 3 ) .
E x t e n s i o n s t o s u m s o f m o r e t h a n t w o v e c t o r s
a r e i m m e d i a t e ( s e e P r o b l e m 4 ) .
F i g . 4
4 . T h e d i f f e r e n c e o f v e c t o r s A a n d B , r e p r e s e n t e d b y A - B , i s t h a t v e c t o r C w h i c h a d d e d t o B
y i e l d s v e c t o r A .
E q u i v a l e n t l y , A - B c a n b e d e f i n e d a s t h e s u m A + ( - B ) .
I f A = B , t h e n A - B i s d e f i n e d a s t h e n u l l o r z e r o v e c t o r a n d i s r e p r e s e n t e d b y t h e s y m -
b o l 0 o r s i m p l y 0 .
I t h a s z e r o m a g n i t u d e a n d n o s p e c i f i c d i r e c t i o n . A v e c t o r w h i c h i s n o t
n u l l i s a p r o p e r v e c t o r . A l l v e c t o r s w i l l b e a s s u m e d p r o p e r u n l e s s o t h e r w i s e s t a t e d .
5 . T h e p r o d u c t o f a v e c t o r A b y a s c a l a r m i s a v e c t o r m A w i t h m a g n i t u d e
I m f t i m e s t h e m a g n i -
t u d e o f A a n d w i t h d i r e c t i o n t h e s a m e a s o r o p p o s i t e t o t h a t o f A , a c c o r d i n g a s m i s p o s i t i v e
o r n e g a t i v e . I f m = 0 , m A i s t h e n u l l v e c t o r .
L A W S O F V E C T O R A L G E B R A . I f A , B a n d C a r e v e c t o r s a n d m a n d n a r e s c a l a r s , t h e n
1 . A + B = B + A
C o m m u t a t i v e L a w f o r A d d i t i o n
2 . A + ( B + C ) _ ( A + B ) + C
A s s o c i a t i v e L a w f o r A d d i t i o n
3 . m A = A m
C o m m u t a t i v e L a w f o r M u l t i p l i c a t i o n
4 . m ( n A ) _ ( m n ) A
A s s o c i a t i v e L a w f o r M u l t i p l i c a t i o n
5 .
( m + n ) A = m A + n A
D i s t r i b u t i v e L a w
6 . m ( A + B ) = m A + m B
D i s t r i b u t i v e L a w
N o t e t h a t i n t h e s e l a w s o n l y m u l t i p l i c a t i o n o f a v e c t o r b y o n e o r m o r e s c a l a r s i s u s e d . I n C h a p -
t e r 2 , p r o d u c t s o f v e c t o r s a r e d e f i n e d .
T h e s e l a w s e n a b l e u s t o t r e a t v e c t o r e q u a t i o n s i n t h e s a m e w a y a s o r d i n a r y a l g e b r a i c e q u a t i o n s .
F o r e x a m p l e , i f A + B = C t h e n b y t r a n s p o s i n g A = C - B .
A U N I T V E C T O R i s a v e c t o r h a v i n g u n i t m a g n i t u d e , i f
A i s a v e c t o r w i t h m a g n i t u d e A 0 ,
t h e n A / A i s a u n i t v e c t o r h a v i n g t h e s a m e - - d i r e c t i o n a s
A .
A n y v e c t o r A c a n b e r e p r e s e n t e d b y a u n i t v e c t o r a
i n t h e d i r e c t i o n o f A m u l t i p l i e d b y t h e m a g n i t u d e o f A . I n
s y m b o l s , A = A a .
T H E R E C T A N G U L A R U N I T V E C T O R S i , j , k . A n i m p o r -
t a n t s e t o f
u n i t v e c t o r s a r e t h o s e h a v i n g t h e d i r e c t i o n s o f t h e p o s -
i t i v e x , y , a n d z a x e s o f a t h r e e d i m e n s i o n a l r e c t a n g u -
l a r c o o r d i n a t e s y s t e m , a n d a r e d e n o t e d r e s p e c t i v e l y b y
i , j , a n d k ( F i g . 5 ) .
W e s h a l l u s e r i g h t - h a n d e d r e c t a n g u l a r c o o r d i n a t e
s y s t e m s u n l e s s o t h e r w i s e s t a t e d . S u c h a s y s t e m d e r i v e s
z
F i g . 5
Y
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V E C T O R S a n d S C A L A R S
i t s n a m e f r o m t h e f a c t t h a t a r i g h t t h r e a d e d s c r e w r o t a t -
e d t h r o u g h 9 0 0 f r o m O x t o O y w i l l a d v a n c e i n t h e p o s -
i t i v e z d i r e c t i o n , a s i n F i g . 5 a b o v e .
I n
g e n e r a l , t h r e e v e c t o r s A , B a n d C w h i c h h a v e
c o i n c i d e n t i n i t i a l p o i n t s a n d a r e n o t c o p l a n a r , i . e . d o
n o t l i e i n o r a r e n o t p a r a l l e l t o t h e s a m e p l a n e , a r e s a i d
t o f o r m a r i g h t - h a n d e d s y s t e m o r d e x t r a l s y s t e m i f a
r i g h t t h r e a d e d s c r e w r o t a t e d t h r o u g h a n a n g l e l e s s t h a n
1 8 0 ° f r o m A t o B w i l l a d v a n c e i n t h e d i r e c t i o n C a s
s h o w n i n F i g . 6 .
C O M P O N E N T S O F A V E C T O R . A n y v e c t o r A i n 3 d i -
m e n s i o n s c a n
a r e p r e -
s e n t e d w i t h i n i t i a l p o i n t a t t h e o r i g i n 0 o f a r e c a n g u l a r
c o o r d i n a t e
s y s t e m
( F i g . 7 ) . L e t ( A l , A 2 , A 3 )
b e t h e
r e c t a n g u l a r c o o r d i n a t e s o f t h e t e r m i n a l p o i n t o f v e c t o r A
w i t h i n i t i a l p o i n t a t 0 . T h e v e c t o r s A l i , A 2 j , a n d A 3 k
a r e c a l l e d t h e r e c t a l a r c o m p o n e n t v e c t o r s o r s i m p l y
c o m p o n e n t v e c t o r s o f A i n t h e x , y a n d z d i r e c t i o n s r e -
s p e c t i v e l y .
A 1 , A 2 a n d A 3 a r e c a l l e d t h e r e c t a n g u l a r
c o m p o n e n t s o r s i m p l y c o m p o n e n t s o f A i n t h e x , y a n d z
d i r e c t i o n s r e s p e c t i v e l y .
T h e s u m o r r e s u l t a n t o f
A l i , A 2 j
v e c t o r A s o t h a t w e c a n w r i t e
a n d A 3 k
i s t h e
A = A 1 i + A 2 I
+ A k
T h e m a g n i t u d e o f A i s
A =
I A I
A l + A 2 + A 3
F i g . 6
F i g . 7
I n p a r t i c u l a r , t h e p o s i t i o n v e c t o r o r r a d i u s v e c t o r r f r o m 0 t o t h e p o i n t ( x , y , z ) i s w r i t t e n
r
= x i + y j + z k
a n d h a s m a g n i t u d e r
=
I r I
=
x 2 + y 2 + z 2 .
3
. y 0 , 1 t o $
.
S C A L A R F I E L D . I f t o e a c h p o i n t ( x , y , z ) o f a r e g i o n R i n s p a c e t h e r e c o r r e s p o n d s a n u m b e r o r s c a l a r
t h e n
i s c a l l e d a s c a l a r f u n c t i o n o f p o s i t i o n
o r s c a l a r p o i n t f u n c t i o n
a n d w e s a y t h a t a s c a l a r f i e l d 0 h a s b e e n d e f i n e d i n R .
E x a m p l e s . ( 1 ) T h e t e m p e r a t u r e a t a n y p o i n t w i t h i n o r o n t h e e a r t h ' s s u r f a c e a t a c e r t a i n t i m e
d e f i n e s a s c a l a r f i e l d .
( 2 ) c t ( x , y , z ) = x 3 y - z 2
d e f i n e s a s c a l a r f i e l d .
A s c a l a r f i e l d w h i c h i s i n d e p e n d e n t o f t i m e i s c a l l e d a s t a t i o n a r y o r s t e a d y - s t a t e s c a l a r f i e l d .
V E C T O R F I E L D . I f t o e a c h p o i n t ( x , y , z ) o f a r e g i o n R i n s p a c e t h e r e c o r r e s p o n d s a v e c t o r V ( x , y , z ) ,
t h e n V i s c a l l e d a v e c t o r f u n c t i o n o f p o s i t i o n o r v e c t o r p o i n t f u n c t i o n a n d w e s a y
t h a t a v e c t o r f i e l d V h a s b e e n d e f i n e d i n R .
E x a m p l e s . ( 1 ) I f t h e v e l o c i t y a t a n y p o i n t ( x , y , z ) w i t h i n a m o v i n g f l u i d i s k n o w n a t a c e r t a i n
t i m e , t h e n a v e c t o r f i e l d i s d e f i n e d .
( 2 ) V ( x , y , z )
= x y 2 i - 2 y z 3 j + x 2 z k d e f i n e s a v e c t o r f i e l d .
A v e c t o r f i e l d w h i c h i s i n d e p e n d e n t o f t i m e i s c a l l e d a s t a t i o n a r y o r s t e a d y - s t a t e v e c t o r f i e l d .
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4
V E C T O R S a n d S C A L A R S
S O L V E D P R O B L E M S
1 . S t a t e w h i c h o f t h e f o l l o w i n g a r e s c a l a r s a n d w h i c h a r e v e c t o r s .
( a ) w e i g h t
( c ) s p e c i f i c h e a t
( e ) d e n s i t y
( g ) v o l u m e
( i ) s p e e d
( b ) c a l o r i e
( d ) m o m e n t u m
( f ) e n e r g y ( h ) d i s t a n c e ( j ) m a g n e t i c f i e l d i n t e n s i t y
A n s . ( a ) v e c t o r
( c ) s c a l a r
( e ) s c a l a r
( g ) s c a l a r
( i ) s c a l a r
( b ) s c a l a r
( d ) v e c t o r ( f ) s c a l a r
( h ) s c a l a r
( j ) v e c t o r
2 . R e p r e s e n t g r a p h i c a l l y ( a ) a f o r c e o f 1 0 l b i n a d i r e c t i o n 3 0 ° n o r t h o f e a s t
( b ) a f o r c e o f 1 5 l b i n a d i r e c t i o n 3 0 ° e a s t o f n o r t h .
N
N
U n i t = 5 l b
W
S
E
W
F i g . ( a )
S
F i g . ( b )
C h o o s i n g t h e u n i t o f m a g n i t u d e s h o w n , t h e r e q u i r e d v e c t o r s a r e a s i n d i c a t e d a b o v e .
F
3 . A n a u t o m o b i l e t r a v e l s 3 m i l e s d u e n o r t h , t h e n 5 m i l e s n o r t h e a s t . R e p r e s e n t t h e s e d i s p l a c e m e n t s
g r a p h i c a l l y a n d d e t e r m i n e t h e r e s u l t a n t d i s p l a c e m e n t ( a ) g r a p h i c a l l y , ( b ) a n a l y t i c a l l y .
V e c t o r O P o r A r e p r e s e n t s d i s p l a c e m e n t o f 3 m i d u e n o r t h .
V e c t o r P Q o r B r e p r e s e n t s d i s p l a c e m e n t o f 5 m i n o r t h e a s t .
V e c t o r O Q o r C r e p r e s e n t s t h e r e s u l t a n t d i s p l a c e m e n t o r
s u m o f v e c t o r s A a n d B ,
i . e . C = A + B . T h i s , i s t h e t r i a n g l e
l a w o f v e c t o r a d d i t i o n .
T h e r e s u l t a n t v e c t o r O Q c a n a l s o b e o b t a i n e d b y c o n -
s t r u c t i n g t h e d i a g o n a l o f t h e p a r a l l e l o g r a m O P Q R h a v i n g v e c t o r s
O P = A a n d O R ( e q u a l t o v e c t o r P Q o r B ) a s s i d e s . T h i s i s t h e
p a r a l l e l o g r a m l a w o f v e c t o r a d d i t i o n .
( a ) G r a p h i c a l D e t e r m i n a t i o n o f R e s u l t a n t . L a y o f f t h e 1 m i l e
u n i t o n v e c t o r O Q t o f i n d t h e m a g n i t u d e 7 . 4 m i ( a p p r o x i m a t e l y ) .
A n g l e
E O Q = 6 1 . 5 ° , u s i n g a p r o t r a c t o r . T h e n v e c t o r O Q h a s
m a g n i t u d e 7 . 4 m i a n d d i r e c t i o n 6 1 . 5 ° n o r t h o f e a s t .
( b ) A n a l y t i c a l D e t e r m i n a t i o n o f R e s u l t a n t . F r o m t r i a n g l e O P Q ,
d e n o t i n g t h e m a g n i t u d e s o f
A , B . C b y A , B , C , w e h a v e b y
t h e l a w o f c o s i n e s
C 2 = A 2 + B 2 - 2 A B c o s
L O P Q
= 3 2 +
5 2
- 2 ( 3 ) ( 5 ) c o s 1 3 5 °
= 3 4 + 1 5 V 2 = 5 5 . 2 1
a n d C = 7 . 4 3 ( a p p r o x i m a t e l y ) .
B y t h e l a w o f s i n e s ,
A
C
T h e n
s i n L O Q P
s i n L O P Q
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V E C T O R S a n d S C A L A R S
5
s i n L O Q P =
A s i n L O P Q
_
3 ( 0 . 7 0 7 )
°
,
C
7 . 4 3
T h u s v e c t o r O Q h a s m a g n i t u d e 7 . 4 3 m i a n d d i r e c t i o n ( 4 5 ° + 1 6 ° 3 5 ' ) = 6 1 ° 3 5 ' n o r t h o f e a s t .
4 . F i n d t h e s u m o r r e s u l t a n t o f t h e f o l l o w i n g d i s p l a c e m e n t s :
A , 1 0 f t n o r t h w e s t ; B , 2 0 f t 3 0 ° n o r t h o f e a s t ; C , 3 5 f t d u e s o u t h .
S e e F i g . ( a ) b e l o w .
A t t h e t e r m i n a l p o i n t o f A p l a c e t h e i n i t i a l p o i n t o f B .
A t t h e t e r m i n a l p o i n t o f B p l a c e t h e i n i t i a l p o i n t o f C .
T h e r e s u l t a n t D i s f o r m e d b y j o i n i n g t h e i n i t i a l p o i n t o f A t o t h e t e r m i n a l p o i n t o f C , i . e . D = A + B + C .
G r a p h i c a l l y t h e r e s u l t a n t i s m e a s u r e d t o h a v e m a g n i t u d e o f 4 . 1 u n i t s = 2 0 . 5 f t a n d d i r e c t i o n 6 0 0 s o u t h o f E .
F o r a n a n a l y t i c a l m e t h o d o f a d d i t i o n o f 3 o r m o r e v e c t o r s , e i t h e r i n a p l a n e o r i n s p a c e s e e P r o b l e m 2 6 .
F i g . ( a )
F i g . ( b )
5 . S h o w t h a t a d d i t i o n o f v e c t o r s i s c o m m u t a t i v e , i . e . A + B = B + A .
S e e F i g . ( b ) a b o v e .
O P + P Q = O Q o r
A + B
=
C ,
a n d
O R + R Q = O Q
o r
B + A
=
C .
T h e n
A + B
= B + A .
6 .
S h o w t h a t t h e a d d i t i o n o f v e c t o r s i s a s s o c i a t i v e , i . e . A + ( B + C ) = ( A + B ) + C C .
a n d
O P + P Q
P Q + Q R
= O Q =
= P R =
( A + B ) ,
( B + C ) .
O P + P R
= O R = D , i . e . A + ( B + C )
= D .
O Q + Q R
= O R =
D , i . e . ( A + B ) + C =
D .
T h e n
A + ( B + C )
=
( A + B ) + C .
E x t e n s i o n s o f t h e r e s u l t s o f P r o b l e m s 5 a n d 6 s h o w
t h a t t h e o r d e r o f a d d i t i o n o f a n y n u m b e r o f v e c t o r s i s i m -
m a t e r i a l .
= =
0 . 2 8 5 5
a n d
L O Q P
= 1 6 3 5
.
7 . F o r c e s F 1 , F 2 0
. . .
, F 6 a c t a s s h o w n o n o b j e c t P . W h a t f o r c e i s n e e d e d t o p r e v e n t P f r o m m o v -
i n g ?
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s
V E C T O R S a n d S C A L A R S
S i n c e t h e o r d e r o f a d d i t i o n o f v e c t o r s i s i m m a t e r i a l , w e m a y s t a r t w i t h a n y v e c t o r , s a y F l . T o F l a d d
F 2 , t h e n F 3 ,
e t c . T h e v e c t o r d r a w n f r o m t h e i n i t i a l p o i n t o f F l t o t h e t e r m i n a l p o i n t o f F 6 i s t h e r e s u l t a n t
R , i . e . R = F 1 + F 2 + F 3 + F µ + F 5 + F 6 .
T h e f o r c e n e e d e d t o p r e v e n t P f r o m m o v i n g i s - R w h i c h i s a v e c t o r e q u a l i n m a g n i t u d e t o R b u t o p p o s i t e
i n d i r e c t i o n a n d s o m e t i m e s c a l l e d t h e e q u i l i b r a n t .
F 4
8 . G i v e n v e c t o r s A , B a n d C ( F i g . 1 a ) , c o n s t r u c t ( a ) A - B + 2 C ( b ) 3 C - - z ( 2 A - B ) .
( a )
F i g . 1 ( a )
( b )
F i g . 2 ( a )
F i g . 1 ( b )
F i g . 2 ( b )
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V E C T O R S a n d S C A L A R S
9 . A n a i r p l a n e m o v e s i n a n o r t h w e s t e r l y d i r e c t i o n a t
1 2 5 m i / h r r e l a t i v e t o t h e g r o u n d , d u e t o t h e f a c t
t h e r e i s a w e s t e r l y w i n d o f 5 0 m i / h r r e l a t i v e t o
t h e g r o u n d . H o w f a s t a n d i n w h a t d i r e c t i o n w o u l d
t h e p l a n e h a v e t r a v e l e d i f t h e r e w e r e n o w i n d ?
L e t W
= w i n d v e l o c i t y
V a = v e l o c i t y o f p l a n e w i t h w i n d
V b = v e l o c i t y o f p l a n e w i t h o u t w i n d
- w
T h e n V a =
V b + W
o r
V b
= V a - W =
V a + ( - W )
V b h a s m a g n i t u d e 6 . 5 u n i t s = 1 6 3 m i / h r a n d d i r e c t i o n 3 3 ° n o r t h o f w e s t .
7
1 0 . G i v e n t w o n o n - c o l l i n e a r v e c t o r s a a n d b , f i n d a n e x p r e s s i o n f o r a n y v e c t o r r l y i n g i n t h e p l a n e d e -
t e r m i n e d b y a a n d b .
N o n - c o l l i n e a r v e c t o r s a r e v e c t o r s w h i c h a r e n o t p a r a l l e l t o
t h e s a m e l i n e .
H e n c e w h e n t h e i r i n i t i a l p o i n t s c o i n c i d e , t h e y
d e t e r m i n e a p l a n e . L e t r b e a n y v e c t o r l y i n g i n t h e p l a n e o f a
a n d b a n d h a v i n g i t s i n i t i a l p o i n t c o i n c i d e n t w i t h t h e i n i t i a l
p o i n t s o f a a n d b a t O . F r o m t h e t e r m i n a l p o i n t R o f r c o n s t r u c t
l i n e s p a r a l l e l t o t h e v e c t o r s a a n d b a n d c o m p l e t e t h e p a r a l l e l -
o g r a m O D R C b y e x t e n s i o n o f t h e l i n e s o f a c t i o n o f a a n d b i f
n e c e s s a r y . F r o m t h e a d j o i n i n g f i g u r e
O D = x ( O A ) = x a , w h e r e x i s a s c a l a r
O C = y ( O B ) = y b , w h e r e y i s a s c a l a r .
B u t b y t h e p a r a l l e l o g r a m l a w o f v e c t o r a d d i t i o n
O R = O D + O C
o r
r = x a + y b
w h i c h i s t h e r e q u i r e d e x p r e s s i o n . T h e v e c t o r s x a a n d y b a r e c a l l e d c o m p o n e n t v e c t o r s o f r i n t h e d i r e c t i o n s
a a n d b r e s p e c t i v e l y . T h e s c a l a r s x a n d y m a y b e p o s i t i v e o r n e g a t i v e d e p e n d i n g o n t h e r e l a t i v e o r i e n t a t i o n s
o f t h e v e c t o r s . F r o m t h e m a n n e r o f c o n s t r u c t i o n i t i s c l e a r t h a t x a n d y a r e u n i q u e f o r a g i v e n a , b , a n d r .
T h e v e c t o r s a a n d b a r e c a l l e d b a s e v e c t o r s i n a p l a n e .
1 1 . G i v e n t h r e e n o n - c o p l a n a r v e c t o r s a , b , a n d c , f i n d a n e x p r e s s i o n f o r a n y v e c t o r r i n t h r e e d i m e n -
s i o n a l s p a c e .
N o n - c o p l a n a r v e c t o r s a r e v e c t o r s w h i c h a r e n o t p a r a l -
l e l t o t h e s a m e p l a n e .
H e n c e w h e n t h e i r i n i t i a l p o i n t s c o -
i n c i d e t h e y d o n o t l i e i n t h e s a m e p l a n e .
L e t r b e a n y v e c t o r i n s p a c e h a v i n g i t s i n i t i a l p o i n t c o -
i n c i d e n t w i t h t h e i n i t i a l p o i n t s o f a , b a n d c a t O . T h r o u g h
t h e t e r m i n a l p o i n t o f r p a s s p l a n e s p a r a l l e l r e s p e c t i v e l y
t o t h e p l a n e s d e t e r m i n e d b y a a n d b , b a n d c , a n d a a n d c ;
a n d c o m p l e t e t h e p a r a l l e l e p i p e d P Q R S T U V b y e x t e n s i o n o f
t h e l i n e s o f a c t i o n o f a , b a n d c i f n e c e s s a r y .
F r o m t h e
a d j o i n i n g f i g u r e ,
O V
= x ( O A ) = x a
w h e r e x i s a s c a l a r
O P = y ( O B ) = y b
w h e r e y i s a s c a l a r
O T = z ( O C ) = z c
w h e r e z i s a s c a l a r .
B u t O R = O V + V Q + Q R = O V + O P + O T
o r
r
= x a + y b + z c .
F r o m t h e m a n n e r o f c o n s t r u c t i o n i t i s c l e a r t h a t x , y a n d z a r e u n i q u e f o r a g i v e n a ,
b , c a n d r .
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8
V E C T O R S a n d S C A L A R S
T h e v e c t o r s x a , y b a n d z c a r e c a l l e d c o m p o n e n t v e c t o r s o f r i n d i r e c t i o n s a , b a n d c r e s p e c t i v e l y . T h e
v e c t o r s a , b a n d c a r e c a l l e d b a s e v e c t o r s i n t h r e e d i m e n s i o n s .
A s a s p e c i a l c a s e , i f a , b a n d c a r e t h e u n i t v e c t o r s i , j a n d k , w h i c h a r e m u t u a l l y p e r p e n d i c u l a r , w e
s e e t h a t a n y v e c t o r r c a n b e e x p r e s s e d u n i q u e l y i n t e r m s o f i , j , k b y t h e e x p r e s s i o n r = x i + y j + z k .
A l s o , i f c = 0 t h e n r m u s t l i e i n t h e p l a n e o f a a n d b s o t h e r e s u l t o f P r o b l e m 1 0 i s o b t a i n e d .
1 2 . P r o v e t h a t i f a a n d b a r e n o n - c o l l i n e a r t h e n x a + y b = 0 i m p l i e s x = y = 0 .
S u p p o s e x / 0 . T h e n x a + y b = 0 i m p l i e s x a = - y b o r a = - ( y / x ) b ,
i . e . a a n d b m u s t b e p a r a l l e l t o
t o t h e s a m e l i n e ( c o l l i n e a r ) c o n t r a r y t o h y p o t h e s i s . T h u s x = 0 ; t h e n y b = 0 , f r o m w h i c h y = 0 .
1 3 . I f x l a + y l b = x 2 a + y 2 b ,
w h e r e a a n d b a r e n o n - c o l l i n e a r , t h e n x 1 = x 2 a n d y l = y 2
x 1 a + y l b = x 2 a + y 2 b c a n b e w r i t t e n
x 1 a + y 1 b - ( x 2 a + y 2 b ) = 0
o r
( x 1 - - x 2 ) a + ( y l - y 2 ) b = 0 .
H e n c e b y P r o b l e m 1 2 ,
x l - x 2 = 0 , y 1 - y 2 = 0 o r
x l = x 2 , y i = y 2
.
1 4 . P r o v e t h a t i f a , b a n d c a r e n o n - c o p l a n a r t h e n x a + y b + z c = 0 i m p l i e s x = y = z = 0 .
S u p p o s e
x / 0 .
T h e n x a + y b + z c = 0
i m p l i e s
x a = - y b - z c
o r a = - ( y / x ) b - ( z / x ) c .
B u t
- ( y / x ) b - ( z / x ) c i s a v e c t o r l y i n g i n t h e p l a n e o f b a n d c ( P r o b l e m 1 0 ) , i . e . a l i e s i n t h e
p l a n e o f b a n d c
w h i c h i s c l e a r l y a c o n t r a d i c t i o n t o t h e h y p o t h e s i s t h a t a , b a n d c a r e n o n - c o p l a n a r . H e n c e x = 0 . B y s i m -
i l a r r e a s o n i n g , c o n t r a d i c t i o n s a r e o b t a i n e d u p o n s u p p o s i n g y / 0 a n d z / 0 .
1 5 . I f x 1 a + y 1 b + z l c
= x 2 a + y 2 b + z 2 c , w h e r e a , b a n d c a r e n o n - c o p l a n a r , t h e n x 1 = x 2 , y 1 = y 2 ,
z 1 = z 2 .
T h e e q u a t i o n c a n b e w r i t t e n ( x 1 - x 2 ) a + ( y 1 - y 2 ) b + ( z l - z 2 ) c = 0 . T h e n b y P r o b l e m 1 4 , x l - x 2 = 0 ,
y 1 - y 2 = 0 , z 1 - z 2 = 0
o r x 1 = x 2 , y 1 = y 2 , z 1 = z 2 .
1 6 . P r o v e t h a t t h e d i a g o n a l s o f a p a r a l l e l o g r a m b i s e c t e a c h o t h e r .
L e t A B C D b e t h e g i v e n p a r a l l e l o g r a m w i t h d i a g o n a l s i n -
t e r s e c t i n g a t P .
S i n c e B D + a = b , B D = b - a . T h e n B P = x ( b - a ) .
S i n c e A C = a + b , A P = y ( a + b ) .
B u t
A B = A P + P B = A P - B P ,
i . e . a = y ( a + b ) - x ( b - a ) = ( x + y ) a + ( y - x ) b .
S i n c e a a n d b a r e n o n - c o l l i n e a r w e h a v e b y P r o b l e m 1 3 ,
x + y = 1
a n d y - x = 0 ,
i . e .
x = y = 2
a n d P i s t h e m i d -
p o i n t o f b o t h d i a g o n a l s .
1 7 . I f t h e m i d p o i n t s o f t h e c o n s e c u t i v e s i d e s o f a n y q u a d r i l a t e r a l a r e c o n n e c t e d b y s t r a i g h t l i n e s ,
p r o v e t h a t t h e r e s u l t i n g q u a d r i l a t e r a l i s a p a r a l l e l o g r a m .
L e t A B C D b e t h e g i v e n q u a d r i l a t e r a l a n d P , Q , R , S t h e m i d p o i n t s o f i t s s i d e s . R e f e r t o F i g . ( a ) b e l o w .
T h e n P Q = 2 ( a + b ) ,
Q R = 2 ( b + c ) ,
R S = 2 ( c + d ) ,
S P = 2 ( d + a ) .
B u t a + b + c + d = 0 .
T h e n
P Q = 2 ( a + b ) = - 2 ( c + d ) = S R
a n d
Q R = 2 ( b + c ) 2 ( d + a ) = P S
T h u s o p p o s i t e s i d e s a r e e q u a l a n d p a r a l l e l a n d P Q R S i s a p a r a l l e l o g r a m .
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V E C T O R S a n d S C A L A R S
g
1 8 . L e t P . , P 2
1
P 3 b e p o i n t s f i x e d r e l a t i v e t o a n o r i g i n 0 a n d l e t r 1 , r 2 , r 3 b e p o s i t i o n v e c t o r s f r o m
0 t o e a c h p o i n t . S h o w t h a t i f t h e v e c t o r e q u a t i o n a l r l + a 2 r 2 + a 3 r 3 = 0 h o l d s w i t h r e s p e c t t o
o r i g i n 0 t h e n i t w i l l h o l d w i t h r e s p e c t t o a n y o t h e r o r i g i n 0 ' i f a n d o n l y i f
a l + a 2 + a 3 = 0 .
L e t
r 3 b e t h e p o s i t i o n v e c t o r s o f P I , P 2 a n d P 3 w i t h r e s p e c t t o 0 ' a n d l e t v b e t h e p o s i t i o n
v e c t o r o f 0 ' w i t h r e s p e c t t o 0 . W e s e e k c o n d i t i o n s u n d e r w h i c h t h e e q u a t i o n a , r + a r ' + a r `
= 0
w i l l
h o l d i n t h e n e w r e f e r e n c e s y s t e m .
F r o m F i g . ( b ) b e l o w , i t i s c l e a r t h a t
r 1 = v + r i , r 2 = v + r 2 , r 3 = v + r 3
s o t h a t
a 1 r 1 + a 2 r 2 + a
3
r
3
= 0
b e c o m e s
a l r l + a 2 r 2 + a 3 r 3
= a , ( v + r ' ) + a 2 ( v + r 2 ) + a 3 ( v + r 3 )
_ ( a l + a 2 + a 3 ) v + a l r 1 + a 2 r 2 + a 3 r 3
=
0
T h e r e s u l t
a l r j + a 2 r 2 + a 3 r 3 = 0
w i l l h o l d i f a n d o n l y i f
( a l + a 2 + a 3 ) v = 0 ,
i . e .
a l + a 2 + a 3
=
0 .
T h e r e s u l t c a n b e g e n e r a l i z e d .
O '
F i g . ( a ) F i g . ( b )
1 9 . F i n d t h e e q u a t i o n o f a s t r a i g h t l i n e w h i c h p a s s e s t h r o u g h t w o g i v e n p o i n t s A a n d B h a v i n g p o s i -
t i o n v e c t o r s a a n d b w i t h r e s p e c t t o a n o r i g i n 0 .
L e t r b e t h e p o s i t i o n v e c t o r o f a n y p o i n t P o n t h e l i n e
t h r o u g h A a n d B .
F r o m t h e a d j o i n i n g f i g u r e ,
O A + A P = O P
o r
a + A P = r ,
i . e . A P = r - a
a n d O A + A B = O B o r
a + A B = b ,
i . e .
A B = b - a
S i n c e A P a n d A B a r e c o l l i n e a r ,
A P = t A B o r
r - a = t ( b - - a ) .
T h e n t h e r e q u i r e d e q u a t i o n i s
r = a + t ( b - a )
o r
r = ( 1 - t ) a + t b
I f t h e e q u a t i o n i s w r i t t e n
( 1 - t ) a + t b - r = 0 , t h e s u m
o f t h e c o e f f i c i e n t s o f a , b a n d r i s 1 - t + t - 1 = 0 . H e n c e b y
P r o b l e m 1 8 i t i s s e e n t h a t t h e p o i n t P i s a l w a y s o n t h e l i n e
j o i n i n g A a n d B a n d d o e s n o t d e p e n d o n t h e c h o i c e o f o r i g i n
0 , w h i c h i s o f c o u r s e a s i t s h o u l d b e .
A n o t h e r M e t h o d . S i n c e A P a n d P B a r e c o l l i n e a r , w e h a v e f o r s c a l a r s m a n d n :
S o l v i n g , r
m a + n b
m + n
m A P = n P B o r
m ( r - a ) = n ( b - r )
w h i c h i s c a l l e d t h e s y m m e t r i c f o r m .
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1 0
2 0 .
V E C T O R S a n d S C A L A R S
( a ) F i n d t h e p o s i t i o n v e c t o r s r 1 a n d r 2 f o r t h e
p o i n t s P ( 2 , 4 , 3 ) a n d Q ( 1 , - 5 , 2 ) o f a r e c t a n g u l a r
c o o r d i n a t e s y s t e m i n t e r m s o f t h e u n i t v e c t o r s
i , j , k .
( b ) D e t e r m i n e g r a p h i c a l l y a n d a n a l y t i -
c a l l y t h e r e s u l t a n t o f t h e s e p o s i t i o n v e c t o r s .
( a ) r 1 = O P = O C + C B + B P = 2 i + 4 j + 3 k
r 2 = O Q = O D + D E + E Q =
i - 5 j + 2 k
( b ) G r a p h i c a l l y , t h e r e s u l t a n t o f r 1 a n d r 2 i s o b t a i n e d
a s t h e d i a g o n a l O R o f P a r a l l e l o g r a m O P R Q . A n a -
l y t i c a l l y , t h e r e s u l t a n t o f r 1 a n d r 2 i s g i v e n b y
r 1 + r 2 =
( 2 i + 4 j + 3 k ) + ( i - 5 j + 2 k ) =
2 1 . P r o v e t h a t t h e m a g n i t u d e A o f t h e v e c t o r A =
A 1 i + A 2 j + A 3 k i s A =
A 1 + A 2 + A 3
.
B y t h e P y t h a g o r e a n t h e o r e m ,
_
( O P ) 2 =
( O Q ) 2 + ( Q P ) 2
w h e r e O P d e n o t e s t h e m a g n i t u d e o f v e c t o r O P , e t c .
S i m i l a r l y ,
( O Q ) 2 = ( O R ) 2 + ( R Q ) 2 .
T h e n
( 5 P ) 2 =
( O R ) 2 + ( R Q ) 2 + ( Q P ) 2 o r
A 2 = A i + A 2 + A 2 , i . e . A =
A l + A 2 + A .
2 2 . G i v e n
r 1 = 3 i - 2 j + k ,
r 2 = 2 i - 4 j - 3 k ,
r 3 = - i + 2 j + 2 k ,
( a ) r 3 ,
( b ) r 1 + r 2 + r 3 ,
( c ) 2 r 1 - 3 r 2 - - 5 r 3 .
( a )
I r 3 I =
I - i
+ 2 j + 2 k I
= V ' ( - 1 ) 2 + ( 2 ) 2 + ( 2 ) 2
=
3 .
f i n d t h e m a g n i t u d e s o f
( b ) r 1 + r 2 + r 3 =
( 3 i - 2 j + k ) + ( 2 i - 4 ; j - 3 k ) + ( - i + 2 j + 2 k )
= 4 i - 4 j + O k =
4 i - 4 j
T h e n
I r 1 + r 2 + r 3 I = 1 4 i
- 4 j + 0 k
( 4 ) 2 + ( - 4 ) 2 + ( 0 ) 2
= 3 2 = 4 / 2 .
( c )
2 r 1 - 3 r 2 - 5 r 3
=
2 ( 3 i - 2 j + k ) - - 3 ( 2 i - 4 j - 3 k ) - 5 ( - i + 2 j + 2 k )
= 6 i - 4 j + 2 k - 6 i + 1 2 j + 9 k + 5 i - 1 0 j - 1 0 k = 5 i - 2 j + k .
T h e n
I 2 r 1 - 3 r 2 - 5 r 3
I
= 1 5 i - 2 j + k I
= V ' ( 5 ) 2 + ( - 2 ) 2 + ( 1 ) 2 =
V 1 3 0 .
Y
2 3 . I f
r 1 = 2 i - j + k ,
r 2 = i + 3 j - 2 k ,
r 3 = - 2 1 + j - - 3 k a n d r 4 = 3 i + 2 j + 5 k , f i n d s c a l a r s a , b , c s u c h
t h a t r 4 = a r t + b r 2 + c r 3 .
W e r e q u i r e 3 i + 2 j + 5 k
= a ( 2 i - j + k ) + b ( i + 3 j - 2 k ) + c ( - 2 i + j - 3 k )
_ ( 2 a + b - 2 c ) i + ( - a + 3 b + c ) j + ( a - 2 b - 3 c ) k .
S i n c e
i , j , k a r e n o n - c o p l a n a r w e h a v e b y P r o b l e m 1 5 ,
2 a + b - 2 c = 3 ,
- a + 3 b + c = 2 , a - 2 b - 3 c = 5 .
S o l v i n g ,
a = - 2 , b = 1 ,
c = - 3 a n d
r 4 = - 2 r 1 + r 2 - 3 r 3 .
T h e v e c t o r r 4 i s s a i d t o b e l i n e a r l y d e p e n d e n t o n r 1 , r 2 , a n d r 3 ; i n o t h e r w o r d s r 1 , r 2 , r 3 a n d r 4 c o n s t i t u t e a
l i n e a r l y d e p e n d e n t s e t o f v e c t o r s . O n t h e o t h e r h a n d a n y t h r e e ( o r f e w e r ) o f t h e s e v e c t o r s a r e l i n e a r l y i n -
d e p e n d e n t .
I n g e n e r a l t h e v e c t o r s
A , B , C , . . .
a r e c a l l e d l i n e a r l y d e p e n d e n t i f w e c a n f i n d a s e t o f s c a l a r s ,
a , b , c , . . . ,
n o t a l l z e r o , s o t h a t a A + b B + c C + . . . = 0 . o t h e r w i s e t h e y a r e l i n e a r l y i n d e p e n d e n t .
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V E C T O R S a n d S C A L A R S
2 4 . F i n d a u n i t v e c t o r p a r a l l e l t o t h e r e s u l t a n t o f v e c t o r s r 1 = 2 i + 4 j - 5 k , r 2 = i + 2 j + 3 k
.
R e s u l t a n t R = r 1 + r 2 = ( 2 i + 4 j - 5 k ) + ( i + 2 j + 3 k ) = 3 i + 6 j - 2 k .
R =
P . I
=
1 3 i + 6 j - 2 k I
=
/ ( - 3 7 + ( 6 ) 2 + ( - 2 ) 2
= 7 .
T h e n a u n i t v e c t o r p a r a l l e l t o R i s
R =
3 i + 6 j - 2 k
=
3 i + 6 j - 2 k .
R
7 7 7 7
3 i
6 j
2
V 3 ) 2
6 2
2 2
C h e c k :
I i
+
i
-
7
7 7
1
=
( 3 + ( 6 ) + ( - i )
=
1 .
2 5 . D e t e r m i n e t h e v e c t o r h a v i n g i n i t i a l p o i n t P ( x 1 ,
y 1 , z 1 )
a n d t e r m i n a l p o i n t Q ( x 2 , y 2 , z 2 ) a n d f i n d i t s m a g n i t u d e .
T h e p o s i t i o n v e c t o r o f P i s
r 1 = x 1 i + y 1 j + z 1 k .
T h e p o s i t i o n v e c t o r o f Q i s
r 2 = x 2 i + y 9 j + z 2 k .
r 1 + P Q = r 2
o r
P Q = r 2 - r 1 = ( x 2 i + y 2 j + z 2 k ) - ( x l i + y l j + z l k )
( x 2 - x 1 ) i + ( y 2 - ) j + ( z 2 - z 1 ) k .
M a g n i t u d e o f P Q = P Q = ( x 2 - x 1 ) 2 + ( y 2
-
y 1 ) 2 + ( z 2 - z 1
N o t e t h a t t h i s i s t h e d i s t a n c e b e t w e e n p o i n t s P a n d Q .
1 1
2 6 . F o r c e s A , B a n d C a c t i n g o n a n o b j e c t a r e g i v e n i n t e r m s o f t h e i r c o m p o n e n t s b y t h e v e c t o r e q u a -
t i o n s
A = A 1 i + A 2 j + A 3 k ,
B = B l i + B 2 j + B 3 k ,
C = C l i + C 2 j + C 3 k .
F i n d t h e m a g n i t u d e o f t h e
r e s u l t a n t o f t h e s e f o r c e s .
.
R e s u l t a n t f o r c e
R = A + B + C = ( A 1 + B 1 + C 1 ) i + ( A 2 + B 2 + C 2 ) j + ( A 3 + B 3
+ C 3 ) k .
2 7 .
M a g n i t u d e o f r e s u l t a n t -
( A 1 + B 1 + C 1 ) 2 + ( A 2 + B 2 + C 2 ) 2 + ( A 3 + B 3 + C 3 ) 2 .
T h e r e s u l t i s e a s i l y e x t e n d e d t o m o r e t h a n t h r e e f o r c e s .
D e t e r m i n e t h e a n g l e s a , ( 3 a n d y w h i c h t h e v e c t o r
r = x i + y j + z k
m a k e s w i t h t h e p o s i t i v e d i r e c -
t i o n s o f t h e c o o r d i n a t e a x e s a n d s h o w t h a t
c o s t a + c o s t r 3 + c o s t y =
1 .
R e f e r r i n g t o t h e f i g u r e , t r i a n g l e O A P i s a r i g h t
t r i a n g l e w i t h r i g h t a n g l e a t A ; t h e n c o s a =
I r l
.
S i m -
i l a r l y f r o m r i g h t t r i a n g l e s O B P a n d O C P , c o s ( 3 =
Y
a n d c o s y = z .
A l s o ,
I r l
I r l
I r I = r = v x 2 + + y 2 + z 2
T h e n c o s a =
x
,
c o s p =
y
,
c o s y =
z
f r o m
w h i c h a , 0 , y c a n b e o b t a i n e d . F r o m t h e s e i t f o l l o w s
t h a t
c o s t a + c o s t ( 3 + c o s t y =
x 2 + y 2 + z 2
r 2
=
1 .
T h e n u m b e r s c o s a , c o s ( 3 , c o s y a r e c a l l e d t h e d i r e c t i o n
x
z
c o s i n e s o f t h e v e c t o r O P .
2 8 . D e t e r m i n e a s e t o f e q u a t i o n s f o r t h e s t r a i g h t l i n e p a s s i n g t h r o u g h t h e p o i n t s
P ( x 1 , y 1 , z 1 ) a n d
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1 2
V E C T O R S a n d S C A L A R S
L e t r 1 a n d r 2 b e t h e p o s i t i o n v e c t o r s o f P a n d Q r e s p e c -
t i v e l y , a n d r t h e p o s i t i o n v e c t o r o f a n y p o i n t R o n t h e l i n e
j o i n i n g P a n d Q .
r 1 + P R =
r
o r
P R =
r - r 1
r 1 + P Q =
r 2
o r
P Q =
r 2 - r 1
B u t
P R = t P Q w h e r e t
i s a s c a l a r .
T h e n r - r 1 =
t ( r 2 - r 1 )
i s t h e r e q u i r e d v e c t o r e q u a t i o n o f t h e s t r a i g h t l i n e
( c o m p a r e w i t h P r o b l e m 1 9 ) .
I n r e c t a n g u l a r c o o r d i n a t e s w e h a v e , s i n c e
r
= x i + y j + z k ,
( x i + y j + z k ) - ( x 1 i + y 1 ) + z 1 k )
=
t [ ( x 2 i + y 2 j + z 2 k ) - ( x 1 i + y 1 j + z 1 k ) ]
o r
( x - x 1 ) i + ( y - y 1 ) j + ( z - z 1 ) k
=
t [ ( x 2 - x 1 ) i + ( y 2 - y 1 ) j + ( z 2 - z 1 ) k ]
S i n c e i , j , k a r e n o n - c o p l a n a r v e c t o r s w e h a v e b y P r o b l e m 1 5 ,
x - x 1 =
t
( x 2 - x 1 ) ,
y - y 1 =
t ( y 2 - y 1 ) ,
z - z 1
=
t ( z 2 - z 1 )
a s t h e p a r a m e t r i c e q u a t i o n s o f t h e l i n e , t b e i n g t h e p a r a m e t e r . E l i m i n a t i n g t , t h e e q u a t i o n s b e c o m e
X - x
x 2 - x
Y - Y 1
z - z 1
Y 2 ` Y 1
z 2 - z 1
2 9 . G i v e n t h e s c a l a r f i e l d d e f i n e d b y
( x , y , z )
=
3 x 2 2 - x y 3 + 5 ,
f i n d
a t t h e p o i n t s
( a )
( 0 , 0 , 0 ) ,
( b ) ( 1 , - 2 , 2 )
( c ) ( - 1 , - 2 , - 3 ) .
( a ) 0 ( 0 , 0 , 0 )
=
3 ( 0 ) 2 ( 0 ) - ( 0 ) ( 0 ) 3 + 5
=
0 - 0 + 5
= 5
( b ) 0 0 , - 2 , 2 )
=
3 ( 1 ) 2 ( 2 ) - ( 1 ) ( - 2 ) 3 + 5
=
6 + 8 + 5
= 1 9
( c )
) ( - 1 , - 2 , - 3 ) =
3 ( - 1 ) 2 ( - 3 ) - ( - 1 ) ( - 2 ) 3 + 5
=
- 9 - 8 + 5 - 1 2
3 0 . G r a p h t h e v e c t o r f i e l d s d e f i n e d b y :
( a ) V ( x , y ) = x i + y j ,
( b ) V ( x , y ) _ - x i - y j ,
( c ) V ( x , y , z )
= x i + y j + A .
( a ) A t e a c h p o i n t ( x , y ) , e x c e p t ( 0 , 0 ) , o f t h e x y p l a n e t h e r e i s d e f i n e d a u n i q u e v e c t o r x i + y j o f m a g n i t u d e
h a v i n g d i r e c t i o n p a s s i n g t h r o u g h t h e o r i g i n a n d o u t w a r d f r o m i t . T o s i m p l i f y g r a p h i n g p r o c e -
d u r e s , n o t e t h a t a l l v e c t o r s a s s o c i a t e d w i t h p o i n t s o n t h e c i r c l e s x 2 + y 2 = a 2 a > 0 h a v e m a g n i t u d e
a . T h e f i e l d t h e r e f o r e a p p e a r s a s i n F i g u r e ( a ) w h e r e a n a p p r o p r i a t e s c a l e i s u s e d .
Y
F i g . ( a )
F i g . ( b )
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1 4
V E C T O R S a n d S C A L A R S
3 7 .
I f A B C D E F a r e t h e v e r t i c e s o f a r e g u l a r h e x a g o n , f i n d t h e r e s u l t a n t o f t h e f o r c e s r e p r e s e n t e d b y t h e v e c -
t o r s A B , A C , A D , A E a n d A F .
A n s . 3 A D
3 8 .
I f A a n d B a r e g i v e n v e c t o r s s h o w t h a t ( a )
I A + B I
I A I + I B I , ( b ) I A - B I
I A I - I B I .
3 9 . S h o w t h a t I A + B + C I " S J A I + I B I + I C I .
4 0 . T w o t o w n s A a n d B a r e s i t u a t e d d i r e c t l y o p p o s i t e e a c h o t h e r o n t h e b a n k s o f a r i v e r w h o s e w i d t h i s 8 m i l e s
a n d w h i c h f l o w s a t a s p e e d o f 4 m i / h r . A m a n l o c a t e d a t A w i s h e s t o r e a c h t o w n C w h i c h i s 6 m i l e s u p -
s t r e a m f r o m a n d o n t h e s a m e s i d e o f t h e r i v e r a s t o w n B .
I f h i s b o a t c a n t r a v e l a t a m a x i m u m s p e e d o f 1 0
m i / h r a n d i f h e w i s h e s t o r e a c h C i n t h e s h o r t e s t p o s s i b l e t i m e w h a t c o u r s e m u s t h e f o l l o w a n d h o w l o n g
w i l l t h e t r i p t a k e )
A n s . A s t r a i g h t l i n e c o u r s e u p s t r e a m m a k i n g a n a n g l e o f 3 4 ° 2 8 ` w i t h t h e s h o r e l i n e .
1 h r 2 5 m i n .
4 1 . A m a n t r a v e l l i n g s o u t h w a r d a t 1 5 m i / h r o b s e r v e s t h a t t h e w i n d a p p e a r s t o b e c o m i n g f r o m t h e w e s t . O n i n -
c r e a s i n g h i s s p e e d t o 2 5 m i / h r i t a p p e a r s t o b e c o m i n g f r o m t h e s o u t h w e s t . F i n d t h e d i r e c t i o n a n d s p e e d o f
t h e w i n d .
A n s . T h e w i n d i s c o m i n g f r o m a d i r e c t i o n 5 6 ° 1 8 ' n o r t h o f w e s t a t 1 8 m i / h r .
4 2 . A 1 0 0 l b w e i g h t i s s u s p e n d e d f r o m t h e c e n t e r o f a r o p e
a s s h o w n i n t h e a d j o i n i n g f i g u r e .
D e t e r m i n e t h e t e n -
s i o n T i n t h e r o p e .
A n s . 1 0 0 l b
4 3 . S i m p l i f y 2 A + B + 3 C - { A - 2 B - 2 ( 2 A - 3 B - C ) } .
A n s . 5 A - 3 B + C
4 4 . I f a a n d b a r e n o n - c o l l i n e a r v e c t o r s a n d A = ( x + 4 y ) a +
( 2 x + y + 1 ) b
a n d
B = ( y - 2 x + 2 ) a + ( 2 x - 3 y - 1 ) b ,
f i n d x a n d y s u c h t h a t 3 A = 2 B .
A n s . x = 2 , y = - 1
1 0 0 1 b
4 5 . T h e b a s e v e c t o r s a 1 , a 2 , a 3 a r e g i v e n i n t e r m s o f t h e b a s e v e c t o r s b 1 , b 2 , b 3 b y t h e r e l a t i o n s
a 1
=
2 b 1 + 3 b 2 - b 3 ,
a 2 =
b 1 - 2 b 2 + 2 b 3 ,
a 3
= - 2 b 1 + b 2 - 2 b 3
I f F = 3 b 1 - b 2 + 2 b 3 ,
e x p r e s s F i n t e r m s o f a 1 , a 2 a n d a 3 .
A n s .
2 a 1 + 5 a 2 + 3 a 3
4 6 .
I f a , b , c a r e n o n - c o p l a n a r v e c t o r s d e t e r m i n e w h e t h e r t h e v e c t o r s r 1 = 2 a - 3 b + c , r 2 = 3 a - 5 b + 2 c ,
a n d
r 3 = 4 a - 5 b + c a r e l i n e a r l y i n d e p e n d e n t o r d e p e n d e n t .
A n s . L i n e a r l y d e p e n d e n t s i n c e r 3 = 5 r 1 - 2 r 2 .
4 7 .
I f A a n d B a r e g i v e n v e c t o r s r e p r e s e n t i n g t h e d i a g o n a l s o f a p a r a l l e l o g r a m , c o n s t r u c t t h e p a r a l l e l o g r a m .
4 8 . P r o v e t h a t t h e l i n e j o i n i n g t h e m i d p o i n t s o f t w o s i d e s o f a t r i a n g l e i s p a r a l l e l t o t h e t h i r d s i d e a n d h a s o n e
h a l f o f i t s m a g n i t u d e .
4 9 . ( a ) I f 0 i s a n y p o i n t w i t h i n t r i a n g l e A B C a n d P , Q , R a r e m i d p o i n t s o f t h e s i d e s A B , B C , C A r e s p e c t i v e l y ,
p r o v e t h a t O A + O B + O C = O P + O Q + O R .
( b ) D o e s t h e r e s u l t h o l d i f 0 i s a n y p o i n t o u t s i d e t h e t r i a n g l e ? P r o v e y o u r r e s u l t .
A n s . Y e s
5 0 . I n t h e a d j o i n i n g f i g u r e , A B C D i s a p a r a l l e l o g r a m w i t h
P a n d Q t h e m i d p o i n t s o f s i d e s B C a n d C D r e s p e c -
t i v e l y .
P r o v e t h a t A P a n d A Q t r i s e c t d i a g o n a l B D a t
t h e p o i n t s E a n d F .
5 1 . P r o v e t h a t t h e m e d i a n s o f a t r i a n g l e m e e t i n a c o m m o n
p o i n t w h i c h i s a p o i n t o f t r i s e c t i o n o f t h e m e d i a n s .
5 2 . P r o v e t h a t t h e a n g l e b i s e c t o r s o f a t r i a n g l e m e e t i n a
c o m m o n p o i n t .
5 3 . S h o w t h a t t h e r e e x i s t s a t r i a n g l e w i t h s i d e s w h i c h a r e
e q u a l a n d p a r a l l e l t o t h e m e d i a n s o f a n y g i v e n t r i a n g l e .
5 4 . L e t t h e p o s i t i o n v e c t o r s o f p o i n t s P a n d Q r e l a t i v e t o a n o r i g i n 0 b e g i v e n b y p a n d q r e s p e c t i v e l y . I f R i s
a p o i n t w h i c h d i v i d e s l i n e P Q i n t o s e g m e n t s w h i c h a r e i n t h e r a t i o m : n s h o w t h a t t h e p o s i t i o n v e c t o r o f R
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V E C T O R S a n d S C A L A R S
1 5
i s g i v e n b y
r =
' n P + n q
a n d t h a t t h i s i s i n d e p e n d e n t o f t h e o r i g i n .
+ n
5 5 . I f r 1 , r 2 , . . . , r n a r e t h e p o s i t i o n v e c t o r s o f m a s s e s m 1 , m 2 , . . . , m n r e s p e c t i v e l y r e l a t i v e t o a n o r i g i n 0 ,
s h o w t h a t t h e p o s i t i o n v e c t o r o f t h e c e n t r o i d i s g i v e n b y
r =
a n d t h a t t h i s i s i n d e p e n d e n t o f t h e o r i g i n .
m 1 r 1 + m 2 r 2 + . . . + m n r n
I n 1 + m 2 + . . . + I n n
5 6 . A q u a d r i l a t e r a l A B C D h a s m a s s e s o f
1 , 2 , 3 a n d 4 u n i t s l o c a t e d r e s p e c t i v e l y a t i t s v e r t i c e s A ( - 1 , - 2 , 2 ) ,
B ( 3 , 2 , - 1 ) ,
C ( 1 , - 2 , 4 ) , a n d D ( 3 , 1 , 2 ) . F i n d t h e c o o r d i n a t e s o f t h e c e n t r o i d .
A n s .
( 2 , 0 , 2 )
5 7 . S h o w t h a t t h e e q u a t i o n o f a p l a n e w h i c h p a s s e s t h r o u g h t h r e e g i v e n p o i n t s A , B , C n o t i n t h e s a m e s t r a i g h t
l i n e a n d h a v i n g p o s i t i o n v e c t o r s a , b , c r e l a t i v e t o a n o r i g i n 0 , c a n b e w r i t t e n
r
m a + n b + p c
=
m + n + p
w h e r e i n , n , p a r e s c a l a r s . V e r i f y t h a t t h e e q u a t i o n i s i n d e p e n d e n t o f t h e o r i g i n .
5 8 . T h e p o s i t i o n v e c t o r s o f p o i n t s P a n d Q a r e g i v e n b y r 1 = 2 i + 3 j - k ,
r 2 = 4 i - 3 j + 2 k . D e t e r m i n e P Q i n
t e r m s o f i , j , k a n d f i n d i t s m a g n i t u d e . A n s .
2 i - 6 j + 3 k , 7
5 9 .
I f A = 3 i - j - 4 k , B = - 2 i + 4 j - 3 k , C = i + 2 j - k ,
f i n d
( a ) 2 A - B + 3 C ,
( b )
f A + B + C I ,
( c ) 1 3 A - 2 B + 4 C 1 ,
( d ) a u n i t v e c t o r p a r a l l e l t o 3 A - 2 B + 4 C .
( a ) 1 1 i - 8 k
( b )
( c )
( d ) 3 A -
2 B + 4 C
A n s
.
6 0 . T h e f o l l o w i n g f o r c e s a c t o n a p a r t i c l e P :
F 1 = 2 i + 3 j - 5 k ,
F 2 = - 5 i + j + 3 k ,
F 3 =
i - 2 j + 4 k , F 4 = 4 i -
3 j - 2 k , m e a s u r e d i n p o u n d s . F i n d ( a ) t h e r e s u l t a n t o f t h e f o r c e s , ( b ) t h e m a g n i t u d e o f t h e r e s u l t a n t .
A n s .
( a ) 2 i - j
( b ) y r
6 1 . I n e a c h c a s e d e t e r m i n e w h e t h e r t h e v e c t o r s a r e l i n e a r l y i n d e p e n d e n t o r l i n e a r l y d e p e n d e n t :
( a ) A = 2 1 + j - 3 k , B = i - 4 k , C = 4 i + 3 j - k ,
( b ) A = i - 3 j + 2 k , B = 2 i - 4 j - k , C = 3 i + 2 j - k .
A n s . ( a ) l i n e a r l y d e p e n d e n t , ( b ) l i n e a r l y i n d e p e n d e n t
6 2 . P r o v e t h a t a n y f o u r v e c t o r s i n t h r e e d i m e n s i o n s m u s t b e l i n e a r l y d e p e n d e n t .
6 3 . S h o w t h a t a n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n t h a t t h e v e c t o r s A = A 1 i + A
2 j + A 3 k , B = B 1 i + B 2 j + B 3 k ,
A l A 2 A 3
C = C
I
i + C 2 j + C 3 k b e l i n e a r l y i n d e p e n d e n t i s t h a t t h e d e t e r m i n a n t
B 1 B 2 B .
b e d i f f e r e n t f r o m z e r o .
C 1 C 2 C 3
6 4 .
( a ) P r o v e t h a t t h e v e c t o r s A = 3 i + j - 2 k , B = - i + 3 j + 4 k , C = 4 i - 2 j
- 6 k c a n f o r m t h e s i d e s o f a t r i a n g l e .
( b ) F i n d t h e l e n g t h s o f t h e m e d i a n s o f t h e t r i a n g l e .
A n s . ( b ) v i m ,
2 v 4 , 2 V - 1 - - 5 0
6 5 . G i v e n t h e s c a l a r f i e l d d e f i n e d b y c ( x , y , z ) = 4 y z 3 + 3 x y z - z 2 + 2 . F i n d ( a ) 0 ( 1 , - 1 , - 2 ) , ( b ) 4 ( 0 , - 3 , 1 ) .
A n s . ( a ) 3 6 ( b ) - 1 1
6 6 . G r a p h t h e v e c t o r f i e l d s d e f i n e d b y
( a ) V ( x , y ) = x i - y j ,
( b ) V ( x , y ) = y i - x j , ( c ) V ( x , y , z ) =
x i + y i + z k
x 2 + y 2 + z 2
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T H E D O T O R S C A L A R P R O D U C T o f t w o v e c t o r s A a n d B , d e n o t e d b y
A d o t B ) , i s d e -
f i n e d a s t h e p r o d u c t o f t h e m a g n i t u d e s o f A a n d B a n d t h e c o s i n e
o f t h e a n g l e 6 b e t w e e n t h e m .
I n s y m b o l s ,
A B
N o t e ' t h a t A . B i s a s c a l a r a n d n o t a v e c t o r .
T h e f o l l o w i n g l a w s a r e v a l i d :
1 . A B = B A
C o m m u t a t i v e L a w f o r D o t P r o d u c t s
2 . A ( B + C ) = A B + A C
D i s t r i b u t i v e L a w
3 . m ( A B ) = ( m A ) B = A - ( m B ) = ( A B ) m ,
w h e r e m i s a s c a l a r .
4 .
j . j = 1 ,
0
5 .
I f
A = A l i + A 2 j + A 3 k a n d B = B l i + B 2 j + B 3 k , t h e n
A 1 B 1 + A 2 B 2 + A 3 8 3
A 2 = A i + A 2 + A 3
B - B = 8 2 = B i + B 2 + B 3
6 .
I f A - B = 0 a n d A a n d B a r e n o t n u l l v e c t o r s , t h e n A a n d B a r e p e r p e n d i c u l a r .
T H E C R O S S O R V E C T O R P R O D U C T o f A a n d B i s a v e c t o r C = A x B ( r e a d A c r o s s B ) . T h e m a g -
n i t u d e o f A x B i s d e f i n e d a s t h e p r o d u c t o f t h e m a g n i t u d e s o f
A a n d B a n d t h e s i n e o f t h e a n g l e 6 b e t w e e n t h e m . T h e d i r e c t i o n o f t h e v e c t o r C = A x B i s p e r p e n -
d i c u l a r t o t h e p l a n e o f A a n d B a n d s u c h t h a t A , B a n d C f o r m a r i g h t - h a n d e d s y s t e m . I n s y m b o l s ,
A x B = A B s i n O u ,
0
r c
w h e r e u i s a u n i t v e c t o r i n d i c a t i n g t h e d i r e c t i o n o f A x B .
I f A = B , o r i f A i s p a r a l l e l t o B , t h e n
s i n O = 0 a n d w e d e f i n e A x B = 0 .
T h e f o l l o w i n g l a w s a r e v a l i d :
1 . A X B
2 . A x ( B + C ) = A x B + A x C
( C o m m u t a t i v e L a w f o r C r o s s P r o d u c t s F a i l s . )
D i s t r i b u t i v e L a w
3 . m ( A x B ) = ( m A ) x B = A x ( m B ) = ( A x B ) m ,
w h e r e m i s a s c a l a r .
4 . i x i = j x j = k x k = 0 ,
i x j = 1 L ) j x k = ( i 3 k x i =
5 .
I f A = A l i + A 2 j + A 3 k
a n d B = B l i + 8 2 j + B 3 k , t h e n
1 6
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T h e D O T a n d C R O S S P R O D U C T
1 7
A x B =
i
j
k
A l
A 2
A 3
B 1
B 2
B 3
6 . T h e m a g n i t u d e o f A x B i s t h e s a m e a s t h e a r e a o f a p a r a l l e l o g r a m w i t h s i d e s A a n d B .
7 .
I f A x B = 0 , a n d A a n d B a r e n o t n u l l v e c t o r s , t h e n A a n d B a r e p a r a l l e l .
T R I P L E P R O D U C T S . D o t a n d c r o s s m u l t i p l i c a t i o n o f t h r e e v e c t o r s A , B a n d C m a y p r o d u c e m e a n -
i n g f u l p r o d u c t s o f t h e f o r m ( A B ) C , A - ( B x C ) a n d A x ( B x C ) . T h e f o l l o w -
i n g l a w s a r e v a l i d :
1 .
2 . A - ( B x C ) = B . ( C x A ) = C ( A x B ) = v o l u m e o f a p a r a l l e l e p i p e d h a v i n g A , B a n d C a s
e d g e s ,
o r t h e n e g a t i v e o f t h i s v o l u m e , a c c o r d i n g a s A , B a n d C d o o r d o n o t f o r m a r i g h t - h a n d e d s y s -
t e m .
I f A = A 1 i + A 2 j + A s k , B = B 1 i + B 2 j + B 3 k a n d C = C 1 i + C 2 j + C 3 k , t h e n
A . ( B x C )
=
3 . A x ( B x C ) / ( A x B ) x C
4 . A x ( B x C ) =
( A . B ) C
( A x B ) x C =
A l A 2
A s
B 1
B 2
B 3
C 1 C 2
C 3
( A s s o c i a t i v e L a w f o r C r o s s P r o d u c t s F a i l s . )
T h e p r o d u c t A ( B x C ) i s s o m e t i m e s c a l l e d t h e s c a l a r t r i p l e p r o d u c t o r b o x p r o d u c t a n d m a y b e
d e n o t e d b y [ A B C ] .
T h e p r o d u c t A x ( B x C ) i s c a l l e d t h e v e c t o r t r i p l e p r o d u c t .
I n A ( B x C ) p a r e n t h e s e s a r e s o m e t i m e s o m i t t e d a n d w e w r i t e A B x C ( s e e P r o b l e m 4 1 ) . H o w -
e v e r , p a r e n t h e s e s m u s t b e u s e d i n A x ( B x C ) ( s e e P r o b l e m s 2 9 a n d 4 7 ) .
R E C I P R O C A L S E T S O F V E C T O R S . T h e s e t s o f v e c t o r s a , b , c a n d a ' , b ' , c ' a r e c a l l e d r e c i p r o c a l
s e t s o r s y s t e m s o f v e c t o r s i f
1
a b = a ' c = b ' a = b ' c = c ' a = c ' b = 0
T h e s e t s a , b , c a n d a ' , b ' , c ' a r e r e c i p r o c a l s e t s o f v e c t o r s i f a n d o n l y i f
a '
b ,
_
_ c x a
a . b x c
a . b x c
b x c
c '
a x b
a b x c
w h e r e a b x c 4 0 .
S e e P r o b l e m s 5 3 a n d 5 4 .
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1 8
T h e D O T a n d C R O S S P R O D U C T
S O L V E D P R O B L E M S
T H E D O T O R S C A L A R P R O D U C T .
1 . P r o v e A B = B A .
A B = A B c o s 8 = B A c o s 6 = B A
T h e n t h e c o m m u t a t i v e l a w f o r d o t p r o d u c t s i s v a l i d .
2 . P r o v e t h a t t h e p r o j e c t i o n o f A o n B i s e q u a l t o A b , w h e r e
b i s a u n i t v e c t o r i n t h e d i r e c t i o n o f B .
T h r o u g h t h e i n i t i a l a n d t e r m i n a l p o i n t s o f A p a s s p l a n e s p e r - E
p e n d i c u l a r t o B a t G a n d H r e s p e c t i v e l y a s i n t h e a d j a c e n t f i g u r e ;
t h e n
P r o j e c t i o n o f A o n B = G H = E F = A c o s B = A b
3 . P r o v e A ( B + C ) = A B + A - C .
L e t a b e a u n i t v e c t o r i n t h e d i r e c t i o n o f A ; t h e n
P r o j e c t i o n o f ( B + C ) o n A = p r o j . o f B o n A + p r o j . o f C o n A
( B + C ) a
=
M u l t i p i v i n g b y A ,
( B + C ) . A a =
a n d
T h e n b y t h e c o m m u t a t i v e l a w f o r d o t p r o d u c t s ,
a n d t h e d i s t r i b u t i v e l a w i s v a l i d .
4 . P r o v e t h a t
G
F
E
H B
B y P r o b l e m 3 ,
( A + B ) - ( C + D ) = A - ( C + D ) + B - ( C + D )
= A C + A D + B C + B D
T h e o r d i n a r y l a w s o f a l g e b r a a r e v a l i d f o r d o t p r o d u c t s .
I
5 . E v a l u a t e e a c h o f t h e f o l l o w i n g .
( a )
I i i
I i I c o s 0 0
( 1 ) ( 1 ) ( 1 )
=
1
( b )
I i i
I k J c o s 9 0 °
_ ( 1 ) ( 1 ) ( 0 )
= 0
( c )
I k I I i i c o s 9 0 ° _ ( 1 ) ( 1 ) ( 0 ) = 0
( d )
j - ( 2 i - 3 j + k ) =
0 - 3 + 0 = - 3
( e ) ( 2 i - j )
( 3 i + k ) = 2 i
( 3 i + k ) - j
( 3 i + k )
= 6 1
i + 2 i k - 3 j
i - j k = 6 + 0 - 0 - 0
=
6
6 . I f A = A 1 i + A 2 j + A 3 k
a n d
B = B 1 i + B 2 j + B 3 k , p r o v e t h a t A B = A 1 B 1 + A 2 B 2 + A 3 B 3
A B =
( A 1 i + A 2 i + A 3 k ) . ( B 1 i + B 2 j + B 3 k )
=
A 1 B 1 i i + A 1 B 2 i j + A 1 B 3 i k + A 2 B 1 j i + A 2 B 2
A 2 B 3 j k + A 3 B 1 k i + A 3 B 2 k j + A 3 B 3 k k
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T h e D O T a n d C R O S S P R O D U C T
=
A 1 B 1 + A 2 B 2 + A 3 B 3
s i n c e
i i
= j j = k k = 1
a n d a l l o t h e r d o t p r o d u c t s a r e z e r o .
7 .
I f A = A 1 i + A 2 j + A 3 k , s h o w t h a t A = A =
A l + A 2
( A ) ( A ) c o s 0 ° = A 2 .
T h e n A = V I A A .
A l s o , A A = ( A 1 i + A 2 j + A 3 k ) ( A 1 i + A 2 j + A 3 k )
_ ( A 1 ) ( A 1 ) + ( A 2 ) ( A 2 ) + ( A 3 ) ( A o )
= A 2 + A 2 + A s
b y P r o b l e m 6 , t a k i n g B = A .
+ A 2
T h e n A = / A A =
A 2 + A 2
3
i s t h e m a g n i t u d e o f A .
S o m e t i m e s A A . A i s w r i t t e n A 2 .
8 . F i n d t h e a n g l e b e t w e e n A =
2 i + 2 j - k
a n d
B =
6 i - 3 j + 2 k .
A - B
= A B
c o s 8 ,
A =
( 2 ) 2 + ( 2 ) 2 + ( - 1 ) 2 = 3
B =
( 6 ) 2 + ( - 3 ) 2 + ( 2 ) 2 = 7
A - B = ( 2 ) ( 6 ) + ( 2 ) ( - 3 ) + ( - 1 ) ( 2 ) = 1 2 - 6 - 2 = 4
T h e n c o s 8
=
A B
( 3 ) ( 7 )
4
0 . 1 9 0 5
a n d
8 = 7 9 0 a p p r o x i m a t e l y .
2 1
9 . I f A B = 0 a n d i f A a n d B a r e n o t z e r o , s h o w t h a t A i s p e r p e n d i c u l a r t o B .
I f A B
c o s 6 = 0 , t h e n
c o s 6 = 0
o r 8 = 9 0 ° .
C o n v e r s e l y , i f 6 = 9 0 ° ,
0 .
1 0 . D e t e r m i n e t h e v a l u e o f a s o t h a t A = 2 i + a j + k a n d B = 4 i - 2 j - 2 k a r e p e r p e n d i c u l a r .
F r o m P r o b l e m 9 , A a n d B a r e p e r p e n d i c u l a r i f A B = 0 .
T h e n A B = ( 2 ) ( 4 ) + ( a ) ( - 2 ) + ( 1 ) ( - 2 ) = 8 - 2 a - 2 = 0
f o r
a = 3 .
1 1 . S h o w t h a t t h e v e c t o r s A = 3 i - 2 j + k , B = i - 3 j + 5 k , C = 2 i + j - 4 k f o r m a r i g h t t r i a n g l e .
W e f i r s t h a v e t o s h o w t h a t t h e v e c t o r s f o r m a t r i a n g l e .
( a )
I
( b )
1 9
F r o m t h e f i g u r e s i t i s s e e n t h a t t h e v e c t o r s w i l l f o r m a t r i a n g l e i f
( a )
o n e o f t h e v e c t o r s , s a y ( 3 ) , i s t h e r e s u l t a n t o r s u m o f ( 1 ) a n d ( 2 ) ,
( b ) t h e s u m o r r e s u l t a n t o f t h e v e c t o r s ( 1 ) + ( 2 ) + ( 3 ) i s z e r o ,
a c c o r d i n g a s ( a ) t w o v e c t o r s h a v e a c o m m o n t e r m i n a l p o i n t o r ( b ) n o n e o f t h e v e c t o r s h a v e a c o m m o n t e r m i n a l
p o i n t . B y t r i a l w e f i n d A = B + C s o t h a t t h e v e c t o r s d o f o r m a t r i a n g l e .
S i n c e A - B = ( 3 ) ( 1 ) + ( - 2 ) ( - 3 ) + ( 1 ) ( 5 ) = 1 4 , A C = ( 3 ) ( 2 ) + ( - 2 ) ( 1 ) + ( 1 ) ( - 4 ) = 0 ,
a n d
B C = ( 1 ) ( 2 ) + ( - 3 ) ( 1 ) + ( 5 ) ( - 4 )
2 1 ,
i t f o l l o w s t h a t A a n d C a r e p e r p e n d i c u l a r a n d t h e t r i a n g l e i s a
r i g h t t r i a n g l e .
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2 0
T h e D O T a n d C R O S S P R O D U C T
1 2 . F i n d t h e a n g l e s w h i c h t h e v e c t o r A = 3 i - 6 j + 2 k m a k e s w i t h t h e c o o r d i n a t e a x e s .
L e t a , P . y b e t h e a n g l e s w h i c h A m a k e s w i t h t h e p o s i t i v e x , y , z a x e s r e s p e c t i v e l y .
A i
= ( A ) ( 1 ) c o s a = ( 3 ) 2 + ( - 6 ) 2 + ( 2 ) 2 c o s a = 7 c o s a
3
T h e n
c o s a = 3 / 7 = 0 . 4 2 8 6 ,
a n d
a = 6 4 . 6 ° a p p r o x i m a t e l y .
S i m i l a r l y , c o s 0 = - 6 / 7 , R = 1 4 9 °
a n d
c o s y = 2 / 7 , y = 7 3 . 4 ° .
T h e c o s i n e s o f a , ( 3 , a n d y a r e c a l l e d t h e d i r e c t i o n c o s i n e s o f A . ( S e e P r o b . 2 7 , C h a p . 1 ) .
1 3 . F i n d t h e p r o j e c t i o n o f t h e v e c t o r A = i - 2 j + k o n t h e v e c t o r B = 4 i - 4 j + 7 k .
4 4
.
A u n i t v e c t o r i n t h e d i r e c t i o n B i s b =
B
B =
4 i - 4 j + 7 k
=
4
1 - 9 j +
? k
9
( 4 ) 2 + ( - 4 ) 2 + ( 7 ) 2
P r o j e c t i o n o f A o n t h e v e c t o r B = A . b = ( i - 2 j + k )
( 4
i -
9
j + 9 k )
( 1 ) ( 9 ) + ( - 2 ) ( -
9 ) + ( 1 ) ( 9 ) =
1 9
1 4 . P r o v e t h e l a w o f c o s i n e s f o r p l a n e t r i a n g l e s .
F r o m F i g . ( a ) b e l o w ,
B + C = A
o r
C = A - B .
T h e n
( A - B ) ( A - B ) =
a n d
C 2 = A 2 + B 2 - 2 A B c o s 8 .
F i g . ( a )
F i g . ( b )
1 5 . P r o v e t h a t t h e d i a g o n a l s o f a r h o m b u s a r e p e r p e n d i c u l a r . R e f e r t o F i g . ( b ) a b o v e .
O Q = O P + P Q = A + B
O R + R P = O P
o r
B + R P = A
a n d R P = A - B
T h e n O Q R P = ( A + B ) ( A - B ) = A 2 - B 2 = 0 ,
s i n c e A = B .
H e n c e O Q i s p e r p e n d i c u l a r t o R P .
1 6 . D e t e r m i n e a u n i t v e c t o r p e r p e n d i c u l a r t o t h e p l a n e o f A = 2 i - 6 j - 3 k a n d B = 4 i + 3 j - k
.
L e t v e c t o r C = c 1 i + c 2 j + c 3 k b e p e r p e n d i c u l a r t o t h e p l a n e o f A a n d B . T h e n C i s p e r p e n d i c u l a r t o A
a n d a l s o t o B . H e n c e ,
C A = 2 c 1 - 6 c 2 - 3 c 3 = 0
o r ( 1 ) 2 c 1 - 6 c 2 = 3 c 3
C B = 4 c 1 + 3 c 2 - c 3 = 0
o r
( 2 ) 4 c 1 + 3 c 2 = c 3
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T h e D O T a n d C R O S S P R O D U C T
S o l v i n g ( 1 ) a n d ( 2 ) s i m u l t a n e o u s l y :
c l = 2 c s ,
c 2 = - 3 G 3 ,
C = c 3 ( 2 i -
3
i + k ) .
c 3 ( 1 i - i j + k )
T h e n a u n i t v e c t o r i n t h e d i r e c t i o n o f C i s
C
= 2 3
= ± ( 7 i -
7 + ? k ) .
C
/ C 3 2 2 ) 2 + ( - 3 ) 2 + ( 1 ) 2 1
2 1
1 7 . F i n d t h e w o r k d o n e i n m o v i n g a n o b j e c t a l o n g a v e c t o r
r = 3 i + 2 j - 5 k
i f t h e a p p l i e d f o r c e i s
F = 2 i - j - k .
R e f e r t o F i g - ( a ) b e l o w .
W o r k d o n e
=
( m a g n i t u d e o f f o r c e i n d i r e c t i o n o f m o t i o n ) ( d i s t a n c e m o v e d )
=
( F c o s 6 ) ( r )
=
F r
=
6 - 2 + 5 =
9 .
z
r
F i g . ( a )
F i g . ( b )
1 8 . F i n d a n e q u a t i o n f o r t h e p l a n e p e r p e n d i c u l a r t o t h e v e c t o r A = 2 i + 3 j + 6 k a n d p a s s i n g t h r o u g h t h e
t e r m i n a l p o i n t o f t h e v e c t o r B = i + 5 j + 3 k ( s e e F i g . ( b ) a b o v e ) .
L e t r b e t h e p o s i t i o n v e c t o r o f p o i n t P , a n d Q t h e t e r m i n a l p o i n t o f B .
S i n c e P Q = B - r i s p e r p e n d i c u l a r t o A , ( B - r ) A = 0
o r r A = B A i s t h e r e q u i r e d e q u a t i o n o f t h e
p l a n e i n v e c t o r f o r m . I n r e c t a n g u l a r f o r m t h i s b e c o m e s
o r
( x i + y j + z k ) ( 2 i + 3 j + 6 k )
=
( i + 5 j + 3 k ) ( 2 i + 3 j + 6 k )
2 x + 3 y + 6 z
=
( 1 ) ( 2 ) + ( 5 ) ( 3 ) + ( 3 ) ( 6 )
=
3 5
1 9 . I n P r o b l e m 1 8 f i n d t h e d i s t a n c e f r o m t h e o r i g i n t o t h e p l a n e .
T h e d i s t a n c e f r o m t h e o r i g i n t o t h e p l a n e i s t h e p r o j e c t i o n o f B o n A .
A u n i t v e c t o r i n d i r e c t i o n A i s
a
= A
2 i + 3 j + 6 k 2 i
+ 3 .
+ 6 k
A
( 2 ) 2 + ( 3 ) 2 + ( 6 ) 2
7
7
7
T h e n , p r o j e c t i o n o f B o n A = B a = ( i + 5 j + 3 k ) ( ? i +
- a
j +
6
k ) = 1 ( 2 ) + 5 ( 3 ) + 3 ( s ) = 5 .
7 7 7 7
7
7
2 0 . I f A i s a n y v e c t o r , p r o v e t h a t A = ( A . i ) i + ( A - j ) j + ( A - k ) k .
S i n c e A = A 1 i + A 2 j + 4 3 k , A A . i = A 1 i i + A 2 j i +
A
A
A
j
+ A s k = ( A . i ) i + ( A j ) j + ( A k ) k .
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2 2
T h e D O T a n d C R O S S P R O D U C T
T H E C R O S S O R V E C T O R P R O D U C T .
2 1 . P r o v e A x B = - B x A .
F i g . ( a )
F i g . ( b )
A x B = C h a s m a g n i t u d e A B s i n 8 a n d d i r e c t i o n s u c h t h a t
A , B a n d C f o r m a r i g h t - h a n d e d s y s t e m
( F i g . ( a ) a b o v e ) .
B X A = D h a s m a g n i t u d e B A s i n 8 a n d d i r e c t i o n s u c h t h a t B , A a n d D f o r m a r i g h t - h a n d e d s y s t e m
( F i g . ( b ) a b o v e ) .
T h e n D h a s t h e s a m e m a g n i t u d e a s C b u t i s o p p o s i t e i n d i r e c t i o n , i . e . C = - D o r
A x B = - B X A .
T h e c o m m u t a t i v e l a w f o r c r o s s p r o d u c t s i s n o t v a l i d .
2 2 . I f A x B = 0 a n d i f A a n d B a r e n o t z e r o , s h o w t h a t A i s p a r a l l e l t o B .
I f A x B = A B s i n e u = 0 , t h e n s i n 8 = 0 a n d e = 0 ° o r 1 8 0 ° .
2 3 . S h o w t h a t
I A x B 1 2
+
I A - B l 2
=
I A 1 2 1 B I 2 .
I A x B 1 2 + I A - B I
2
=
I A B s i n 8 u 1 2 + I A B
c o s 8 1 2
A 2 B 2 s i n g 8 + A 2 B 2 c o s 2 8
A 2 B 2
_
J A I ' I B 1 2
2 4 . E v a l u a t e e a c h o f t h e f o l l o w i n g .
( a ) i x j = k
( f ) j x j = 0
( b ) j x k = i
( g ) i x k = - k x i = - j
( c ) k x i = j
( h ) ( 2 j ) x ( 3 k ) = 6 j x k =
6 1
( d ) k x j = - j x k = - i
( i ) ( 3 i ) x ( - 2 k ) _ - 6 i x k = 6 j
( e )
i x i = 0
( j )
2 j x i - 3 k = - 2 k - 3 k = - 5 k
2 5 . P r o v e t h a t A x ( B + C ) = A x B + A x C f o r t h e
c a s e w h e r e A i s p e r p e n d i c u l a r t o B a n d a l s o t o
C .
S i n c e A i s p e r p e n d i c u l a r t o B , A x B i s a v e c t o r
p e r p e n d i c u l a r t o t h e p l a n e o f A a n d B a n d h a v i n g m a g -
n i t u d e A B s i n 9 0 ° = A B o r m a g n i t u d e o f A B . T h i s
i s e q u i v a l e n t t o m u l t i p l y i n g v e c t o r B b y A a n d r o t a t i n g
t h e
r e s u l t a n t v e c t o r t h r o u g h
9 0 °
t o t h e p o s i t i o n
s h o w n i n t h e a d j o i n i n g d i a g r a m .
S i m i l a r l y , A x C i s t h e v e c t o r o b t a i n e d b y m u l t i -
p l y i n g C b y A a n d r o t a t i n g t h e r e s u l t a n t v e c t o r t h r o u g h
9 0 ° t o t h e p o s i t i o n s h o w n .
I n l i k e m a n n e r , A x ( B + C ) i s t h e v e c t o r o b t a i n e d
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T h e D O T a n d C R O S S P R O D U C T
2 3
b y m u l t i p l y i n g B + C b y A a n d r o t a t i n g t h e r e s u l t a n t v e c t o r t h r o u g h 9 0 ° t o t h e p o s i t i o n s h o w n .
S i n c e A x ( B + C )
i s t h e d i a g o n a l o f t h e p a r a l l e l o g r a m w i t h A x B a n d A x C a s s i d e s , w e h a v e
A x ( B + C ) = A x B + A x C .
2 6 . P r o v e t h a t A x ( B + C ) = A x B + A x C i n t h e g e n -
e r a l c a s e w h e r e A , B a n d C a r e n o n - c o p l a n a r .
R e s o l v e B i n t o t w o c o m p o n e n t v e c t o r s , o n e p e r p e n -
d i c u l a r t o A a n d t h e o t h e r p a r a l l e l t o A , a n d d e n o t e t h e m
b y B 1 a n d B r e s p e c t i v e l y . T h e n B = B l + B .
I f h i s t h e a n g l e b e t w e e n A a n d B , t h e n B 1 = B s i n e .
T h u s t h e m a g n i t u d e o f A x B 1 i s A B s i n B , t h e s a m e a s
t h e m a g n i t u d e o f A X B . A l s o , t h e d i r e c t i o n o f A x B 1 i s
t h e s a m e a s t h e d i r e c t i o n o f A x B . H e n c e A X B 1 = A x B .
S i m i l a r l y i f C i s r e s o l v e d i n t o t w o c o m p o n e n t v e c -
t o r s C , i a n d C 1 , p a r a l l e l a n d p e r p e n d i c u l a r r e s p e c t i v e l y
t o A , t h e n A x C , = A x C .
A l s o , s i n c e
B + C = B . + B + C 1 + C = ( B l + C 1 ) + ( B , , +
i t f o l l o w s t h a t
A x ( B 1 + C 1 ) = A x ( B + C ) .
N o w B 1 a n d C 1 a r e v e c t o r s p e r p e n d i c u l a r t o A a n d s o b y P r o b l e m 2 5 ,
A x ( B 1 + C 1 ) = A X B 1 + A X C 1
T h e n
A x ( B + C )
= A x B + A x C
a n d t h e d i s t r i b u t i v e l a w h o l d s . M u l t i p l y i n g b y - 1 , u s i n g P r o b . 2 1 , t h i s b e c o m e s ( B + C ) x A = B X A + C x A .
N o t e t h a t t h e o r d e r o f f a c t o r s i n c r o s s p r o d u c t s i s i m p o r t a n t . T h e u s u a l l a w s o f a l g e b r a a p p l y o n l y i f p r o p -
e r o r d e r i s m a i n t a i n e d .
2 7 . I f
A = A l i + A 2 j + A 3 k
a n d B = B 1 i + B 2 j + B 3 k ,
p r o v e t h a t
A x B =
i j
k
A ,
A 2
A s
B 1
B 2
B 3
A x B =
( A l i + A 2 j + A 3 k ) x ( B 1 i + B 2 j + B 3 k )
=
A l i x ( B i t + B 2 j + B 3 k ) + A 2 j x ( B 1 i + B 2 j + B 3 k ) + A s k x ( B i t + B 2 j + B 3 k )
=
A 1 B 1 i x i + A 1 B 2 i x j + A 1 B 3 i x k + A 2 B 1 j x i + A 2 B 2 j x j + A 2 B 3 j x k + A 3 B 1 k x i + A 3 B 2 k x j + A 3 B 3 k x k
_ - ( A 2 B 3 - A 3 B 2 ) i + ( A 3 B 1 - A 1 B 3 ) j + ( A 1 B 2 - A 2 B 1 ) k
=
i i
k
A l
A 2 A s
B 1
B 2 B 3
2 8 . I f A = 2 i - 3 j - k a n d B = i + 4 j - 2 k ,
f i n d ( a ) A x B ,
( b ) B x A , ( c ) ( A + B ) x ( A - B ) .
( a ) A x B =
( 2 i - 3 j - k ) x ( i + 4 j - 2 k ) =
i
j
k
2
- 3
- 1
1
4
- 2
r i
4
- 2 1 _
j
1 2
- 2 I + k l l
- 4 I
= 1 0 i + 3 j + 1 1 k
1
A n o t h e r M e t h o d .
( 2 i - 3 j - k ) x ( i + 4 j - 2 k ) = 2 i x ( i + 4 j - 2 k ) - 3 j x ( i + 4 j - 2 k ) -
k x ( i + 4 j - 2 k )
= 2 i x i + 8 i x j - 4 i x k - 3 j x i - 1 2 j x j + 6 j x k - k x i - 4 k x j + 2 k x k
= 0 + 8 k + 4 i + 3 k - 0 + 6 i - j + 4 1 + 0
= 1 0 i + 3 j + 1 1 k
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2 4
T h e D O T a n d C R O S S P R O D U C T
i j
k
( b ) B x A = ( i + 4 j - 2 k ) x ( 2 1 - 3 j - - k ) =
1
4
- 2
2 - 3
- I 1
4
- 2
1
- 2
- 1
-
- 1 1 + k 1 2
3
= - 1 0 i - 3 j - I l k .
3
C o m p a r i n g w i t h ( a ) , A x B = - B x A . N o t e t h a t t h i s i s e q u i v a l e n t t o t h e t h e o r e m : I f t w o r o w s o f
a d e t e r m i n a n t a r e i n t e r c h a n g e d , t h e d e t e r m i n a n t c h a n g e s s i g n .
( c ) A + B = ( 2 i - 3 j - k ) + ( i + 4 j - 2 k ) = 3 i + j - 3 k
A - B = ( 2 i - 3 j - k ) - ( i + 4 j - 2 k ) = i - 7 j + k
T h e n
( A + B ) x ( A - B ) = ( 3 1 + j - 3 k ) x ( i - 7 j + k )
_
1
- 7
- 3
1
( 3
j l
i j
k
3
1 - 3
1 - 7 1
`
1
I
+ k 1 1
- 7 1
= - 2 0 i - 6 j - 2 2 k .
A n o t h e r M e t h o d .
( A + B ) x ( A - B ) = A x ( A - B ) + B x ( A - B )
= A x A - - - A x B + B x A - - B x B = O - A x B - A x B - 0 = - 2 A X B
_ - 2 ( 1 0 i + 3 j + I l k ) _ - 2 0 i - 6 j - 2 2 k ,
u s i n g ( a ) .
2 9 . I f
A = 3 i - j + 2 k , B = 2 i + j - k , a n d C = i - 2 j + 2 k ,
f i n d
( a ) ( A x B ) x C , ( b ) A x ( B x C ) .
( a ) A x B =
i
i k
3 - 1 2
2
1
- 1
= - i + 7 j + 5 k .
T h e n ( A x B ) x C = ( - i + 7 j + 5 k ) x ( i - 2 j + 2 k )
=
( b ) B x C =
i i
k
2
1 - 1
1
- 2
2
= O i - 5 j - 5 k
= - 5 j - 5 k .
T h e n A x ( B x C ) _ ( 3 1 - i + 2 k ) x ( - 5 j - 5 k ) _
i
j k
- 1
7
5
1
- 2
2
i i k
3
- 1 2
- 5
- 5
= 2 4 1 + 7 j - 5 k .
= 1 5 i + 1 5 j - 1 5 k .
T h u s ( A x B ) x C i A x ( B x C ) , s h o w i n g t h e n e e d f o r p a r e n t h e s e s i n A x B x C t o a v o i d a m b i g u i t y .
3 0 . P r o v e t h a t t h e a r e a o f a p a r a l l e l o g r a m w i t h s i d e s A
a n d B i s j A x B I .
A r e a o f p a r a l l e l o g r a m = h I B
_
J A S s i n 6 J B {
= J A x B .
N o t e t h a t t h e a r e a o f t h e t r i a n g l e w i t h s i d e s A a n d
B = 2 1 A x B I .
3 1 . F i n d t h e a r e a o f t h e t r i a n g l e h a v i n g v e r t i c e s a t P ( 1 , 3 , 2 ) , Q ( 2 , - 1 , 1 ) , R ( - 1 , 2 , 3 ) .
P Q = ( 2 - 1 ) i + ( - 1 - 3 ) j + ( 1 - 2 ) k =
i - 4 j - k
P R = ( - 1 - 1 ) i + ( 2 - 3 ) j + ( 3 - 2 ) k = - 2 i - j + k
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T h e D O T a n d C R O S S P R O D U C T
F r o m P r o b l e m 3 0 ,
a r e a o f t r i a n g l e =
2 1 1 P Q x P R I
=
1 2 1 ( i - 4 j - k ) x ( - 2 i - j + k )
i
j
k
= 2 1 - 4
- 1
= 2 I - 5 i + j - 9 k l
=
z
( - 5 ) 2 + ( 1 ) 2 + ( - 9 ) 2
=
2 1 0 7 .
- 2
- 1
1
3 2 . D e t e r m i n e a u n i t v e c t o r p e r p e n d i c u l a r t o t h e p l a n e o f A = 2 i - 6 j - 3 k a n d B = 4 i + 3 j - k
.
A x B i s a v e c t o r p e r p e n d i c u l a r t o t h e p l a n e o f A a n d B .
i j k
A x B =
2
- 6
- 3
=
1 5 i - I O j + 3 0 k
4 3
- 1
A u n i t v e c t o r p a r a l l e l t o A X B i s
A X B
I A x B
1 5 i - 1 0 j + 3 0 k
( 1 5 ) 2 + ( - 1 0 ) 2 +
( 3 0 ) 2
3 2
=
7 i - 7 j + 7 k
A n o t h e r u n i t v e c t o r , o p p o s i t e i n d i r e c t i o n , i s ( - 3 i + 2 j - 6 k ) / 7 .
C o m p a r e w i t h P r o b l e m 1 6 .
3 3 . P r o v e t h e l a w o f s i n e s f o r p l a n e t r i a n g l e s .
L e t a , b a n d c r e p r e s e n t t h e s i d e s o f t r i a n g l e A B C
a s s h o w n i n t h e a d j o i n i n g f i g u r e ; t h e n a + b + c = 0 .
M u l -
t i p l y i n g b y a x , b x a n d c x i n s u c c e s s i o n , w e f i n d
a x b = b x c = c x a
i . e .
a b s i n C =
b e s i n A
= c a s i n B
s i n A
s i n B
s i n C
n r
= -
_
a b
c
3 4 . C o n s i d e r a t e t r a h e d r o n w i t h f a c e s
F l , F 2 , F 3 , F 4 .
L e t V 1 , V 2 , V 3 , V 4 b e v e c t o r s w h o s e m a g n i t u d e s a r e
r e s p e c t i v e l y e q u a l t o t h e a r e a s o f F l , F 2 , F 3 , F 4 a n d
w h o s e d i r e c t i o n s a r e p e r p e n d i c u l a r t o t h e s e f a c e s
i n t h e o u t w a r d d i r e c t i o n . S h o w t h a t V 1 + V 2 + V 3 + V 4 = 0 .
B y P r o b l e m 3 0 , t h e a r e a o f a t r i a n g u l a r f a c e d e t e r -
m i n e d b y R a n d S i s
2 I R x S I .
T h e v e c t o r s a s s o c i a t e d w i t h e a c h o f t h e f a c e s o f
t h e t e t r a h e d r o n a r e
V 1 = 2 A x B ,
V 2 = 2 B x C ,
V 3 = 2 C x A ,
V 4 = 2 ( C - A ) x ( B - A )
T h e n V 1 + V 2 + V 3 + V 4 =
2
[ A x B + B x C + C x A + ( C - A ) x ( B - A ) ]
=
2 [ A x B + B x C + C x A + C x B - C x A - A x B + A x A ]
0 .
2 5
T h i s r e s u l t c a n b e g e n e r a l i z e d t o c l o s e d p o l y h e d r a a n d i n t h e l i m i t i n g c a s e t o a n y c l o s e d s u r f a c e .
B e c a u s e o f t h e a p p l i c a t i o n p r e s e n t e d h e r e i t i s s o m e t i m e s c o n v e n i e n t t o a s s i g n a d i r e c t i o n t o a r e a a n d
w e s p e a k o f t h e v e c t o r a r e a .
3 5 . F i n d a n e x p r e s s i o n f o r t h e m o m e n t o f a f o r c e F a b o u t a p o i n t P .
T h e m o m e n t M o f F a b o u t P i s i n m a g n i t u d e e q u a l t o F t i m e s t h e p e r p e n d i c u l a r d i s t a n c e f r o m P t o t h e
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2 6
T h e D O T a n d C R O S S P R O D U C T
l i n e o f a c t i o n o f F . T h e n i f r i s t h e v e c t o r f r o m P t o t h e i n i -
t i a l p o i n t Q o f F ,
M = F ( r s i n 8 ) = r F s i n 8 =
( r x F I
I f w e t h i n k o f a r i g h t - t h r e a d e d s c r e w a t P p e r p e n d i c u l a r
t o t h e p l a n e o f r a n d F , t h e n w h e n t h e f o r c e F a c t s t h e s c r e w
w i l l m o v e i n t h e d i r e c t i o n o f r x F . B e c a u s e o f t h i s i t i s c o n -
v e n i e n t t o d e f i n e t h e m o m e n t a s t h e v e c t o r M = r x F .
3 6 . A r i g i d b o d y r o t a t e s a b o u t a n a x i s t h r o u g h p o i n t 0 w i t h
a n g u l a r s p e e d w . P r o v e t h a t t h e l i n e a r v e l o c i t y v o f a
p o i n t P o f t h e b o d y w i t h p o s i t i o n v e c t o r r i s g i v e n b y
v = , w x r , w h e r e w i s t h e v e c t o r w i t h m a g n i t u d e w w h o s e
d i r e c t i o n i s t h a t i n w h i c h a r i g h t - h a n d e d s c r e w w o u l d
a d v a n c e u n d e r t h e g i v e n r o t a t i o n .
S i n c e P t r a v e l s i n a c i r c l e o f r a d i u s r s i n 0 , t h e m a g n i -
t u d e o f t h e l i n e a r v e l o c i t y v i s w ( r s i n 0 ) _ j c v x r I .
A l s o , v
m u s t b e p e r p e n d i c u l a r t o b o t h w a n d r a n d i s s u c h t h a t r , 4 ) a n d
v f o r m a r i g h t - h a n d e d s y s t e m .
T h e n v a g r e e s b o t h i n m a g n i t u d e a n d d i r e c t i o n w i t h w x r ;
h e n c e v = 6 ) x r .
T h e v e c t o r C a i s c a l l e d t h e a n g u l a r v e l o c i t y .
T R I P L E P R O D U C T S .
3 7 . S h o w t h a t A ( B x C ) i s i n a b s o l u t e v a l u e e q u a l
t o t h e v o l u m e o f a p a r a l l e l e p i p e d w i t h s i d e s
A , B a n d C .
L e t n b e a u n i t n o r m a l t o p a r a l l e l o g r a m 1 ,
h a v i n g t h e d i r e c t i o n o f B x C , a n d l e t h b e t h e
h e i g h t o f t h e t e r m i n a l p o i n t o f A a b o v e t h e p a r -
a l l e l o g r a m 1 .
V o l u m e o f p a r a l l e l e p i p e d =
( h e i g h t h ) ( a r e a o f p a r a l l e l o g r a m 1 )
_
A { J B x C j n } =
I f A , B a n d C d o n o t f o r m a r i g h t - h a n d e d s y s t e m , A . n < 0 a n d t h e v o l u m e =
I A A . ( B x C )
3 8 . I f A = A 1 i + A 2 j + A s k , B = B 1 i + B 2 j + B 3 k ,
C = C 1 i + C 2 j + C 3 k
s h o w t h a t
A - ( B x C )
=
A
i
i
k
B 1 B 2 B 3
C 1
C 2 C 3
A l
A 2
B 1
B 2
C l
C 2
A s
B 3
C 3
= ( A 1 i + A 2 j + A 3 k ) ' l ( B 2 C 3 - B 3 C 2 ) i + ( B 3 C 1 - B 1 C 3 ) j + ( B 1 C 2 - B 2 C 1 ) k
A l A 2 A s
= A 1 ( B 2 C 3 - B 3 C 2 ) + A 2 ( B 3 C 1 - B 1 C 3 ) + A 3 ( B 1 C 2 - B 2 C 1 )
=
B 1
B 2 B 3
C 1
C 2
C 3
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T h e D O T a n d C R O S S P R O D U C T
2 7
3 9 . E v a l u a t e
( 2 i - 3 j )
[ ( i + j - k ) x ( 3 i - k ) ]
.
B y P r o b l e m 3 8 , t h e r e s u l t i s
2 - 3
0
1
1 - 1
3
0 - 1
= 4 .
A n o t h e r M e t h o d . T h e r e s u l t i s e q u a l t o
( 2 i - 3 j ) . [ i x ( 3 1 - k ) + j x ( 3 i - k ) - k x ( 3 i - k ) ]
=
( 2 i - 3 j ) - [ 3 i x i - i x k + 3 j x i - j x k - 3 k x i + k x k ]
=
( 2 i -
j - 3 k - i - 3 j + 0 )
= ( 2 i - 3 j ) ( - i - 2 j - 3 k ) =
( 2 ) ( - 1 ) + ( - 3 ) ( - 2 ) + ( 0 ) ( - 3 ) =
4 .
4 0 . P r o v e t h a t A ( B x C ) = B ( C x A )
= C ( A x B ) .
B y P r o b l e m 3 8 ,
A ( B x C )
=
A l A 2 A 3
B 1
B 2 B 3
C 1 C 2
C 3
B y a t h e o r e m o f d e t e r m i n a n t s w h i c h s t a t e s t h a t i n t e r c h a n g e o f t w o r o w s o f a d e t e r m i n a n t c h a n g e s i t s
s i g n , w e h a v e
A l A 2 A 3
B 1
B 2 B 3
B 1 B 2 B 3
B 1 B 2 B 3
A l A 2 A s
C 1
C 2 C 3
=
C l
C 2
C 3
C 1
C 2
C 3
A l A 2 A 3
A l A 2 A 3 C 1 C 2 C 3
C l
C 2 C 3
B 1
B 2 B 3
B 1
B 2 B 3
A l A 2 A 3
=
C 1
C 2
C 3
A l A 2 A 3
B 1
B 2
B 3
4 1 .
S h o w t h a t
A - ( B x C ) = ( A x B ) C
F r o m P r o b l e m 4 0 ,
A ( B x C ) = C . ( A x B ) =
( A x B ) C
O c c a s i o n a l l y A ( B x C ) i s w r i t t e n w i t h o u t p a r e n t h e s e s a s A B x C .
I n s u c h c a s e t h e r e c a n n o t b e
a n y a m b i g u i t y s i n c e t h e o n l y p o s s i b l e i n t e r p r e t a t i o n s a r e A ( B x C ) a n d ( A B ) x C . T h e l a t t e r h o w e v e r
h a s n o m e a n i n g s i n c e t h e c r o s s p r o d u c t o f a s c a l a r w i t h a v e c t o r i s u n d e f i n e d .
T h e r e s u l t A B x C = A x B C i s s o m e t i m e s s u m m a r i z e d i n t h e s t a t e m e n t t h a t t h e d o t a n d c r o s s c a n
b e i n t e r c h a n g e d w i t h o u t a f f e c t i n g t h e r e s u l t .
4 2 . P r o v e t h a t
A ( A x C ) = 0 .
F r o m P r o b l e m 4 1 ,
A . ( A x C ) =
( A x A ) . C =
0 .
4 3 . P r o v e t h a t a n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n f o r t h e v e c t o r s A , B a n d C t o b e c o p l a n a r i s t h a t
A B x C = 0 .
N o t e t h a t A A . B x C c a n h a v e n o m e a n i n g o t h e r t h a n A ( B x C ) .
I f A , B a n d C a r e c o p l a n a r t h e v o l u m e o f t h e p a r a l l e l e p i p e d f o r m e d b y t h e m i s z e r o . T h e n b y P r o b l e m
3 7 ,
A B x C = 0
t h e v o l u m e o f t h e p a r a l l e l e p i p e d f o r m e d b y v e c t o r s A , B a n d C i s z e r o ,
a n d s o t h e v e c t o r s m u s t l i e i n a p l a n e .
4 4 . L e t
r 1 = x 1 i + y 1 j + z 1 k ,
r 2 = x 2 i + y 2 i + z 2 k
a n d
r 3 = x 3 i + y 3 j + z 3 k
b e t h e p o s i t i o n v e c t o r s o f
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2 8
T h e D O T a n d C R O S S P R O D U C T
p o i n t s P i ( x 1 , y i , z 1 ) ,
P 2 ( x 2 , y 2 , z 2 ) a n d P 3 ( x 3 , y 3 , z 3 ) .
F i n d a n e q u a t i o n f o r t h e p l a n e p a s s i n g t h r o u g h P 1 ,
P 2 a n d P 3
.
W e a s s u m e t h a t P i , P 2 a n d P 3 d o n o t l i e i n t h e s a m e
s t r a i g h t l i n e ; h e n c e t h e y d e t e r m i n e a p l a n e .
L e t r = x i + y j + z k d e n o t e t h e p o s i t i o n v e c t o r o f a n y
p o i n t P ( x , y , z )
i n t h e p l a n e . C o n s i d e r v e c t o r s P I P 2 =
r 2 - r 1 , P i P 3 = r 3 - r 1 a n d P 1 P = r - r i w h i c h a l l l i e i n
t h e p l a n e .
B y P r o b l e m 4 3 ,
P I P
P i P 2 X P 1 P 3 = 0
o r
( r - r i ) . ( r 2 - r i ) x ( r 3 - r 1 )
=
0
I n t e r m s o f r e c t a n g u l a r c o o r d i n a t e s t h i s b e c o m e s
[ ( x - x i ) i + ( y - y 1 ) i + ( z - z 1 ) k ]
[ ( x 2 _ x 1 ) i + ( y 2 - Y 1 ) i + ( z 2 - z 1 ) k ] x [ ( x 3 - x 1 ) i + ( y 3 - Y 1 ) j + ( z 3 - z i ) k ] = 0
o r , u s i n g P r o b l e m 3 8 ,
- x 1
X 2 - X I
Y - Y 1
Y 2 - Y i
x 3 - x 1 Y 3 - y 1
= 0 .
4 5 . F i n d a n e q u a t i o n f o r t h e p l a n e d e t e r m i n e d b y t h e p o i n t s P 1 ( 2 , - 1 , 1 ) ,
P 2 ( 3 , 2 , - 1 ) a n d P 3 ( , - 1 , 3 , 2 ) .
T h e p o s i t i o n v e c t o r s o f P 1 , P 2 , P 3 a n d a n y p o i n t P ( x , y , z ) a r e r e s p e c t i v e l y r 1 = 2 1 - j + k ,
r 2 = 3 i + 2 j - k ,
r 3 = - i + 3 j + 2 k a n d r = x i + y j + z k .
T h e n P P 1 = r - r 1 ,
P 2 P 1 = r 2 - r 1 ,
P 3 P 1 = r 3 - r 1
a l l l i e i n t h e r e q u i r e d p l a n e , s o t h a t
( r - r 1 )
( r 2 - r 1 ) x ( r 3 - r 1 )
=
0
i . e .
[ ( x - 2 ) i + ( y + 1 ) j + ( z - 1 ) k ]
[ i + 3 j - 2 k ] x [ - 3 i + 4 j + k ]
=
0
[ ( x - 2 ) i + ( y + 1 ) j + ( z - l ) k ]
[ 1 l i + 5 j + 1 3 k ]
=
0
1 1 ( x - 2 ) + 5 ( y + 1 ) + 1 3 ( z - - 1 ) = 0
o r
1 1 x + 5 y + 1 3 z =
3 0 .
4 6 . I f t h e p o i n t s P , Q a n d R , n o t a l l l y i n g o n t h e s a m e s t r a i g h t l i n e , h a v e p o s i t i o n v e c t o r s a , b a n d c
r e l a t i v e t o a g i v e n o r i g i n , s h o w t h a t a x b + b x c + c x a i s a v e c t o r p e r p e n d i c u l a r t o t h e p l a n e
o f P , Q a n d R .
L e t r b e t h e p o s i t i o n v e c t o r o f a n y p o i n t i n t h e p l a n e o f P . Q a * 1 R . T h e n t h e v e c t o r s r - a , b -
a a n d
c - a a r e c o p l a n a r , s o t h a t b y P r o b l e m 4 3
( r - a )
( b - a ) x ( C - a )
= 0
o r
( r - a )
( a x b + b x c + c x a ) =
0 .
T h u s a x b + b x c + c x a i s p e r p e n d i c u l a r t o r - a a n d i s t h e r e f o r e p e r p e n d i c u l a r t o t h e p l a n e o f P , Q
a n d R .
4 7 . P r o v e : ( a )
A x ( B x C ) = B ( A C ) - C ( A B ) ,
( b )
( A x B ) x C = B ( A C ) - A ( B C ) .
( a ) L e t A = A i i + A 2 j + A s k ,
B = B 1 i + B 2 j + B 3 k , C = C i i + C 2 j + C 3 k .
i
j
k
T h e n
A x ( B x C ) _ ( A l l + A 2 j + A s k ) x
B 1 B 2 B 3
C 1 C 2 C 3
= ( A 1 i + A 2 j + A 3 k ) x ( [ B 2 C 3 - B 3 C 2 ] i + [ B S C 1 - B I C 3 ] i + [ B 1 C 2 - B 2 C 1 ] k )
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T h e D O T a n d C R O S S P R O D U C T
2 9
i j
k
A l A 2
A s
B 2 C 3 - B 3 C 2
B 3 C 1 - B 1 C 3
B 1 C 2 - B 2 C 1
_
( A 2 B 1 C 2 - A 2 B 2 C 1 - A 3 B 2 C 1 + A 3 B 1 C 3 ) i + ( A 3 B 2 C 3 - A 3 B 3 C 2 - A 1 B 1 C 2 + A 1 B 2 C 1 ) j
+ ( A 1 B 3 C 1 - A 1 B 1 C 3 - A 2 B 2 C 3 + A 2 B 3 C 2 ) k
A l s o
B ( A C ) - C ( A B )
( B 1 i + B 2 j + B 3 k ) ( A 1 C 1 + A 2 C 2 + A 3 C 3 ) - ( C 1 i + C 2 j + C 3 k ) ( A 1 B 1 + A 2 B 2 + A 4 B 3 )
( A 2 B 1 C 2 + A 3 B 1 C 3 - A 2 C 1 B 2 - A 3 C 1 B 3 ) i
+ ( B 2 A j C j + B 2 A 3 C 3 - C 2 A j . B j - C 2 A s B 3 ) j
+ ( B 3 A 1 C 1 + B 3 A 2 C 2 - C 3 A 1 B 1 - C 3 A 2 B 2 ) k
a n d t h e r e s u l t f o l l o w s .
( b ) ( A x B ) x C = - C x ( A x B ) = - { A ( C B ) - B ( C A ) } = B ( A C ) - A ( B C )
u p o n r e p l a c i n g A , B a n d
C i n ( a ) b y C , A a n d B r e s p e c t i v e l y .
N o t e t h a t
A x ( B x C ) / ( A x B ) x C ,
i . e . t h e a s s o c i a t i v e l a w f o r v e c t o r c r o s s p r o d u c t s i s n o t
v a l i d f o r a l l v e c t o r s A , B , C .
4 8 . P r o v e : ( A x B ) ( C X D ) =
F r o m P r o b l e m 4 1 ,
X . ( C X D ) _ ( X X C ) D .
L e t X = A X B ;
t h e n
( A x B ) ( C x D ) _ { ( A x B ) x C } D = { B ( A C ) - A ( B C ) }
D
_
( A C ) ( B D ) - ( A D ) ( B C ) ,
u s i n g P r o b l e m 4 7 ( b ) .
4 9 . P r o v e : A x ( B x C ) + B x ( C x A ) + C x ( A x B ) =
0 .
B y P r o b l e m 4 7 ( a ) ,
A d d i n g , t h e r e s u l t f o l l o w s .
A x ( B x C ) =
B
x ( C x A )
= C ( B A ) - A ( B C )
C x ( A x B ) = A ( C B ) - B ( C A )
5 0 . P r o v e : ( A x B ) x ( C x D ) =
B ( A C x D ) - A ( B C x D ) = C ( A B x D ) - D ( A B x C ) .
B y P r o b l e m 4 7 ( a ) ,
X x ( C x D ) = C ( X D ) - D ( X C ) .
L e t X = A x B ;
t h e n
( A x B ) x ( C x D ) = C ( A x B D ) - D ( A x B C )
= C ( A B X D ) - D ( A B X C )
B y P r o b l e m 4 7 ( b ) ,
( A x B ) x Y = B ( A Y ) - A ( B Y ) . L e t Y = C x D ; t h e n
( A x B ) x ( C x D ) = B ( A C x D ) - A ( B C x D )
5 1 . L e t P Q R b e a s p h e r i c a l t r i a n g l e w h o s e s i d e s p , q , r a r e a r c s o f g r e a t c i r c l e s . P r o v e t h a t
s i n P
s i n p
s i n Q
s i n q
s i n R
s i n r
S u p p o s e t h a t t h e s p h e r e ( s e e f i g u r e b e l o w ) h a s u n i t r a d i u s , a n d l e t u n i t v e c t o r s A , B a n d C b e d r a w n
f r o m t h e c e n t e r 0 o f t h e s p h e r e t o P , Q a n d R r e s p e c t i v e l y . F r o m P r o b l e m 5 0 ,
( 1 )
( A x B ) x ( A x C )
=
( A B x C ) A
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3 0
T h e D O T a n d C R O S S P R O D U C T
A u n i t v e c t o r p e r p e n d i c u l a r t o A x B a n d A x C i s A , s o
t h a t ( 1 ) b e c o m e s
( 2 ) s i n r s i n q s i n P A
=
( A . B x C ) A
o r
( 3 )
s i n r
s i n q s i n P
= A B x C
B y c y c l i c p e r m u t a t i o n o f p , q , r , P , Q , R a n d A , B , C w e
o b t a i n
( 4 )
s i n p s i n r
s i n Q
= B C x A
( 5 )
s i n q s i n p s i n R = C A x B
T h e n s i n c e t h e r i g h t h a n d s i d e s o f ( 3 ) , ( 4 ) a n d ( 5 ) a r e
e q u a l ( P r o b l e m 4 0 )
s i n r s i n q s i n P =
s i n p s i n r s i n Q = s i n q s i n p s i n R
f r o m w h i c h w e f i n d
s i n P
s i n Q
s i n R
s i n p
s i n q
s i n r
T h i s i s c a l l e d t h e l a w o f s i n e s f o r s p h e r i c a l t r i a n g l e s .
5 2 . P r o v e :
( A x B ) ( B x C ) x ( C x A )
( A B x C ) 2 .
B y P r o b l e m 4 7 ( a ) ,
X x ( C x A ) = C ( X A ) - A ( X . C ) .
L e t X = B x C ; t h e n
( B x C ) x ( C x A )
=
C ( B x C A ) - A ( B x C C )
= C ( A B x C ) - A ( B C x C ) = C ( A B x C )
T h u s
( A x B ) ( B x C ) x ( C x A ) =
( A x B ) C ( A B x C )
( A x B C ) ( A B x C ) _ ( A B x C ) 2
5 3 . G i v e n t h e v e c t o r s
a ' =
b x c
b ' =
c x a
a n d c ' =
a x b
, s h o w t h a t i f a b x c X 0 ,
a b x c '
a - b x c
a b x c
( a ) a ' a = b ' b = c ' c = 1 ,
( b ) a ' b = a ' c = 0 , b ' - a = b ' c = 0 ,
c ' a = c ' b = 0 ,
( c ) i f a b x c = V
t h e n a b ' x c ' = 1 / V ,
( d ) a ' , b ' , a n d c ' a r e n o n - c o p l a n a r i f a , b a n d c a r e n o n - c o p l a n a r .
( a ) a a = a a = a
b x c
=
a b x c =
1
a b x c
a b x c
_
b ' b = b b ' = b
c x a _ b c x a
=
a . b x c
a b x c
a b x c
a b x c
C
= c c = c -
a x b
=
c a x b
=
a b x c
_
1
a b x c
a b x c
a b x c
( b ) a b = b a = b
b x c
b b x c
b x b c
_ 0
a b x c
a b x c a b x c
S i m i l a r l y t h e o t h e r r e s u l t s f o l l o w . T h e r e s u l t s c a n a l s o b e s e e n b y n o t i n g , f o r e x a m p l e , t h a t a h a s
t h e d i r e c t i o n o f b x c a n d s o m u s t b e p e r p e n d i c u l a r t o b o t h b a n d c , f r o m w h i c h a b = 0 a n d a c = 0 .
F r o m ( a ) a n d ( b ) w e s e e t h a t t h e s e t s o f v e c t o r s a , b , c a n d a ' , b ' , c ' a r e r e c i p r o c a l v e c t o r s . S e e
a l s o S u p p l e m e n t a r y P r o b l e m s 1 0 4 a n d 1 0 6 .
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T h e D O T a n d C R O S S P R O D U C T
( C )
T h e n
a
_
b x c
c x a
c , _ a x b
V V
V
( a b x
V 3
V
V
u s i n g P r o b l e m 5 2 .
3 1
( d ) B y P r o b l e m 4 3 , i f
a , b a n d c a r e n o n - c o p l a n a r a b x c # 0 .
T h e n f r o m p a r t ( c ) i t f o l l o w s t h a t
a
b ' x c X 0
, s o t h a t a ' , b ' a n d c a r e a l s o n o n - c o p l a n a r .
5 4 . S h o w t h a t a n y v e c t o r r c a n b e e x p r e s s e d i n t e r m s o f t h e r e c i p r o c a l v e c t o r s o f P r o b l e m 5 3
r
=
( r - a ' ) a + ( r b ' ) b + ( r - d ) e .
F r o m P r o b l e m 5 0 ,
B ( A C x D ) - A ( B C x D )
=
C ( A B x D ) - D ( A B x C )
T h e n
D
D )
D ) +
D )
=
-
A . B x C
A . B x C
A
= a , B = b , C = c a n d D = r . T h e n
r
r b x c a
+ r c x a b +
r a x b c
a b x c
a b x c a b x c
x
r
( a b b x c ) a
+
r
( a c b x c ) b +
r
( a a b b c ) c
=
( r a ) a + ( r b ) b + ( r c ) c
S U P P L E M E N T A R Y P R O B L E M S
5 5 . E v a l u a t e :
( a ) k ( i + j ) ,
( b )
( i - 2 k )
( j + 3 k ) , ( c ) ( 2 i - j + 3 k ) ( 3 i + 2 j - k ) .
A n s .
( a ) 0
( b ) - 6
( c ) 1
5 6 .
I f A = i + 3 j - 2 k a n d B = 4 i - 2 j + 4 k , f i n d :
( a ) A B , ( b ) A ,
( c ) B ,
( d ) 1 3 A + 2 B ) ,
( e ) ( 2 A + B ) . ( A - 2 B ) .
A n s .
( a ) - 1 0
( b ) 1 4 ( c ) 6
( d )
1 5 0
( e ) - 1 4
a s
5 7 . F i n d t h e a n g l e b e t w e e n : ( a ) A = 3 i + 2 j - 6 k a n d B = 4 i - 3 j + k , ( b ) C = 4 i - 2 j + 4 k a n d D = 3 i - 6 j - 2 k .
A n s . ( a ) 9 0 °
( b ) a r c c o s 8 / 2 1 = 6 7 ° 3 6 '
5 8 . F o r w h a t v a l u e s o f a a r e A = a i - 2 j + k a n d B = 2 a i + a j - 4 k p e r p e n d i c u l a r 9
A n s . a = 2 , - 1
5 9 . F i n d t h e a c u t e a n g l e s w h i c h t h e l i n e j o i n i n g t h e p o i n t s ( 1 , - 3 , 2 ) a n d ( 3 , - 5 , 1 ) m a k e s w i t h t h e c o o r d i n a t e
a x e s .
A n s .
a r e c o s 2 / 3 , a r e c o s 2 / 3 , a r c c o s 1 / 3 o r 4 8 ° 1 2 ' , 4 8 ° 1 2 ' , 7 0 0 3 2 '
6 0 . F i n d t h e d i r e c t i o n c o s i n e s o f t h e l i n e j o i n i n g t h e p o i n t s ( 3 , 2 , - 4 ) a n d ( 1 , - 1 , 2 ) .
A n s . 2 / 7 , 3 / 7 , - 6 / 7 o r - 2 / 7 , - 3 / 7 , 6 / 7
6 1 . T w o s i d e s o f a t r i a n g l e a r e f o r m e d b y t h e v e c t o r s A = 3 i + 6 j - 2 k a n d B = 4 1 - j + 3 k .
D e t e r m i n e t h e a n g l e s
o f t h e t r i a n g l e .
A n s .
a r c c o s 7 / 0 7 - 5 , a r c c o s 2 6 / 7 5 , 9 0 °
o r
3 6 ° 4 ' , 5 3 ° 5 6 ' , 9 0 °
6 2 . T h e d i a g o n a l s o f a p a r a l l e l o g r a m a r e g i v e n b y A = 3 i - 4 j - k a n d B = 2 i + 3 j - 6 k . S h o w t h a t t h e p a r a l l e l o -
g r a m i s a r h o m b u s a n d d e t e r m i n e t h e l e n g t h o f i t s s i d e s a n d i t s a n g l e s .
A n s .
5 v " 3 - / 2 , a r c c o s 2 3 / 7 5 ,
1 8 0 ° - a r e c o s 2 3 / 7 5
o r
4 . 3 3 , 7 2 ° 8 ' ,
1 0 7 ° 5 2 '
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3 2
T h e D O T a n d C R O S S P R O D U C T
6 3 . F i n d t h e p r o j e c t i o n o f t h e v e c t o r
2 i - 3 j + 6 k o n t h e v e c t o r
i + 2 j + 2 k .
A n s . 8 / 3
6 4 . F i n d t h e p r o j e c t i o n o f t h e v e c t o r 4 1 - 3 J + k o n t h e l i n e p a s s i n g t h r o u g h t h e p o i n t s ( 2 , 3 , - 1 ) a n d ( - 2 , - 4 , 3 ) .
A n s .
1
6 5 . I f A = 4 i - j + 3 k a n d B = - 2 i + j - 2 k , f i n d a u n i t v e c t o r p e r p e n d i c u l a r t o b o t h A
a n d B .
A n s . ± ( i - 2 j - 2 k ) / 3
6 6 . F i n d t h e a c u t e a n g l e f o r m e d b y t w o d i a g o n a l s o f a c u b e .
A n s .
a r c c o s 1 / 3 o r
7 0 ° 3 2 `
6 7 . F i n d a u n i t v e c t o r p a r a l l e l t o t h e x y p l a n e a n d p e r p e n d i c u l a r t o t h e v e c t o r 4 i - 3 j + k .
A n s . ± ( 3 i + 4 j ) / 5
6 8 . S h o w t h a t A = ( 2 i - 2 j + k ) / 3 , B = ( i + 2 j + 2 k ) / 3 a n d C = ( 2 i + j - 2 k ) / 3 a r e m u t u a l l y o r t h o g o n a l u n i t
v e c t o r s .
6 9 . F i n d t h e w o r k d o n e i n m o v i n g a n o b j e c t a l o n g a s t r a i g h t l i n e f r o m ( 3 , 2 , - 1 ) t o ( 2 , - 1 , 4 ) i n a f o r c e f i e l d g i v e n
b y F = 4 1 - 3 j + 2 k .
A n s . 1 5
7 0 . L e t F b e a c o n s t a n t v e c t o r f o r c e f i e l d . S h o w t h a t t h e w o r k d o n e i n m o v i n g a n o b j e c t a r o u n d a n y c l o s e d p o l -
y g o n i n t h i s f o r c e f i e l d i s z e r o .
7 1 . P r o v e t h a t a n a n g l e i n s c r i b e d i n a s e m i - c i r c l e i s a r i g h t a n g l e .
7 2 . L e t A B C D b e a p a r a l l e l o g r a m . P r o v e t h a t A B 2 + B C 2 + C D 2 + D A 2 = A C 2 +
i f
7 3 .
I f A B C D i s a n y q u a d r i l a t e r a l a n d P a n d Q a r e t h e m i d p o i n t s o f i t s d i a g o n a l s , p r o v e t h a t
A B 2 + B C 2 + C D - 2 + D A 2 = A C 2 + Y D - 2 + 4 P Q 2
T h i s i s a g e n e r a l i z a t i o n o f t h e p r e c e d i n g p r o b l e m .
7 4 .
( a ) F i n d a n e q u a t i o n o f a p l a n e p e r p e n d i c u l a r t o a g i v e n v e c t o r A a n d d i s t a n t p f r o m t h e o r i g i n .
( b ) E x p r e s s t h e e q u a t i o n o f ( a ) i n r e c t a n g u l a r c o o r d i n a t e s .
A n s . ( a ) r n = p , w h e r e n = A / A ; ( b ) A 1 x + A 2 y + A 3 z = A p
7 5 . L e t r 1 a n d r 2 b e u n i t v e c t o r s i n t h e x y p l a n e m a k i n g a n g l e s a a n d R w i t h t h e p o s i t i v e x - a x i s .
( a ) P r o v e t h a t r 1 = c o s a i
+ s i n a j ,
r 2 = c o s ( 3 i
+ s i n I 3 j .
( b ) B y c o n s i d e r i n g r 1 . r 2 p r o v e t h e t r i g o n o m e t r i c f o r m u l a s
c o s ( a - ( 3 ) = c o s a c o s a + s i n a s i n ( 3 ,
c o s ( ( % + S ) =
c o s a c o s ( 3 - s i n a s i n R
7 6 . L e t a b e t h e p o s i t i o n v e c t o r o f a g i v e n p o i n t ( x 1 , y 1 , z 1 ) , a n d r t h e p o s i t i o n v e c t o r o f a n y p o i n t ( x , y , z ) .
D e -
s c r i b e t h e l o c u s o f r i f ( a )
I r - a I
= 3 , ( b ) ( r - a ) . a = 0 ,
( c ) ( r - a ) . r = 0 .
A n s . ( a ) S p h e r e , c e n t e r a t ( x 1 , y 1 , z 1 ) a n d r a d i u s 3 .
( b ) P l a n e p e r p e n d i c u l a r t o a a n d p a s s i n g t h r o u g h i t s t e r m i n a l p o i n t .
( c ) S p h e r e w i t h c e n t e r a t ( x 1 / 2 , y 1 / 2 , z 1 / 2 ) a n d r a d i u s i
x i + y 1 + z 1 , o r a s p h e r e w i t h a a s d i a m e t e r .
7 7 . G i v e n t h a t A = 3 i + j + 2 k a n d B = i - 2 j - 4 k a r e t h e p o s i t i o n v e c t o r s o f p o i n t s P a n d Q r e s p e c t i v e l y .
( a ) F i n d a n e q u a t i o n f o r t h e p l a n e p a s s i n g t h r o u g h Q a n d p e r p e n d i c u l a r t o l i n e P Q .
( b ) W h a t i s t h e d i s t a n c e f r o m t h e p o i n t ( - 1 , 1 , 1 ) t o t h e p l a n e ?
A n s .
( a )
0
o r
2 x + 3 y + 6 z = - 2 8 ;
( b ) 5
7 8 . E v a l u a t e e a c h o f t h e f o l l o w i n g :
( a ) 2 j x ( 3 i - 4 k ) , ( b ) ( i + 2 j ) x k , ( c ) ( 2 i - 4 k ) x ( i + 2 j ) , ( d ) ( 4 i + j - 2 k ) x ( 3 i + k ) , ( e ) ( 2 i + j - k ) x ( 3 i - 2 j + 4 k ) .
A n s . ( a ) - 8 i - 6 k ,
( b ) 2 i - j ,
( c ) 8 i - 4 j + 4 k ,
( d ) i - l O j - 3 k ,
( e ) 2 i - l l j - 7 k
7 9 . I f A = 3 i - j - 2 k a n d B = 2 i + 3 j + k , f i n d :
( a ) I A x B I ,
( b ) ( A + 2 B ) x ( 2 A - B ) ,
( c )
I ( A + B ) x ( A - B )
.
A n s . ( a ) ,
( b ) - 2 5 i + 3 5 j - 5 5 k ,
( c ) 2
1 9 5
8 0 .
I f A = i - 2 j - 3 k , B = 2 1 + j - k a n d C = i + 3 j - 2 k , f i n d :
( a )
I ( A x B ) x C I ,
( c ) A ( B x C ) ,
( e ) ( A x B ) x ( B x C )
( b ) I A x ( B x C ) I ,
( d )
( f )
A n s .
( a ) 5
2 6 ,
( b ) 3
1 6 , ( c ) - 2 0 ,
( d ) - 2 0 ,
( e ) - 4 0 1 - 2 0 j + 2 0 k , ( ( f ) 3 5 i - 3 5 j + 3 5 k
8 1 . S h o w t h a t i f A 0 a n d b o t h o f t h e c o n d i t i o n s ( a )
a n d ( b ) A x B = A x C h o l d s i m u l t a n e o u s l y
t h e n B = C , b u t i f o n l y o n e o f t h e s e c o n d i t i o n s h o l d s t h e n B # C n e c e s s a r i l y .
8 2 . F i n d t h e a r e a o f a p a r a l l e l o g r a m h a v i n g d i a g o n a l s A = 3 i + J - 2 k a n d B = i - 3 j + 4 k .
A n s .
5 0 3 -
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T h e D O T a n d C R O S S P R O D U C T
3 3
8 3 . F i n d t h e a r e a o f a t r i a n g l e w i t h v e r t i c e s a t ( 3 , - 1 , 2 ) , ( 1 , - 1 , - 3 ) a n d ( 4 , - 3 , 1 ) .
A n s . 2
6 1
8 4 .
I f A = 2 t + j - 3 k a n d B = i - 2 j + k , f i n d a v e c t o r o f m a g n i t u d e 5 p e r p e n d i c u l a r t o
b o t h A a n d B .
A n s . ±
5 3
( i + j + k )
8 5 . U s e P r o b l e m 7 5 t o d e r i v e t h e f o r m u l a s
s i n ( a - ( 3 ) = s i n a c o s ( 3 - c o s a s i n Q ,
s i n ( a + ( 3 )
= s i n a c o s S + c o s a s i n R
8 6 . A f o r c e g i v e n b y F = 3 i + 2 j - 4 k i s a p p l i e d a t t h e p o i n t ( 1 , - 1 , 2 ) . F i n d t h e m o m e n t o f F
a b o u t t h e p o i n t
( 2 , - 1 , 3 ) .
A n s .
2 1 - 7 j - 2 k
8 7 . T h e a n g u l a r v e l o c i t y o f a r o t a t i n g r i g i d b o d y a b o u t a n a x i s o f r o t a t i o n i s g i v e n b y w = 4 i + j - 2 k . F i n d t h e
l i n e a r v e l o c i t y o f a p o i n t P o n t h e b o d y w h o s e p o s i t i o n v e c t o r r e l a t i v e t o a p o i n t o n t h e a x i s o f r o t a t i o n i s
2 i - 3 j + k .
A n s . - 5 i - 8 i - - 1 4 k
8 8 . S i m p l i f y ( A + B ) ( B + C ) x ( C + A ) .
A n s . 2 A B x C
8 9 . P r o v e t h a t
( A
B x C ) ( a b x c ) _
A a A b A c
B a B b B c
C - a C b C c
9 0 . F i n d t h e v o l u m e o f t h e p a r a l l e l e p i p e d w h o s e e d g e s a r e r e p r e s e n t e d b y
A = 2 t - 3 j + 4 k , B = i + 2 j - k '
C = 3 i - j + 2 k .
A n s . 7
9 1 .
I f A . B x C = 0 , s h o w t h a t e i t h e r ( a ) A , B a n d C a r e c o p l a n a r b u t n o t w o o f t h e m a r e c o l l i n e a r , o r ( b )
t w o
o f t h e v e c t o r s A , B a n d C a r e c o l l i n e a r , o r ( c ) a l l o f t h e v e c t o r s A , B a n d C a r e c o l l i n e a r .
9 2 . F i n d t h e c o n s t a n t a s u c h t h a t t h e v e c t o r s 2 i - j + k , i + 2 j - 3 k a n d 3 i + a j + 5 k a r e c o p l a n a r .
A n s . a =
9 3 .
I f A = x 1 a + y i b + z i c , B = x 2 a + y 2 b + z 2 c a n d C = x 3 a + y 3 b + z 3 c ,
p r o v e t h a t
A B x C
x i
Y 1
Z i
X 2
Y 2
Z 2
X 3
Y 3
Z 3
( a b x c )
- 4
9 4 . P r o v e t h a t a n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n t h a t A x ( B x C ) = ( A x B ) x C i s ( A x C ) x B = 0 .
D i s -
c u s s t h e c a s e s w h e r e A - B = 0 o r B C = 0 .
9 5 . L e t p o i n t s P . Q a n d R h a v e p o s i t i o n v e c t o r s r 1 = 3 i - 2 j - k , r 2 = i + 3 j + 4 k a n d r 3 = 2 1 + j - 2 k r e l a t i v e t o
a n o r i g i n 0 . F i n d t h e d i s t a n c e f r o m P t o t h e p l a n e O Q R .
A n s .
3
9 6 . F i n d t h e s h o r t e s t d i s t a n c e f r o m ( 6 , - 4 , 4 ) t o t h e l i n e j o i n i n g ( 2 , 1 , 2 ) a n d ( 3 , - 1 , 4 ) .
A n s .
3
9 7 . G i v e n p o i n t s P ( 2 , 1 , 3 ) , Q ( 1 , 2 , 1 ) , R ( - 1 , - 2 , - 2 ) a n d S ( 1 , - 4 , 0 ) , f i n d t h e s h o r t e s t d i s t a n c e b e t w e e n l i n e s P Q a n d
R S .
A n s . 3 v 2
9 8 . P r o v e t h a t t h e p e r p e n d i c u l a r s f r o m t h e v e r t i c e s o f a t r i a n g l e t o t h e o p p o s i t e s i d e s ( e x t e n d e d i f n e c e s s a r y )
m e e t i n a p o i n t ( t h e o r t h o c e n t e r o f t h e t r i a n g l e ) .
9 9 . P r o v e t h a t t h e p e r p e n d i c u l a r b i s e c t o r s o f t h e s i d e s o f a t r i a n g l e m e e t i n a p o i n t ( t h e c i r c u m c e n t e r o f t h e t r i -
a n g l e ) .
1 0 0 . P r o v e t h a t ( A x B ) ( C x D ) + ( B x C ) ( A x D ) + ( C x A ) ( B x D ) = 0 .
1 0 1 . L e t P Q R b e a s p h e r i c a l t r i a n g l e w h o s e s i d e s p , q , r a r e a r c s o f g r e a t c i r c l e s . P r o v e t h e l a w o f c o s i n e s f o r
s p h e r i c a l t r i a n g l e s ,
c o s p = c o s q c o s r + s i n q s i n r c o s P
w i t h a n a l o g o u s f o r m u l a s f o r c o s q a n d c o s r o b t a i n e d b y c y c l i c p e r m u t a t i o n o f t h e l e t t e r s .
[ H i n t : I n t e r p r e t b o t h s i d e s o f t h e i d e n t i t y ( A x B ) ( A x C ) = ( B C ) ( A A ) - ( A C ) ( B A ) . ]
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3 4
T h e D O T a n d C R O S S P R O D U C T
1 0 2 . F i n d a s e t o f v e c t o r s r e c i p r o c a l t o t h e s e t 2 1 + 3 j - k , i - j - 2 k , - i + 2 j + 2 k .
A n s . 2 i + l k
- 8 i + j - ? k , - ? i + j - 5 k
3 3 3 3 3 3
b x c
b
,
c x a
,
a x b
1 0 3 . I f
a ' =
a . b x c '
a b x c
a n d
c =
a b x c '
p r o v e t h a t
b ' x c '
c ' x
a
a x b '
a ' b x c '
b
a b ' x c '
c
a b ' x c '
1 0 4 . I f a , b , c a n d a ' , b ' , c ' a r e s u c h t h a t
a ' a =
b ' b = c ' c
a ' b = a ' c = b ' a = b ' c = c ' a = c ' b = 0
p r o v e t h a t i t n e c e s s a r i l y f o l l o w s t h a t
a
=
b x c
b , =
c x a
C l =
a x b
a b x c
a b x c a b x c
1 0 5 . P r o v e t h a t t h e o n l y r i g h t - h a n d e d s e l f - r e c i p r o c a l s e t s o f v e c t o r s a r e t h e u n i t v e c t o r s i , j , k .
1 0 6 . P r o v e t h a t t h e r e i s o n e a n d o n l y o n e s e t o f v e c t o r s r e c i p r o c a l t o a g i v e n s e t o f n o n - c o p l a n a r v e c t o r s a , b , c .
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O R D I N A R Y D E R I V A T I V E S O F V E C T O R S . L e t R ( u )
b e a v e c t o r d e p e n d i n g o n a s i n g l e s c a l a r v a r i a b l e u .
T h e n
L R
_
R ( u + A u ) - R ( u )
A u
A u
w h e r e A u d e n o t e s a n i n c r e m e n t i n u ( s e e a d j o i n i n g
f i g u r e ) .
T h e o r d i n a r y d e r i v a t i v e o f t h e v e c t o r R ( u ) w i t h r e s p e c t t o t h e s c a l a r u i s g i v e n b y
d R
=
l i m
A R
=
l i m
R ( u + A u ) - R ( u )
d u
A u - ' o A u
A u - . o
A u
i f t h e l i m i t e x i s t s .
S i n c e d R i s i t s e l f a v e c t o r d e p e n d i n g o n u , w e c a n c o n s i d e r i t s d e r i v a t i v e w i t h r e s p e c t t o u . I f
t h i s d e r i v a t i v e e x i s t s i t i s d e n o t e d b y
a R .
I n l i k e m a n n e r h i g h e r o r d e r d e r i v a t i v e s a r e d e s c r i b e d .
S P A C E C U R V E S . I f i n p a r t i c u l a r R ( u ) i s t h e p o s i t i o n v e c t o r r ( u ) j o i n i n g t h e o r i g i n 0 o f a c o o r d i n a t e
s y s t e m a n d a n y p o i n t ( x , y , z ) , t h e n
r ( u )
=
x ( u ) i + y ( u ) j + z ( u ) k
a n d s p e c i f i c a t i o n o f t h e v e c t o
u n c t i o n r ( u d e f i n e s x , y a n d z a s f u n c t i o n s o f
A s u c h a n g e s , t h e t e r m i n a l p o i n t o f r d e s c r i b e s
a s p a c e c u r v e h a v i n g p a r a m e t r i c e q u a t i o n s
x = x ( u ) ,
y = y ( u ) ,
z = z ( u )
T h e n
Q u
=
r ( u + A u u )
A u
- r ( u )
i s a v e c t o r i n
O r
t h e d i -
r e c t i o n o f A r ( s e e a d j a c e n t f i g u r e ) .
I f
l i m
= d r
A U - 0 A u
d u
e x i s t s , t h e l i m i t w i l l b e a v e c t o r i n t h e d i r e c t i o n o f
t h e t a n g e n t t o t h e s p a c e c u r v e a t ( x , y , z ) a n d i s g i v -
e n b y
d r
_
d x
d y d z
d u d u l +
i
u
+ d u k
I f u i s t h e t i m e t ,
d
r e p r e s e n t s t h e v e l o c i t y v w i t h
w h i c h t h e t e r m i n a l p o i n t o f r d e s c r i b e s t h e c u r v e . S i m i l a r l y ,
d
a l o n g t h e c u r v e .
x
d 2 r
d t 2
3 5
r e p r e s e n t s i t s a c c e l e r a t i o n a
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3 6
V E C T O R D I F F E R E N T I A T I O N
C O N T I N U I T Y A N D D I F F E R E N T I A B I L I T Y . A s c a l a r f u n c t i o n
c ( u ) i s c a l l e d c o n t i n u o u s a t u i f
l i m o 4 ) ( u + A u ) _ 0 ( u ) .
E q u i v a l e n t l y , ¢ 6 ( u ) i s c o n t i n u -
A U -
o u s a t u i f f o r e a c h p o s i t i v e n u m b e r a w e c a n f i n d s o m e p o s i t i v e n u m b e r 6 s u c h t h a t
1 g b ( u + A u )
- 0 ( u )
l < E
w h e n e v e r
j A u j
< 8 .
A v e c t o r f u n c t i o n R ( u ) = R 1 ( u ) i + R 2 ( u ) j + R 3 ( u ) k i s c a l l e d c o n t i n u o u s a t u i f t h e t h r e e s c a l a r
m R ( u + A u ) = R ( u ) . E q u i v a l e n t l y , R ( u )
u n c t i o n s R 1 ( u ) , R 2 ( u ) a n d R 3 ( u ) a r e c o n t i n u o u s a t u o r i f
A l u
o
i s c o n t i n u o u s a t u i f f o r e a c h p o s i t i v e n u m b e r e w e c a n f i n d s o m e p o s i t i v e n u m b e r 8 s u c h t h a t
I R ( u + A u )
- R ( u )
I
<
E
w h e n e v e r
I A u f
<
8 .
A s c a l a r o r v e c t o r f u n c t i o n o f u i s c a l l e d d i f f e r e n t i a b l e o f o r d e r n i f i t s n t h d e r i v a t i v e e x i s t s . A
f u n c t i o n w h i c h i s d i f f e r e n t i a b l e i s n e c e s s a r i l y c o n t i n u o u s b u t t h e c o n v e r s e i s n o t t r u e . U n l e s s o t h e r -
w i s e s t a t e d w e a s s u m e t h a t a l l f u n c t i o n s c o n s i d e r e d a r e d i f f e r e n t i a b l e t o a n y o r d e r n e e d e d i n a p a r -
t i c u l a r d i s c u s s i o n .
D I F F E R E N T I A T I O N F O R M U L A S . I f A , B a n d C a r e d i f f e r e n t i a b l e v e c t o r f u n c t i o n s o f a s c a l a r u , a n d
0 i s a d i f f e r e n t i a b l e s c a l a r f u n c t i o n o f u , t h e n
2 .
d u ( A + B )
d u
( A
B )
=
d A +
d u
A d B + d u B
3 .
u ( A x B )
= A x
d B
+
d A x B
4 .
u ( O A )
_
d A
+
L o A
d u
d u
d u ( A - B x C ) _
d u
+
d A B x C
6 . d
u
{ A x ( B x C ) }
=
A X ( x C ) + d u x ( B x C )
T h e o r d e r i n t h e s e p r o d u c t s m a y b e i m p o r t a n t .
P A R T I A L D E R I V A T I V E S O F V E C T O R S . I f A i s a v e c t o r d e p e n d i n g o n m o r e t h a n o n e s c a l a r v a r i a b l e ,
s a y x , y , z
f o r e x a m p l e , t h e n w e w r i t e A = A ( x , y , z ) . T h e
p a r t i a l d e r i v a t i v e o f A w i t h r e s p e c t t o x i s d e f i n e d a s
'
=
l m
A ( x + A x ,
y , z ) - A ( x , y , z )
x
A X - 0
A x
i f t h i s l i m i t e x i s t s . S i m i l a r l y ,
a A
A ( x , y + A y , z ) - A ( x , y , z )
y
y m o A y
a A
l i r a
A ( x , y , z + A z ) - A ( x , y , z )
a z
_
A z
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V E C T O R D I F F E R E N T I A T I O N 3 7
a r e t h e p a r t i a l d e r i v a t i v e s o f A w i t h r e s p e c t t o y a n d z r e s p e c t i v e l y i f t h e s e l i m i t s e x i s t .
T h e r e m a r k s o n c o n t i n u i t y a n d d i f f e r e n t i a b i l i t y f o r f u n c t i o n s o f o n e v a r i a b l e c a n b e e x t e n d e d t o
f u n c t i o n s o f t w o o r m o r e v a r i a b l e s .
F o r e x a m p l e , c ( x , y )
i s c a l l e d c o n t i n u o u s a t
( x , y ) i f
J i m 0 ( x + A x , y + A y ) = q 5 ( x , y ) ,
o r i f f o r e a c h p o s i t i v e n u m b e r e w e c a n f i n d s o m e p o s i t i v e n u m b e r
A Y - 0
8 s u c h t h a t
0 ( x + A x , y + A y ) - g b ( x , y ) 1 < E w h e n e v e r
j A x j
< 8 a n d
I A y I
< 8 .
S i m i l a r d e f i -
n i t i o n s h o l d f o r v e c t o r f u n c t i o n s .
F o r f u n c t i o n s o f t w o o r m o r e v a r i a b l e s w e u s e t h e t e r m d i f f e r e n t i a b l e t o m e a n t h a t t h e f u n c t i o n
h a s c o n t i n u o u s f i r s t p a r t i a l d e r i v a t i v e s . ( T h e t e r m i s u s e d b y o t h e r s i n a s l i g h t l y w e a k e r s e n s e . )
H i g h e r d e r i v a t i v e s c a n b e d e f i n e d a s i n t h e c a l c u l u s . T h u s , f o r e x a m p l e ,
a 2 A
_
a
A
2 A a A
a x e
a x ( a x ) ,
a y e
a y - 6 y )
a 2 A
a a A
a 2 A
a ( a A
a x a y = a x ( a y
a y a x
=
a y a x
a 2 A
a a A
a z 2 a z ( a z )
a 3 A
a
a 2 A
a x a z 2
-
a x a z 2
I f A h a s c o n t i n u o u s p a r t i a l d e r i v a t i v e s o f t h e s e c o n d o r d e r a t l e a s t , t h e n a 2 A ' - a 2 `
a x a y
a y a x
, i . e . t h e
o r d e r o f d i f f e r e n t i a t i o n d o e s n o t m a t t e r .
R u l e s f o r p a r t i a l d i f f e r e n t i a t i o n o f v e c t o r s a r e s i m i l a r t o t h o s e u s e d i n e l e m e n t a r y c a l c u l u s f o r
s c a l a r f u n c t i o n s . T h u s i f A a n d B a r e f u n c t i o n s o f x , y , z t h e n , f o r e x a m p l e ,
1 . a x
( A B ) = A -
a $
+ 2 A . B
a x ( A x B ) = A x
a B
+
a A x
B
=
{ a x ( A . B ) }
=
a
{ A . a B
+ a A . B }
y
y
A
a 2 B
+
a A 3 B
+
a A a B
+
a 2 A
. B
a y a x
a y
a x
a x a y a y a x
'
e t c .
D I F F E R E N T I A L S O F V E C T O R S f o l l o w r u l e s s i m i l a r t o t h o s e o f e l e m e n t a r y c a l c u l u s .
1 .
I f A = A l ' + A 2 j + A 3 k , t h e n d A = d A 1 i + d A 2 j + d A 3 k
2 . d ( A B ) = A d B + d A B
3 . d ( A x B ) = A x d B + d A x B
A
A
d x +
a A
d y + a
d z ,
e t c .
.
I f A = A ( x , y , z ) , t h e n
d A = a
F o r e x a m p l e ,
D I F F E R E N T I A L G E O M E T R Y i n v o l v e s a s t u d y o f s p a c e c u r v e s a n d s u r f a c e s .
I f C i s a s p a c e c u r v e
d e f i n e d b y t h e f u n c t i o n r ( u ) , t h e n w e h a v e s e e n t h a t d u i s a v e c t o r i n
t h e d i r e c t i o n o f t h e t a n g e n t t o C . I f t h e s c a l a r u i s t a k e n a s t h e a r c l e n g t h s m e a s u r e d f r o m s o m e f i x e d
p o i n t o n C , t h e n - d - r -
i s a u n i t t a n g e n t v e c t o r t o C a n d i s d e n o t e d b y T ( s e e d i a g r a m b e l o w ) .
T h e
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3 8
V E C T O R D I F F E R E N T I A T I O N
r a t e a t w h i c h T c h a n g e s w i t h r e s p e c t t o s i s a m e a -
s u r e o f t h e c u r v a t u r e o f C a n d i s g i v e n b y d T .
T h e
d T
a s
d i r e c t i o n o f
d s
a t a n y g i v e n p o i n t o n C i s n o r m a l t o
t h e c u r v e a t t h a t p o i n t ( s e e P r o b l e m 9 ) . I f N i s a
u n i t v e c t o r i n t h i s n o r m a l d i r e c t i o n , i t i s c a l l e d t h e
p r i n c i p a l n o r m a l t o t h e c u r v e . T h e n d s = K N , w h e r e
K i s c a l l e d t h e c u r v a t u r e o f C a t t h e s p e c i f i e d p o i n t .
T h e q u a n t i t y p = 1 / K i s c a l l e d t h e r a d i u s o f c u r v a -
t u r e .
A u n i t v e c t o r B p e r p e n d i c u l a r t o t h e p l a n e o f T a n d N a n d s u c h t h a t B = T x N , i s c a l l e d t h e b i -
n o r m a l t o t h e c u r v e .
I t f o l l o w s t h a t d i r e c t i o n s T , N , B f o r m a l o c a l i z e d r i g h t - h a n d e d r e c t a n g u l a r c o -
o r d i n a t e s y s t e m a t a n y s p e c i f i e d p o i n t o f C . T h i s c o o r d i n a t e s y s t e m i s c a l l e d t h e t r i h e d r a l o r t r i a d
a t t h e p o i n t . A s s c h a n g e s , t h e c o o r d i n a t e s y s t e m m o v e s a n d i s k n o w n a s t h e m o v i n g t r i h e d r a l .
A s e t o f r e l a t i o n s i n v o l v i n g d e r i v a t i v e s o f t h e f u n d a m e n t a l v e c t o r s T , N a n d B i s k n o w n c o l l e c -
t i v e l y a s t h e F r e n e t - S e r r e t f o r m u l a s g i v e n b y
d T
= K N , d N = T B - K T ,
d B
= - T N
w h e r e r i s a s c a l a r c a l l e d t h e t o r s i o n . T h e q u a n t i t y c r = 1 / T i s c a l l e d t h e r a d i u s o f t o r s i o n .
T h e o s c u l a t i n g p l a n e t o a c u r v e a t a p o i n t P i s t h e p l a n e c o n t a i n i n g t h e t a n g e n t a n d p r i n c i p a l
n o r m a l a t P . T h e n o r m a l p l a n e i s t h e p l a n e t h r o u g h P p e r p e n d i c u l a r t o t h e t a n g e n t . T h e r e c t i f y i n g
p l a n e i s t h e p l a n e t h r o u g h P w h i c h i s p e r p e n d i c u l a r t o t h e p r i n c i p a l n o r m a l .
M E C H A N I C S o f t e n i n c l u d e s a s t u d y o f t h e m o t i o n o f p a r t i c l e s a l o n g c u r v e s , t h i s s t u d y b e i n g k n o w n
a s k i n e m a t i c s . I n t h i s c o n n e c t i o n s o m e o f t h e r e s u l t s o f d i f f e r e n t i a l g e o m e t r y c a n b e o f
v a l u e .
A s t u d y o f f o r c e s o n m o v i n g o b j e c t s i s c o n s i d e r e d i n d y n a m i c s .
F u n d a m e n t a l t o t h i s s t u d y i s
N e w t o n ' s f a m o u s l a w w h i c h s t a t e s t h a t i f F i s t h e n e t f o r c e a c t i n g o n a n o b j e c t o f m a s s
m m o v i n g
w i t h v e l o c i t y v , t h e n
F =
d t ( m v )
w h e r e m y i s t h e m o m e n t u m o f t h e o b j e c t . I f m i s c o n s t a n t t h i s b e c o m e s F =
m
d v
= m a , w h e r e a i s
a t
t h e a c c e l e r a t i o n o f t h e o b j e c t .
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V E C T O R D I F F E R E N T I A T I O N
3 9
S O L V E D P R O B L E M S
1 . I f R ( u ) = x ( u ) i + y ( u ) j + z ( u ) k , w h e r e x , y a n d z a r e d i f f e r e n t i a b l e f u n c t i o n s o f a s c a l a r u ,
p r o v e
t h a t
d u R
d u 1 + d j
+ d u
k .
R ( u + A u ) - - - R ( u )
R
=
l i
m
d u
A U - 0
A u
[ x ( u + A u ) i + Y ( u + A u ) i + z ( u + A u ) k ] -
[ x ( u ) i
+
Y ( u ) j + z ( u ) k ]
= J i m
A u - 0
A U
x ( u + A u ) - x ( u )
i
y ( u + A u ) - y ( u )
z ( u + A u ) - z ( u )
k
l i m
A u - 0
+
A u
D u
+
D u
d x
d y
,
d z
d u i +
d d u
+ d u k
2 2
A R
d R
d R
d R
2 . G i v e n R = s i n t i + c o s t j + t k ,
f i n d ( a )
d t
,
( b ) d t 2
,
( c )
I
d t
I
, ( d )
I
d t 2
d R d
d
d
( a )
d t
d t
( s i n t ) i +
d t
( c o s t ) j +
d t
( t ) k = c o s t i - s i n t j + k
d 2 R
d d R
d
d
d
( b )
d t 2 d t ( d t ) = d t
( c o S t ) i -
d t
( s i n t ) j +
d t
( 1 ) k = - s i n t i - c o s t j
( c )
I
d R
I
=
( c o S t ) 2 + ( - s i n t ) 2 + ( 1 ) 2
=
2
( d )
I
d t R I
( - s i n t ) 2 + ( - c o s t ) ' =
1
3 . A p a r t i c l e m o v e s a l o n g a c u r v e w h o s e p a r a m e t r i c e q u a t i o n s a r e x = e - t , y = 2 c o s 3 t , z = 2 s i n 3 t ,
w h e r e t i s t h e t i m e .
( a ) D e t e r m i n e i t s v e l o c i t y a n d a c c e l e r a t i o n a t a n y t i m e .
( b ) F i n d t h e m a g n i t u d e s o f t h e v e l o c i t y a n d a c c e l e r a t i o n a t t = 0 .
( a ) T h e p o s i t i o n v e c t o r r o f t h e p a r t i c l e i s r = x i + y j + z k = e - t i + 2 c o s 3 t j + 2 s i n 3 t k .
T h e n t h e v e l o c i t y i s
v =
d r
= - e - t i - 6 s i n 3 t j + 6 c o s 3 t k
2
a n d t h e a c c e l e r a t i o n i s a = d r
= e ' " t i - 1 8 c o s 3 t j - 1 8 s i n 3 t k
d t 2
2
( b ) A t t = 0 ,
d t
= - i + 6 k
a n d
d t 2 =
i - 1 8 j .
T h e n
m a g n i t u d e o f v e l o c i t y a t t = 0 i s
( - 1 ) 2 + ( 6 ) 2 = 3 7
m a g n i t u d e o f a c c e l e r a t i o n a t t = 0 i s
( 1 ) 2 + ( - 1 8 ) 2 = 4 2 5 .
4 . A p a r t i c l e m o v e s a l o n g t h e c u r v e x = 2 t 2 , y = t 2 - 4 t , z = 3 t - 5 , w h e r e t i s t h e t i m e .
F i n d t h e
c o m p o n e n t s o f i t s v e l o c i t y a n d a c c e l e r a t i o n a t t i m e t = 1 i n t h e d i r e c t i o n i - 3 j + 2 k .
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4 0
V E C T O R D I F F E R E N T I A T I O N
V e l o c i t y
=
d t
d t
[ 2 t 2 i + ( t 2 - 4 t ) j + ( 3 t - 5 ) k ]
= 4 t i + ( 2 t - 4 ) j
+ 3 k
=
4 i - 2 j + 3 k
a t t = 1 .
U n i t v e c t o r i n d i r e c t i o n i - 3 j + 2 k i s
i - 3 j + 2 k
i - 3 i + 2 k
( 1 ) 2 + ( - 3 ) 2 + ( 2 ) 2
T h e n t h e c o m p o n e n t o f t h e v e l o c i t y i n t h e g i v e n d i r e c t i o n i s
( 4 1 - 2 j + 3 k )
( i - 3 j + 2 k )
( 4 ) ( 1 ) + ( - 2 ) ( - 3 ) + ( 3 ) ( 2 )
1 6
8
1 4
V 4 7
A c c e l e r a t i o n
=
d t 2
=
d t ( d t ) d t
[ 4 t i + ( 2 t - 4 ) j + 3 k ] = 4 t + 2 j + O k .
T h e n t h e c o m p o n e n t o f t h e a c c e l e r a t i o n i n t h e g i v e n d i r e c t i o n i s
( 4 1 + 2 j + O k ) ( i - 3 j + 2 k )
( 4 ) ( 1 ) + ( 2 ) ( - 3 ) + ( 0 ) ( 2 )
- 2
- , 1
v / 1 4
V " I 4 Y / 1 4 7
5 . A c u r v e C i s d e f i n e d b y p a r a m e t r i c e q u a t i o n s x = x ( s ) , y = y ( s ) , z = z ( s ) ,
w h e r e s i s t h e a r e
l e n g t h o f C m e a s u r e d f r o m a f i x e d p o i n t o n C .
I f r i s t h e p o s i t i o n v e c t o r o f a n y p o i n t o n C , s h o w
t h a t d r / d s i s a u n i t v e c t o r t a n g e n t t o C .
T h e v e c t o r
d r
=
d
( x i + y j + z k ) =
d x
i
+ - j +
d z
k
d s
d s
d s
d s d s
z = z ( s ) .
T o s h o w t h a t i t h a s u n i t m a g n i t u d e w e n o t e t h a t
f d s l
=
i s t a n g e n t t o t h e c u r v e x = x ( s ) , y = y ( s ) ,
/ ( d x ) 2
+ ( d z ) 2 = / ( d x ) + ( d y ) 2 + ( d z ) 2
d s
d s
d s
/
( d s ) 2
s i n c e
( d s ) 2 = ( d x ) 2 + ( d y ) 2 + ( d z ) 2 f r o m t h e c a l c u l u s .
1
6 . ( a ) F i n d t h e u n i t t a n g e n t v e c t o r t o a n y p o i n t o n t h e c u r v e x = t 2 + 1 , y = 4 t - 3 , z =
2 1 2 - 6 t .
( b ) D e t e r m i n e t h e u n i t t a n g e n t a t t h e p o i n t w h e r e t = 2
.
( a ) A t a n g e n t v e c t o r t o t h e c u r v e a t a n y p o i n t i s
d t
d t
[ ( t 2 + 1 ) i + ( 4 t - 3 ) j + ( 2 t 2 - 6 t ) k ]
=
2 t i + 4 j + ( 4 t - 6 ) k
T h e m a g n i t u d e o f t h e v e c t o r i s
I d
( 2 t ) 2 + ( 4 ) 2 + ( 4 t - - 6 ) 2 .
2 t i + Q + ( 4 t - 6 ) k
T h e n t h e r e q u i r e d u n i t t a n g e n t v e c t o r i s
T =
( 2 t ) 2 + ( 4 ) 2 + ( 4 t - 6 ) 2
d r
d s
, f ,
=
d r / d t
_
d r
N o t e t h a t s i n c e
d t
d t
'
d s / d t
d s
( b ) A t t = 2 ,
t h e u n i t t a n g e n t v e c t o r i s T =
4 i + 4 j + 2 k
=
2
t +
2 2 j
+
1
1 k .
( 4 ) 2 + ( 4 ) 2 + ( 2 ) 2
3 3
3
7 . I f A a n d B a r e d i f f e r e n t i a b l e f u n c t i o n s o f a s c a l a r u , p r o v e :
( a )
d u ( A B ) = A d B + d u B ,
( b ) d u ( A x B ) = A x d B + d A - x B
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V E C T O R D I F F E R E N T I A T I O N
4 1
( a ) l i m
A u - 0
( A
+ A B ) -
A
A B
+
A A
- B +
A . d B +
A u A u A u
A
A B + A A B + A A A B
A u
A n o t h e r M e t h o d . L e t A = A 1 i + A 2 J + A s k , B = B 1 i + B 2 j + B 3 k .
T h e n
d u ( A B )
=
u ( A 1 B 1 + A 2 B 2 + A 3 B 3 )
( b ) d u ( A x B )
_
( A 1 d B 1 + A 2 d B 2 + A 3 d B 3 ) + ( d A 1 B 1 + d A 2 B 2 + d A 3 B 3 )
=
A . d B + d A , B
d u d u d u
d u
d u d u
d u d u
l i m
( A + A A ) x ( B + A n ) - A x B
A X A B + A A X B + A A x A B
A u
l i m
=
l i m
A x
A B
+
D A
x B +
L A - x A B = A x
d B + d A
x B
A u A u
A u
d u
d u
A n o t h e r M e t h o d .
j
A 2
B 2
d u ( A x B )
=
d u
i
A l
B 1
k
A s
B 3
U s i n g a t h e o r e m o n d i f f e r e n t i a t i o n o f a d e t e r m i n a n t , t h i s b e c o m e s
i
j
k
i
j k
4
A
4 4
d B
d A
A l
A 2
A s
d u
d
3
2
u
d u
=
A x
d u
+ d u x B
d B 1
d B 2
d B 3
B 1
B 2
B 3
d u
d u
d u
8 . I f A = 5 t 2 i + t j - t 3 k a n d B = s i n t i - c o s t j
,
f i n d ( a ) d t ( A - B ) , ( b )
d t
( A x B ) ,
( c )
a
( A A ) .
( a )
A . d B + d A B
=
( 5 t 2 i + t j - t A k )
( c o s t i + s i n t j )
=
5 t 2 c o s t + t s i n t + 1 0 t s i n t - c o s t
A n o t h e r M e t h o d .
A . B = 5 t 2 S i n t - t c o s t .
T h e n
d t
( A . B )
+
( l o t i + j - 3 t 2 k )
( s i n s i - c o s t j )
=
( 5 t 2 - 1 ) c o s t + l i t s i n t
= d t ( 5 t 2 s i n t - t c o s t )
=
5 t 2 c o s t + l O t s i n t + t s i n t - c o s t
=
( 5 t 2 - 1 ) c o s t
+ l i t S i n t
i
j
k i j "
k
( b ) d
t
( A x B ) =
A x
d
B +
d A
X B
=
5 t 2
t
- t 3
+
l o t 1
- 3 t 2
c o s t
s i n t
0
s i n t
- c o s t
0
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V E C T O R D I F F E R E N T I A T I O N
( a ) v
d r
d t =
- w s i n w t i + w c o s c o t j
T h e n [ c o s w t i + s i n c o t j ]
[ - c o s i n w t i + w c o s w t j ]
( c o s w t ) ( - w s i n w t ) + ( s i n w t ) ( w c o s w t ) = 0
a n d r a n d v a r e p e r p e n d i c u l a r .
4 3
2
( b ) W d t
=
- C O 2 c o s w t i - C O 2 s i n w t j
= _ w 2 [ c o s w t i + s i n w t j ]
=
- w 2 r
T h e n t h e a c c e l e r a t i o n i s o p p o s i t e t o t h e d i r e c t i o n o f r ,
i . e . i t i s d i r e c t e d t o w a r d t h e o r i g i n . I t s
m a g n i t u d e i s p r o p o r t i o n a l t o
I r I w h i c h i s t h e d i s t a n c e f r o m t h e o r i g i n .
+ w
c o s w t j ]
c ) r x v =
[ c o s w t i + s i n w t j ] x [ - w s i n w t i
i
j
k
c o s w t
s i n w t
0
- w s i n w t
w c o s w t
0
= c o ( c o s 2 w t + s i n e w t ) k = w k ,
a c o n s t a n t v e c t o r .
P h y s i c a l l y , t h e m o t i o n i s t h a t o f a p a r t i c l e m o v i n g o n t h e c i r c u m f e r e n c e o f a c i r c l e w i t h c o n s t a n t
a n g u l a r s p e e d w . T h e a c c e l e r a t i o n , d i r e c t e d t o w a r d t h e c e n t e r o f t h e c i r c l e , i s t h e c e n t r i p e t a l a c c e l -
e r a t i o n .
2
1 3 . P r o v e :
A x
d B _
d A x B
-
d ( A x d B _
d A x B ) .
d t 2
d t 2
d t
d t
d t
d
L B
-
d A
=
d d B
-
d d A
d t
( A x
d t d t x B ) d t ( A x d t )
d t ( d t x
B )
A x
d 2 B
+ d A x d B _ [ d A x d B +
d 2 A x B ]
d t 2
d t
d t d t
d t
d t 2
1 4 . S h o w t h a t
A A . A
t
A
d t
L e t A = A 1 i + A 2 j + A 3 k .
T h e n A =
A l + A 2 + A 3
d A
_
1 ( A 1
+ A 2 +
A 3 ) ` 1 / 2 ( 2 A 1 d A 1
+
2 A 2 d A 2
+
2 A 3 d ' 4 s )
d t
2
d t
d t
d t
d A 1
d A 2 d A 3 d A
A l
d t
+ A 2 d t
+ A 3 d t
A -
d t
( A l + A 2 + A 3 ) 1 / ' 2
A n o t h e r M e t h o d .
A
S i n c e
A . A = A 2 ,
d t ( A . A ) =
d t
( A 2 ) .
i . e .
A d t = A
d A
.
d
=
A
d A d A d A
d
2
d A
d r
( A
A )
d t
+
d t ' A
=
2 A d t
a n d d t ( A ) = 2 A d t
T h e n 2 A
d A =
2 A
d A
o r
A .
d A
= A
d A
d t d t d t d t
A x 6 4 -
d 2 A x B
d t 2
d t 2
N o t e t h a t i f A i s a c o n s t a n t v e c t o r A .
M
= 0 a s i n P r o b l e m 9 .
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4 4
V E C T O R D I F F E R E N T I A T I O N
1 5 . I f A = ( 2 x 2 y - x 4 ) i + ( e - " Y - y s i n x ) j + ( x 2 c o s y ) k , f i n d :
a A a A a 2 A ' 6 2 A - a a 2 A
' 3 2 A
' 3 x ' a y
'
a '
a y ,
x a y a y a x
a A
=
a x ( 2 x 2 y - x 4 ) i +
a ( e x y
- y s i n x ) j
+
a x
( x 2 c o s y ) k
_
( 4 x y - 4 x 3 ) i + ( y e x y - y c o s x ) j + 2 x c o s y k
( e x y - y s i n x ) j +
a
( x 2 c o s y ) k
2 x 2 y - x 4 ) i
+ a
A
- a y
y
a y
2 x 2 i
+ ( x e x y - s i n x ) j - x 2 s i n y k
2
a A
=
a x
( 4 x y - 4 x 3 ) i + a x ( y e x y - y c o s x ) j +
, a
( 2 x c o s y ) k
a x e
a 2 A
a y 2
a 2 A
a x a y
( 4 y - 1 2 x 2 ) i + ( y 2 e x y + y s i n x ) j + 2 c o s y k
a
( 2 x 2 ) i +
( x e x y - s i n x ) j -
a
( x 2 s i n y ) k
a y
a y
a y
0 + x 2 e x y j - x 2 c o s y k = x 2 e x y j - x 2 c o s y k
a x ( a A )
a x
( 2 x 2 )
i + a x
( x e x y - s i n x ) j - a x ( x 2 s i n y ) k
y
=
4 x i + ( x y e x y + e x y - c o s x ) j - 2 x s i n y k
a 2 A
a a A
a a
3
a y a x
a y a x
a y
a y
4 x i
+ ( x y e x y + e x y - c o s x ) j - 2 x s i n y k
2
2
N o t e t h a t
a x = a - A ,
i . e . t h e o r d e r o f d i f f e r e n t i a t i o n i s i m m a t e r i a l . T h i s i s t r u e i n g e n e r a l i f A
- a y
- a y
h a s c o n t i n u o u s p a r t i a l d e r i v a t i v e s o f t h e s e c o n d o r d e r a t l e a s t .
1 6 . I f c ( x , y , z ) = x y 2 z a n d A = x z i - x y 2 j + y z 2 k , f i n d
a
( 4 ) A ) a t t h e p o i n t ( 2 , - 1 , 1 ) .
a x a z
g b A
=
( x y 2 z ) ( x z i - x y 2 j + y z 2 k )
=
x 2 y 2 z 2 i - x 2 y 4 z j + x y 3 z 3 k
( O A )
=
a ( x 2 y 2 z 2
i - . x 2 y 4 z j + x y 3 z 3 k )
=
2 x 2 y 2 z i - x 2 y 4 j
+ 3 x y 3 z 2 k
z
a 2
2 z i - ) , e , 4 j + 3
j + 3 x
3 z 2 k )
= 4 x
3 z 2 k
z i - x 2
( 2 x 2
y
y
y
y
x a z
a x
3
a
a
a z
( O A )
= ( 4 x y 2 z i - 2 x y 4 j + 3 y 3 z 2 k )
=
4 y 2 z i - 2 y 4 j
x
I f x = 2 , y = - 1 , z = 1 t h i s b e c o m e s 4 ( - 1 ) 2 ( 1 ) i - 2 ( - 1 ) 4 j
= 4 i - 2 j .
a
) i
+ - ( y e x y - y c o s x ) j
+
( 4 x
- 4 x
( 2 x c o s y ) k
1 7 . L e t F d e p e n d o n x , y , z , t
w h e r e x , y a n d z d e p e n d o n t .
P r o v e t h a t
d F
a F a F d x
a F d
' 3 F d z
_
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V E C T O R D I F F E R E N T I A T I O N
u n d e r s u i t a b l e a s s u m p t i o n s o f d i f f e r e n t i a b i l i t y .
S u p p o s e t h a t
F = F 1 ( x , y , z , t ) i
+ F 2 ( x , y , z , t ) j + F 3 ( x , y , z , t ) k .
d F
=
d F 1 i
+ d F 2 j + d F 3 k
[ a t d t +
a z d x + a F l d y
y
+
{ f a d t
+
a z 3 d x
+
a F 1
- 6 a F 2
' a a F 3
( 1
a t
a t a t
+
j +
k ) d t
+
(
1
+ - J +
k ) a y
+
(
I
+
T h e n
+ a z d z ] i
+
t 2 d t + a x e d x + a F d y +
a z 2
d z ]
a F 3
d y +
a F 3
d z ] k
y
+
( a F 1
i +
- a a F 2
j
+ ' 3 a F 9
k ) d x
a x
a x
a x
- 3 F ,
.
' 3 F 2
,
- 3 a F s - 6 F ,
- 6 a F 2
+
a F 3
k ) d z
j
a z
y
a y
a y
a Z
a z
a F d t
+ a F d x + a F d y +
3 d z
y
a F
a F d x
a F d y
a F d z
F
+
+
.
=
n d s o
d t
a t
a x d t a y d t
- a Z d t
D I F F E R E N T I A L G E O M E T R Y .
1 8 . P r o v e t h e F r e n e t - S e r r e t f o r m u l a s ( a )
T = K N , ( b )
d B
=
T N ,
c d N
= T B - K T .
4 5
( a ) S i n c e T . T = 1 , i t f o l l o w s f r o m P r o b l e m 9 t h a t T . f - 4
= 0 ,
i . e . d s i s p e r p e n d i c u l a r t o T .
I f N i s a u n i t v e c t o r i n t h e d i r e c t i o n d T , t h e n d s = K N . W e c a l l N t h e p r i n c i p a l n o r m a l , K t h e
c u r v a t u r e a n d p = 1 / K t h e r a d i u s o f c u r v a t u r e .
( b ) L e t B = T x N , s o t h a t
d B
=
T x d N +
d T x N = T x d N + K N x N =
T x d N
T h e n T . d B = T . T x
d N
= 0 , s o t h a t T i s p e r p e n d i c u l a r t o d B
B u t f r o m B B = 1
i t f o l l o w s t h a t B d s B = 0 ( P r o b l e m 9 ) , s o t h a t d B i s p e r p e n d i c u l a r t o B a n d
i s t h u s i n t h e p l a n e o f T a n d N .
S i n c e d B i s i n t h e p l a n e o f T a n d N a n d i s p e r p e n d i c u l a r t o T , i t m u s t b e p a r a l l e l t o N ; t h e n d B =
- - T N .
W e c a l l B t h e b i n o r m a l , ' r t h e t o r s i o n , a n d o - = 1 / T t h e r a d i u s o f t o r s i o n .
( c ) S i n c e T , N , B f o r m a r i g h t - h a n d e d s y s t e m , s o d o N , B a n d T , i . e . N = B x T .
T h e n
d N
=
B x d T + d B X T
= B X K N - - T N X T = - K T + T B = T B - - K T .
1 9 . S k e t c h t h e s p a c e c u r v e x = 3 c o s t , y = 3 s i n t ,
z = 4 t a n d f i n d
( a ) t h e u n i t t a n g e n t T , ( b ) t h e p r i n c i p a l n o r m a l N , c u r v a t u r e K
a n d r a d i u s o f c u r v a t u r e p , ( c ) t h e b i n o r m a l B , t o r s i o n r a n d
r a d i u s o f t o r s i o n c r
.
T h e s p a c e c u r v e i s a c i r c u l a r h e l i x ( s e e a d j a c e n t f i g u r e ) . S i n c e
t = z / 4 , t h e c u r v e h a s e q u a t i o n s x = 3 c o s ( z / 4 ) , y = 3 s i n ( z / 4 ) a n d
t h e r e f o r e l i e s o n t h e c y l i n d e r x 2 + y 2 = 9 .
( a ) T h e p o s i t i o n v e c t o r f o r a n y p o i n t o n t h e c u r v e i s
Y
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4 6
- 3 s i n t i
+ 3 c o s t j + 4 k
r
=
3 c o s t i
+ 3 s i n t j
+ 4 t k
T h e n
( b ) T -
V E C T O R D I F F E R E N T I A T I O N
d r d r
=
( . _ 3 s i n t ) 2 +
( 3 c o s t ) 2 + 4 2
=
5
a t ' d t
I s
I
d r
I
t
I t
=
d r
_
I t
T h u s
T
_ d r
_ d r / d t
d s
d s / d t
-
5
s i n t i
+ 5 c o s t j
+
5
k .
d
t
( - 5 s i n t i +
5
c o s t j
+ 5 k )
I T
d T / d t
3
C o s t i
- -
3 s i n
t j
I s
-
d s / d t
2 5
2 5
c o s t
-
5
s i n t j
S i n c e T = / < N ,
I a T I
=
1 ) < 1 I N I
= K
a s K ? 0 .
T h e n
K =
I T I
2 5 c o s t ) 2
+ ( - 2 5 s i n t ) 2
=
2 5
a n d
p =
1
= 3 5
F r o m d T = K N , w e o b t a i n N =
K
I T
= - c o s t i - s i n t j
i j
k
( c ) B
= T x N
=
5 s i n t
5
3
4
c o s t
5
- C o s t
- s i n t
0
c o s t i
+
5
s i n t j ,
d B
-
4
5 s i n t i -
5
c o s t j +
5
k
d B
_
d B / d t
_ 4
c o s t i
+ 4 s i n s
I s
d s / d t
2 5
2 5
- T N =
- T ( - c o s t i - s i n t j )
_
2 5
c o s t i
+
2 5 s i n t j
o r
T = 2 5
a n d
o - = T =
2 5
2 0 . P r o v e t h a t t h e r a d i u s o f c u r v a t u r e o f t h e c u r v e w i t h p a r a m e t r i c e q u a t i o n s x = x ( s ) , y = y ( s ) , z = z ( s )
i s g i v e n b y p
=
[ ( d 2 2 ) 2 + ( d 2 2 Y
) 2 +
( d 2 2 ) 2 1 - 1 / 2
d s
d s
d s
T h e p o s i t i o n v e c t o r o f a n y p o i n t o n t h e c u r v e i s r = x ( s ) i + y ( s ) j + z ( s ) k .
T h e n
T =
d r
=
d x
i + d y j +
d z k a n d
I T
= d 2 x
i +
d 2 y
+
d 2 z
k
I s
I s
I s I s
I s
d s 2 d s 2
d s 2
B u t
I T
= K N s o t h a t K =
I
I T
s
2
3
2 1 . S h o w t h a t
Y s - a s 2 x a s 3
7 -
P
2
2 2
2
) 2
d s 2
+
( d s 2
) 2
+ ( d s 2
a n d t h e r e s u l t f o l l o w s s i n c e p = K
r
N
d s -
T '
d s 2 =
I T
= K N ,
d s 3 - K
d s v +
d s
N = K ( T B - K T ) +
d s
N = K T B - K 2 T +
I s
2
3
d r
d r x d r
= T K N x ( K T B - K 2 T +
d K N )
d s ' s 2 d s 3
d s
=
K 3 N x T + K d s N x N ) = T . ( K 2 T T + K 3 B ) = K 2 T
T
P 2
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V E C T O R D I F F E R E N T I A T I O N
T h e r e s u l t c a n b e w r i t t e n
T =
[ ( x , t ) 2 + ( y r , ) 2 + ( z , r ) 2 1 - . 1
Y
z
x
I I
y
i t
z
r r
/ ,
I f f f i l l
I x
y
z
w h e r e p r i m e s d e n o t e d e r i v a t i v e s w i t h r e s p e c t t o s , b y u s i n g t h e r e s u l t o f P r o b l e m 2 0 .
2 2 . G i v e n t h e s p a c e c u r v e x = t , y = t 2 ,
z = 3 t 3 ,
f i n d ( a ) t h e c u r v a t u r e K , ( b ) t h e t o r s i o n r .
( a )
T h e p o s i t i o n v e c t o r i s
r =
t j + t 2 j +
3
t 3 k
T h e n
a n d
d o =
i
+ 2 t j
+ 2 t 2 k
d s _
d r
=
d r d _ r
2
2
2 2
2
d t
d t I
d t - d t
( 1 )
+ ( 2 t )
+ ( 2 t
) =
1 + 2 t
T =
d r
=
d r / d t =
i
+ 2 t j + 2 t 2 k
d s
d s / d t
1 + 2 t 2
4 7
d T
( 1 t 2 t 2 ) ( 2 j + 4 t k ) - ( i + 2 t j + 2 t 2 k ) ( 4 t )
- 4 t i + ( 2 - 4 t 2 ) j + 4 t k
d t
( 1 + 2 t 2 ) 2
( 1 + 2 t 2 ) 2
T h e n
d T
=
d T / d t
4 t i
+ ( 2 - 4 t 2 ) j + 4 t k
d s
d s / d t ( 1 + 2 t 2 ) 3
S i n c e d T
= K N K =
I d T I =
( - . 4 t ) 2 + ( 2 - 4 t 2 ) 2 + ( 4 t ) 2
=
2
a s
a s ( 1 + 2 t 2 ) 3
2 2
( 1 + 2 t )
( b )
)
(
=
1 d T = - 2 t i + ( 1 - - - 2 t 2 ) j + 2 t k
N
o m
_
K d s 1
+ 2 t 2
i j
k
1
2 t 2 t 2
2 t 2 i - 2 t j + k
T h e n B = T x N =
1 + 2 t 2
1 + 2 t 2 1 +
2 t 2
1 + 2 t 2
- 2 t
i - 2 t 2
2
t
1 1 + 2 t 2 1 + 2 t 2
N o w
d B
=
4 t i + ( 4 t 2 - 2 ) j - 4 t k
d t
( 1 + 2 t 2 ) 2
a n d
d B
d s
A l s o , - ' T N =
- - r [
- 2 t i
+ ( 1 - 2 t 2 ) j
+ 2 t k
1 + 2 t 2
N o t e t h a t K = T f o r t h i s c u r v e .
1 + 2 t 2
d B / d t
4 t i
+ ( 4 t 2 - 2 ) j - 4 t k
d s / d t
( 1 + 2 1 2 ) 3
S i n c e
d
= - T N ,
w e f i n d
'
2
r =
( 1 + 2 t 2
2 3 . F i n d e q u a t i o n s i n v e c t o r a n d r e c t a n g u l a r f o r m f o r t h e ( a ) t a n g e n t , ( b ) p r i n c i p a l n o r m a l , a n d ( c )
b i n o r m a l t o t h e c u r v e o f P r o b l e m 2 2 a t t h e p o i n t w h e r e t = 1 .
L e t T o , N o a n d B 0 d e n o t e t h e t a n g e n t , p r i n c i p a l n o r m a l a n d b i n o r m a l v e c t o r s a t t h e r e q u i r e d p o i n t .
T h e n f r o m P r o b l e m 2 2 ,
i
+ 2 j + 2 k
- 2 i - j + 2 k
2 i - 2 i + k
T o =
3
,
N o =
3 ,
B o = 3
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4 8
V E C T O R D I F F E R E N T I A T I O N
I f A d e n o t e s a g i v e n v e c t o r w h i l e r o a n d r d e n o t e r e s p e c t i v e l y t h e p o s i t i o n v e c t o r s o f t h e i n i t i a l p o i n t
a n d a n a r b i t r a r y p o i n t o f A , t h e n r - r 0 i s p a r a l l e l t o A a n d s o t h e e q u a t i o n o f A i s ( r - r 0 ) x A = 0 .
E q u a t i o n o f t a n g e n t i s
( r - r o ) x T o = 0
T h e n :
E q u a t i o n o f p r i n c i p a l n o r m a l i s
( r - - r o ) x N o = 0
E q u a t i o n o f b i n o r m a l i s
( r - r o ) x B o = 0
I n r e c t a n g u l a r f o r m , w i t h r = x i
+ y j + z k ,
r o = i
+ j + 3 k
t h e s e b e c o m e r e s p e c t i v e l y
x - 1
y - 1
z - 2 / 3 x - 1
_
y - 1
z - 2 / 3
x - 1
- y - 1
z - 2 / 3
1
2 2
- 2
- 1
2
2 - 2
1
T h e s e e q u a t i o n s c a n a l s o b e w r i t t e n i n p a r a m e t r i c f o r m ( s e e P r o b l e m 2 8 , C h a p t e r 1 ) .
2 4 . F i n d e q u a t i o n s i n v e c t o r a n d r e c t a n g u l a r f o r m f o r t h e ( a ) o s c u l a t i n g p l a n e , ( b ) n o r m a l p l a n e , a n d
( c ) r e c t i f y i n g p l a n e t o t h e c u r v e o f P r o b l e m s 2 2 a n d 2 3 a t t h e p o i n t w h e r e t = 1 .
( a ) T h e o s c u l a t i n g p l a n e i s t h e p l a n e w h i c h c o n t a i n s t h e t a n g e n t a n d p r i n c i p a l n o r m a l .
I f r i s t h e p o s i t i o n
v e c t o r o f a n y p o i n t i n t h i s p l a n e a n d r o i s t h e p o s i t i o n v e c t o r o f t h e p o i n t t = 1 , t h e n r - r o i s p e r p e n d i c -
u l a r t o B o , t h e b i n o r m a l a t t h e p o i n t t = 1 , i . e . ( r - r o ) B o = 0 .
( b ) T h e n o r m a l p l a n e i s t h e p l a n e w h i c h i s p e r p e n d i c u l a r t o t h e t a n g e n t v e c t o r a t t h e g i v e n p o i n t . T h e n
t h e r e q u i r e d e q u a t i o n i s ( r - r o ) T o = 0 .
( c ) T h e r e c t i f y i n g p l a n e i s t h e p l a n e w h i c h i s p e r p e n d i c u -
l a r t o t h e p r i n c i p a l n o r m a l a t t h e g i v e n p o i n t .
T h e
r e q u i r e d e q u a t i o n i s ( r - r o ) N o = 0 .
I n r e c t a n g u l a r f o r m t h e e q u a t i o n s o f ( a ) , ( b ) a n d ( c )
b e c o m e r e s p e c t i v e l y ,
2 ( x - 1 ) - - 2 ( y - 1 ) + 1 ( z - 2 / 3 ) =
0 ,
1 ( x - 1 ) + 2 ( y - 1 ) + 2 ( z - - 2 / 3 ) = 0 ,
- 2 ( x - 1 ) - 1 ( y - 1 ) + 2 ( z - 2 / 3 ) =
0 .
T h e a d j o i n i n g f i g u r e s h o w s t h e o s c u l a t i n g , n o r m a l
a n d r e c t i f y i n g p l a n e s t o a c u r v e C a t t h e p o i n t P .
2 5 . ( a ) S h o w t h a t t h e e q u a t i o n r = r ( u , v ) r e p r e s e n t s a s u r f a c e .
r e p r e s e n t s a v e c t o r n o r m a l t o t h e s u r f a c e .b ) S h o w t h a t a u
x T V
( c ) D e t e r m i n e a u n i t n o r m a l t o t h e f o l l o w i n g s u r f a c e , w h e r e a > 0 :
r
= a c o s u s i n v i + a s i n u s i n v j + a c o s v k
( a )
I f w e c o n s i d e r u t o h a v e a f i x e d v a l u e ,
s a y u o ,
t h e n r = r ( u o , v )
r e p r e s e n t s a
c u r v e w h i c h c a n b e d e n o t e d b y u = u o .
S i m i l a r l y
u = u 1
d e f i n e s a n o t h e r c u r v e
r = r ( u 1 , v ) . A s u v a r i e s , t h e r e f o r e , r =
r ( u , v ) r e p r e s e n t s a c u r v e w h i c h m o v e s i n
s p a c e a n d g e n e r a t e s a s u r f a c e S .
T h e n
r = r ( u , v ) r e p r e s e n t s t h e s u r f a c e S t h u s
g e n e r a t e d , a s s h o w n i n t h e a d j o i n i n g f i g -
u r e .
T h e c u r v e s u = u o , u = u
1 ,
. . .
r e p r e s e n t d e f i n i t e c u r v e s o n t h e s u r f a c e . S i m i l a r l y v = v o , v = v 1 ,
r e p r e s e n t c u r v e s o n t h e s u r f a c e .
B y a s s i g n i n g d e f i n i t e v a l u e s t o u a n d v , w e o b t a i n a p o i n t o n t h e s u r f a c e . T h u s c u r v e s u = u o a n d
v = v o , f o r e x a m p l e , i n t e r s e c t a n d d e f i n e t h e p o i n t ( u o , v o ) o n t h e s u r f a c e . W e s p e a k o f t h e p a i r o f n u m -
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V E C T O R D I F F E R E N T I A T I O N
4 9
b e r s ( u , v ) a s d e f i n i n g t h e c u r v i l i n e a r c o o r d i n a t e s o n t h e s u r f a c e . I f a l l t h e c u r v e s u = c o n s t a n t a n d
v = c o n s t a n t a r e p e r p e n d i c u l a r a t e a c h p o i n t o f i n t e r s e c t i o n , w e c a l l t h e c u r v i l i n e a r c o o r d i n a t e s y s t e m
o r t h o g o n a l . F o r f u r t h e r d i s c u s s i o n o f c u r v i l i n e a r c o o r d i n a t e s s e e C h a p t e r 7 .
( b ) C o n s i d e r p o i n t P h a v i n g c o o r d i n a t e s ( u o , v o )
o n a s u r f a c e S , a s s h o w n i n t h e a d j a c e n t d i a -
g r a m . T h e v e c t o r a r / a u a t P i s o b t a i n e d b y
d i f f e r e n t i a t i n g r w i t h r e s p e c t t o u , k e e p i n g
v = c o n s t a n t = v o . F r o m t h e t h e o r y o f s p a c e
c u r v e s , i t f o l l o w s t h a t a r / a u a t P r e p r e -
s e n t s a v e c t o r t a n g e n t t o t h e c u r v e v = v o a t
P , a s s h o w n i n t h e a d j o i n i n g f i g u r e . S i m i l a r -
l y , a r / a v a t P r e p r e s e n t s a v e c t o r t a n g e n t
t o t h e c u r v e u = c o n s t a n t = u o . S i n c e a r / a u
a n d a r / a v r e p r e s e n t v e c t o r s a t P t a n g e n t
t o c u r v e s w h i c h l i e o n t h e s u r f a c e S a t P , i t
f o l l o w s t h a t t h e s e v e c t o r s a r e t a n g e n t t o t h e
s u r f a c e a t P . H e n c e i t f o l l o w s t h a t
a r
a r
a u
x
a v
i s a v e c t o r n o r m a l t o S a t P .
( c )
a r
= - a s i n u s i n v i
+ a c o s u s i n v j
a u
a r
= a c o s u c o s v i
+ a s i n u c o s v j - a s i n v k
a v
T h e n
a r
x
a r _
a u
a v
i
j
k
- a s i n u s i n v a c o s u s i n v
0
a c o s u c o s v
a s i n u C o s v
- a s i n v
- a
2
C o s u s i n e v i - a 2 s i n u s i n 2 v j - a 2 s i n v c o s v k
r e p r e s e n t s a v e c t o r n o r m a l t o t h e s u r f a c e a t a n y p o i n t ( u , v ) .
A u n i t n o r m a l i s o b t a i n e d b y d i v i d i n g
a u x a v
b y i t s m a g n i t u d e ,
a u x a v I
, g i v e n
b y
a 4 C o s 2 u S i n 4 v + a 4 s i n 2
u s i n 4 v
+ a 4
s i n 2 v
C O S 2
v
=
V a 4 ( c o s 2 u + s i n 2 u ) s i n 4 v
+ a 4 s i n 2 v C O S 2 v
a 4 s i n 2 v ( s i n 2 v + c o s t v )
a 2 s i n v
i f
s i n v > 0
- a 2 s i n v
i f
s i n v < 0
T h e n t h e r e a r e t w o u n i t n o r m a l s g i v e n b y
± ( c o s u s i n v i + s i n u s i n v j
+ c o s v k )
n
I t s h o u l d b e n o t e d t h a t t h e g i v e n s u r f a c e i s d e f i n e d b y x = a c o s u s i n v , y = a s i n u s i n v , z = a
c o s v
f r o m w h i c h i t i s s e e n t h a t x 2 + y 2 + z 2 = 2 , w h i c h i s a s p h e r e o f r a d i u s a . S i n c e r = a n ,
i t f o l l o w s t h a t
n =
c o s u s i n v i
+ s i n u s i n v j
+ c o s v k
i s t h e o u t w a r d d r a w n u n i t n o r m a l t o t h e s p h e r e a t t h e p o i n t ( u , v ) .
2 6 . F i n d a n e q u a t i o n f o r t h e t a n g e n t p l a n e t o t h e s u r f a c e z = x 2 + y 2
a t t h e p o i n t ( 1 , - 1 , 2 ) .
L e t x = u , y = v , z = u 2 + V
2
b e p a r a m e t r i c e q u a t i o n s o f t h e s u r f a c e . T h e p o s i t i o n v e c t o r t o a n y p o i n t
o n t h e s u r f a c e i s
r
= u i
+ v i + ( u 2 + v 2 ) k
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5 0
V E C T O R D I F F E R E N T I A T I O N
T h e n a u = i + 2 u k =
i + 2 k ,
a v =
i + 2 v k = j - 2 k a t t h e p o i n t ( 1 , - 1 , 2 ) , w h e r e u = 1 a n d v = - 1 .
B y P r o b l e m 2 5 , a n o r m a l n t o t h e s u r f a c e a t t h i s p o i n t i s
' a r - 6 r
n
a u
x
a v
=
( i + 2 k ) x ( j
2 k )
2 i + 2 i + k
T h e p o s i t i o n v e c t o r t o p o i n t ( 1 , - 1 , 2 ) i s R o = i - j + 2 k .
T h e p o s i t i o n v e c t o r t o a n y p o i n t o n t h e p l a n e i s
R = x i + y j + z k
T h e n f r o m t h e a d j o i n i n g f i g u r e , R - R o i s p e r p e n d i c u l a r t o
n a n d t h e r e q u i r e d e q u a t i o n o f t h e p l a n e i s ( R - R o ) n = 0
o r [ ( x i + y j + z k ) - ( i - j + 2 k ) ]
[ - 2 i + 2 j + k ]
=
0
i . e . - - - 2 ( x - 1 ) + 2 ( y + l ) + ( z - 2 ) = 0
o r
2 x - 2 y - z = 2 .
M E C H A N I C S
Y
2 7 . S h o w t h a t t h e a c c e l e r a t i o n a o f a p a r t i c l e w h i c h t r a v e l s a l o n g a s p a c e c u r v e w i t h v e l o c i t y v i s
g i v e n b y
a
=
d V
T +
v 2 N
w h e r e T i s t h e u n i t t a n g e n t v e c t o r t o t h e s p a c e c u r v e , N i s i t s u n i t p r i n c i p a l n o r m a l , a n d p i s t h e
r a d i u s o f c u r v a t u r e .
V e l o c i t y v = m a g n i t u d e o f v m u l t i p l i e d b y u n i t t a n g e n t v e c t o r T
o r v
= v T
D i f f e r e n t i a t i n g ,
B u t b y P r o b l e m 1 8 ( a ) ,
T h e n
v ( ) _
d o
T +
P 2 N
T h i s s h o w s t h a t t h e c o m p o n e n t o f t h e a c c e l e r a t i o n i s d v / d t i n a d i r e c t i o n t a n g e n t t o t h e p a t h a n d v 2 / p i n
a d i r e c t i o n o f t h e p r i n c i p a l n o r m a l t o t h e p a t h . T h e l a t t e r a c c e l e r a t i o n i s o f t e n c a l l e d t h e c e n t r i p e t a l a c c e l -
e r a t i o n . F o r a s p e c i a l c a s e o f t h i s p r o b l e m s e e P r o b l e m 1 2 .
2 8 . I f r i s t h e p o s i t i o n v e c t o r o f a p a r t i c l e o f m a s s m r e l a t i v e t o p o i n t 0 a n d F i s t h e e x t e r n a l f o r c e
o n t h e p a r t i c l e , t h e n r x F = M i s t h e t o r q u e o r m o m e n t o f F a b o u t 0 . S h o w t h a t M = d H / d t , w h e r e
H = r x m y a n d v i s t h e v e l o c i t y o f t h e p a r t i c l e .
M
= r x F
r x d t ( m v )
b y N e w t o n ' s l a w .
B u t
d t ( r x m v )
= r x d t ( m v ) +
d r
x m y
a
=
d v
d ( v T )
=
d v
T + v
d T
d t
d t d t
d t
d T
d T d s
d s =
K v N
=
v N
K N
d t = d s
d t
=
d t
p
r x
d t
( m v )
+ v x m y
=
r x d t ( m v )
+ 0
i . e .
M =
d t ( r x m v )
=
d f l
N o t e t h a t t h e r e s u l t h o l d s w h e t h e r i n i s c o n s t a n t o r n o t . H i s c a l l e d t h e a n g u l a r m o m e n t u m . T h e r e s u l t
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V E C T O R D I F F E R E N T I A T I O N
5 1
s t a t e s t h a t t h e t o r q u e i s e q u a l t o t h e t i m e r a t e o f c h a n g e o f a n g u l a r m o m e n t u m .
T h i s r e s u l t i s e a s i l y e x t e n d e d t o a s y s t e m o f n p a r t i c l e s h a v i n g r e s p e c t i v e m a s s e s m 1 , m 2 , . . . . M n
n
a n d p o s i t i o n v e c t o r s r 1 , r 2 , . . .
, r n w i t h e x t e r n a l f o r c e s F 1 , F 2 , . . .
, F n . F o r t h i s c a s e , H = I m k r k x v k
n d H k = 1
a s b e f o r e .
s t h e t o t a l a n g u l a r m o m e n t u m , M = k l r k x F k i s t h e t o t a l t o r q u e , a n d t h e r e s u l t i s M =
a t
2 9 . A n o b s e r v e r s t a t i o n e d a t a p o i n t w h i c h i s f i x e d r e l -
a t i v e t o a n x y z c o o r d i n a t e s y s t e m w i t h o r i g i n 0 , a s
s h o w n i n t h e a d j o i n i n g d i a g r a m , o b s e r v e s a v e c t o r
A = A 1 i + A 2 j + A 3 k a n d c a l c u l a t e s i t s t i m e d e -
r i v a t i v e t o b e
L A 1
i + d t 2 j + d t 3 k .
L a t e r , h e
t
f i n d s o u t t h a t h e a n d h i s c o o r d i n a t e s y s t e m a r e a c -
t u a l l y r o t a t i n g w i t h r e s p e c t t o a n X Y Z c o o r d i n a t e
s y s t e m t a k e n a s f i x e d i n s p a c e a n d h a v i n g o r i g i n
a l s o a t 0 . H e a s k s , ` W h a t w o u l d b e t h e t i m e d e -
r i v a t i v e o f A f o r a n o b s e r v e r w h o i s f i x e d r e l a t i v e
t o t h e X Y Z c o o r d i n a t e s y s t e m ? '
A
l i n
d e n o t e r e s p e c t i v e l y t h e t i m e d e r i v a t i v e s o f A w i t h r e s p e c t t o t h e f i x e d
a ) I f
d A I f
a n d d
a n d m o v i n g s y s t e m s , s h o w t h a t t h e r e e x i s t s a v e c t o r q u a n t i t y c o s u c h t h a t
d A
d A
+
r v x A
d t
( b ) L e t D f a n d D R b e s y m b o l i c t i m e d e r i v a t i v e o p e r a t o r s i n t h e f i x e d a n d m o v i n g s y s t e m s r e -
s p e c t i v e l y . D e m o n s t r a t e t h e o p e r a t o r e q u i v a l e n c e
D f =
D R + C o x
( a ) T o t h e f i x e d o b s e r v e r t h e u n i t v e c t o r s i , j , k a c t u a l l y c h a n g e w i t h t i m e . H e n c e s u c h a n o b s e r v e r w o u l d
c o m p u t e t h e t i m e d e r i v a t i v e o f A a s
d A d A 1
( 1 )
1
+
=
d A 2
j
+
d A 3
d i
k
+
A l
+
d j
A 2 +
d k
A s
i . e .
a t
a t
d
d t
d t
d t d t
( 2 )
d t A I
-
-
A I
+
A l
d i
+ A d + A d k
d t
f
d t d t
2
d t
d t
S i n c e i i s a u n i t v e c t o r , d i / d t i s p e r p e n d i c u l a r t o i ( s e e P r o b l e m 9 ) a n d m u s t t h e r e f o r e l i e i n t h e
p l a n e o f j a n d k . T h e n
( 3 )
d i
-
j + a
k
S i m i l a r l y , ( 4 )
d t
d j
1 2
a
k + a i
( 5 )
d t
d k
s 4
i
+ a
j
d t
6
5
F r o m i . j = 0 , d i f f e r e n t i a t i o n y i e l d s
i I .
d i
+
d i
. i = 0 .
B u t i .
d
= a 4 f r o m ( 4 ) ,
a n d
d t d t
d t
f r o m ( 3 ) ; t h e n
a 4 = - - a 1 .
d i
d t
j = a 1
S i m i l a r l y f r o m i k = 0 ,
i . d t + d t k = 0 a n d a 5 = - a 2 ;
f r o m j k = 0 ,
j
d k + d i
k = 0
a n d
a s = - a 3
T h e n d t
= a 1 j + a 2 k ,
d J :
= a 3 k - a 1 i ,
d k
= - a 2 i -
a s j
a n d
d d
a t
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5 2
V E C T O R D I F F E R E N T I A T I O N
A l d t
+ A 2 d t
+
A i d -
_ ( - a 1 A 2 - a 2 A 3 ) i + ( a 1 A 1 - a 3 A 3 ) j + ( a 2 A 1 + a s A 2 ) k
w h i c h c a n b e w r i t t e n a s
a s
i j
k
- a 2
a 1
I A l
A 2
A s {
T h e n i f w e c h o o s e a s = c v 1 , - a 2 = c v 2 , a 1 = c o s t h e d e t e r m i n a n t b e c o m e s
i j
k
6 0 1 .
C D - 2
6 0 3
= w x A
A l
A 2 A s
w h e r e w = w 1 i + c v 2 j + c v 2 k . T h e q u a n t i t y c o i s t h e a n g u l a r v e l o c i t y v e c t o r o f t h e m o v i n g s y s t e m
w i t h r e s p e c t t o t h e f i x e d s y s t e m .
( b ) B y d e f i n i t i o n
E l f A
d A ( (
= d e r i v a t i v e i n f i x e d s y s t e m
d t I f
D A = A I
= d e r i v a t i v e i n m o v i n g s y s t e m .
i n
d t
I x
F r o m ( a ) ,
D f A = D m A + w x A =
( D . , + c v x ) A
a n d s h o w s t h e e q u i v a l e n c e o f t h e o p e r a t o r s D
f =
D i n + c o x .
3 0 . D e t e r m i n e t h e ( a ) v e l o c i t y a n d ( b ) a c c e l e r a t i o n o f a m o v i n g p a r t i c l e a s s e e n b y t h e t w o o b s e r v -
e r s i n P r o b l e m 2 9 .
( a )
L e t v e c t o r A i n P r o b l e m 2 9 b e t h e p o s i t i o n v e c t o r r o f t h e p a r t i c l e . U s i n g t h e o p e r a t o r n o t a t i o n o f
P r o b l e m 2 9 ( b ) , w e h a v e
( 1 )
D
f
r
=
( D n + w x ) r
=
D . r + w x r
B u t
D
f
r =
v p l f
D n r
=
v p , n
v e l o c i t y o f p a r t i c l e r e l a t i v e t o f i x e d s y s t e m
v e l o c i t y o f p a r t i c l e r e l a t i v e t o m o v i n g s y s t e m
w x r =
v a f f =
v e l o c i t y o f m o v i n g s y s t e m r e l a t i v e t o f i x e d s y s t e m .
T h e n ( 1 ) c a n b e w r i t t e n a s
( 2 )
v p i f =
V p l s + w x r
o r i n t h e s u g g e s t i v e n o t a t i o n
( 3 )
v
v
+
I f
p l n m i f
N o t e t h a t t h e r o l e s o f f i x e d a n d m o v i n g o b s e r v e r s c a n , o f c o u r s e , b e i n t e r c h a n g e d . T h u s t h e f i x e d
o b s e r v e r c a n t h i n k o f h i m s e l f a s r e a l l y m o v i n g w i t h r e s p e c t t o t h e o t h e r . F o r t h i s c a s e w e m u s t i n t e r -
c h a n g e s u b s c r i p t s m a n d f a n d a l s o c h a n g e w t o - w s i n c e t h e r e l a t i v e r o t a t i o n i s r e v e r s e d . I f t h i s i s
d o n e , ( 2 ) b e c o m e s
v
v - w x r
o r
v
V + w x r
0 1 0 1 f
P i f
2
I n
s o t h a t t h e r e s u l t i s v a l i d f o r e a c h o b s e r v e r .
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V E C T O R D I F F E R E N T I A T I O N
5 3
( b )
T h e a c c e l e r a t i o n o f t h e p a r t i c l e a s d e t e r m i n e d b y t h e f i x e d o b s e r v e r a t 0 i s D f r = D f ( D f r ) . T a k e D f
o f b o t h s i d e s o f ( 1 ) , u s i n g t h e o p e r a t o r e q u i v a l e n c e e s t a b l i s h e d i n P r o b l e m 2 9 ( b ) .
T h e n
D f r
D . 2 r
=
D 2 r + D m ( w x r ) + w x D , n r + w x ( w x r )
D t ( D m r + c o x r )
( D I M + w x ) ( D m r + w x r )
D x ( D m r + c v x r ) + c o x ( D , n r + w x r )
a c c e l e r a t i o n o f p a r t i c l e r e l a t i v e t o f i x e d s y s t e m
a c c e l e r a t i o n o f p a r t i c l e r e l a t i v e t o m o v i n g s y s t e m .
2 w x D . r + ( D , n w ) x r + w x ( w x r )
a c c e l e r a t i o n o f m o v i n g s y s t e m r e l a t i v e t o f i x e d s y s t e m
S U P P L E M E N T A R Y P R O B L E M S
3 1 .
I f R = e - t i + I n ( t 2 + 1 ) j - t a n t k ,
f i n d ( a )
d R
, ( b )
d 2 R
,
( c )
I d R
( d )
I d 2 R I
a t t = o .
d t
d t d t
d t 2
A n s .
( a ) - i - k ,
( b ) i + 2 j ,
( c ) V ,
( d ) v 5
3 2 . F i n d t h e v e l o c i t y a n d a c c e l e r a t i o n o f a p a r t i c l e w h i c h m o v e s a l o n g t h e c u r v e x = 2 s i n 3 t
, y = 2 c o s 3 t ,
z = 8 t a t a n y t i m e t > 0 .
F i n d t h e m a g n i t u d e o f t h e v e l o c i t y a n d a c c e l e r a t i o n .
A n s . v = 6 c o s 3 t i - 6 s i n 3 t j + 8 k , a = - 1 8 s i n 3 t i - 1 8 c o s 3 t j , I v 1 0 , I a l = 1 8
3 3 . F i n d a u n i t t a n g e n t v e c t o r t o a n y p o i n t o n t h e c u r v e x = a c o s c v t , y = a s i n w t
,
z = b t w h e r e a , b , c o a r e
- a c v s i n c o t i
+ a w c o s C v t j
+ b k
c o n s t a n t s .
A n s .
Y a 2 c o + b 2
3 4 .
I f A = t 2 i - t j + ( 2 t + 1 ) k a n d B = ( 2 t - 3 ) i +
t k , f i n d
( a )
( A - B ) ,
( b ) d ( A x B ) , ( c )
A + B I ,
( d )
( A x d B )
a t t = 1 .
A n s . ( a ) - 6 , ( b ) 7 j + 3 k , ( c ) 1 ,
d d d
o r
D f ( D f r )
D
f r
D 2 r
+ 2 w x D . r + ( D , n w ) x r + w x ( w x r )
L e t
a p l f =
a p i , n =
T h e n a , n l f =
a n d w e c a n w r i t e
a p t f =
a , , , + a 1 2 I f
.
F o r m a n y c a s e s o f i m p o r t a n c e w i s a c o n s t a n t v e c t o r , i . e . t h e r o t a t i o n p r o c e e d s w i t h c o n s t a n t a n -
g u l a r v e l o c i t y . T h e n
D , n w = 0 a n d
a , n l f =
2 6 ) x D . r + c v x ( w x r )
=
2 w x v . + w x ( w x r )
T h e q u a n t i t y 2 w x v , n i s c a l l e d t h e C o r i o l i s a c c e l e r a t i o n a n d w x ( w x r ) i s c a l l e d t h e c e n t r i p e t a l a c c e l -
e r a t i o n .
N e w t o n ' s l a w s a r e s t r i c t l y v a l i d o n l y i n i n e r t i a l s y s t e m s , i . e . s y s t e m s w h i c h a r e e i t h e r f i x e d o r
w h i c h m o v e w i t h c o n s t a n t v e l o c i t y r e l a t i v e t o a f i x e d s y s t e m . T h e e a r t h i s n o t e x a c t l y a n i n e r t i a l s y s -
t e m a n d t h i s a c c o u n t s f o r t h e p r e s e n c e o f t h e s o c a l l e d ` f i c t i t i o u s ' e x t r a f o r c e s ( C o r i o l i s , e t c . ) w h i c h
m u s t b e c o n s i d e r e d .
I f t h e m a s s o f a p a r t i c l e i s a c o n s t a n t M , t h e n N e w t o n ' s s e c o n d l a w b e c o m e s
( 4 )
M D , 2 n r
=
F - 2 M ( w x D , n r ) - M [ w x ( w x r ) ]
w h e r e D m d e n o t e s d / d t a s c o m p u t e d b y a n o b s e r v e r o n t h e e a r t h , a n d F i s t h e r e s u l t a n t o f a l l r e a l
f o r c e s a s m e a s u r e d b y t h i s o b s e r v e r . T h e l a s t t w o t e r m s o n t h e r i g h t o f ( 4 ) a r e n e g l i g i b l e i n m o s t
c a s e s a n d a r e n o t u s e d i n p r a c t i c e .
T h e t h e o r y o f r e l a t i v i t y d u e t o E i n s t e i n h a s m o d i f i e d q u i t e r a d i c a l l y t h e c o n c e p t s o f a b s o l u t e m o -
t i o n w h i c h a r e i m p l i e d b y N e w t o n i a n c o n c e p t s a n d h a s l e d t o r e v i s i o n o f N e w t o n ' s l a w s .
a t a t t
t
( d ) i + 6 j + 2 k
8/19/2019 M. R. Spiegel, Vector Analysis, Schaum's Series
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5 4
V E C T O R D I F F E R E N T I A T I O N
+ 3 j - k ,
A n s . 7 i + 6 j - 6 k
2 2
3 6 . F i n d d s ( A . d B - d A B ) i f A a n d B a r e d i f f e r e n t i a b l e f u n c t i o n s o f s .
A n s . A . d s B
-
d s A - B
2
3 7 .
I f A ( t ) = 3 t 2 i - ( t + 4 ) j + ( t 2 - 2 t ) k a n d B ( t ) = s i n t i + 3 e t j - 3
c o s t k , f i n d d t 2 ( A x B ) a t t = 0 .
A n s . - 3 0 i + 1 4 j + 2 0 k
.
8 .
I f d t A = 6 t i - 2 4 t 2 j + 4 s i n t k , f i n d A g i v e n t h a t A = 2 i + j a n d d A = -
i - 3 k a t t = 0
A n s . A = ( t 3 - t + 2 ) i + ( 1 - 2 t 4 ) j + ( t - 4 s i n t ) k
3 9 . S h o w t h a t r = e - t ( C 1 c o s 2 t + C 2 s i n 2 t ) ,
w h e r e C 1 a n d C . a r e c o n s t a n t v e c t o r s , i s a s o l u t i o n o f t h e d i f -
2
f e r e n t i a l e q u a t i o n d t 2 + 2 d a + 5 r = 0 .
2
4 0 . S h o w t h a t t h e g e n e r a l s o l u t i o n o f t h e d i f f e r e n t i a l e q u a t i o n d
+ 2 a d + c v 2 r = 0 , w h e r e a a n d c o a r e c o n -
s t a n t s , i s
( a ) r = e - a t ( C
1
a
a 2 - ` ` ' 2 t
+ C
2
e -
a 2 - r ` ' 2 t )
i f
a 2
- c o t > 0
( b )
r =
e - a . t ( C 1 s i n w 2
- a 2 t
+ C 2 C o s 1 / w 2 - a 2 t )
i f a 2 - w 2 < 0 .
( c ) r = e - a t ( C 1 + C 2 t ) i f a 2 - w 2 = 0 ,
w h e r e C 1 a n d C 2 a r e a r b i t r a r y c o n s t a n t v e c t o r s .
2
d 2
4 1 . S o l v e ( a ) 2 - 4 - - 5 r = 0 ,
( b ) 2
2 + 4 r = 0 .
d t r d t
d t r + 2 d a
+ r = 0 ,
( c ) d t r
A n s .
( a ) r = C i e 5 t + C 2 e - t ,
( b ) r = e - t ( C i + C 2 t ) ,
( c )
r = C 1 c o s 2 t + C 2 s i n 2 t
4 2 . S o l v e d Y = X ,
d X = - Y .
A n s . X = C i c o s t
+ C 2 s i n t , Y = C 1 s i n t - C 2 c o s t
2
a A a A a 2 A
' a a 2 A
a 2 A
a 2 A
4 3 .
I f A = c o s x y i + ( 3 x y - 2 x ) j - ( 3 x + 2 y ) k ,
f i n d
a x ' a y '
a x e '
V P
a x a y ' a y a x
A n s . a ` 9 '
_ - - y s i n x y i + ( 3 y - 4 x ) j - 3 k ,
- _ - x s i n x y i + 3 x j - 2 k ,
x 2
2
2
2
a A
= - y 2 c o s x y i - 4 j ,
a
y
A
= - x 2 c o s x y I , 2 a y
a y
a x
= - ( x y c o s x y
+ s i n x y ) i + 3 j
2
4 4 . I f A = x 2 y z i - 2 x z 3 j + x z 2 k a n d B = 2 z i + y j - x 2 k ,
f i n d
a y
x
( A x B ) a t ( 1 , 0 , - 2 ) .
A n s . - 4 i - 8 j
4 5 .
i f C 1 a n d C 2 a r e c o n s t a n t v e c t o r s a n d X i s a c o n s t a n t s c a l a r , s h o w t h a t H =
s i n b y + C 2 c o s X y )
2
2
s a t i s f i e s t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n
a x B + a y
= 0 .
i c ) ( t - - r / c )
4 6 . P r o v e t h a t A = p 0
e
r
,
w h e r e p o i s a c o n s t a n t v e c t o r , w a n d c a r e c o n s t a n t s c a l a r s a n d i = V 1 ,
2
2
A
s a t i s f i e s t h e e q u a t i o n a A +
2
a A = c 2 a t 2
.
T h i s r e s u l t i s o f i m p o r t a n c e i n e l e c t r o m a g n e t i c t h e o r y .
D I F F E R E N T I A L G E O M E T R Y
4 7 . F i n d ( a ) t h e u n i t t a n g e n t T , ( b ) t h e c u r v a t u r e K , ( c ) t h e p r i n c i p a l n o r m a l N , ( d ) t h e b i n o r m a l B , a n d ( e ) t h e
t o r s i o n T f o r t h e s p a c e c u r v e x = t - t 3 / 3 , y = t 2 , z = t + t / 3 .
( 1 - t 2 ) i
+ 2 t j
+ ( 1 + t 2 ) k
2 t
1 - t 2
A n s . ( a ) T =
( c ) N
i +
j
V ( 1 + t 2 )
1 + t 2
1 + t 2
1
( b ) K =
1 ( t 2 - 1 ) i - 2 t J + ( t 2 + 1 ) k
( d ) B
V ' 2 - ( 1 + 1 2 )
( e ) T =
( 1 + t 2 ) 2
8/19/2019 M. R. Spiegel, Vector Analysis, Schaum's Series
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V E C T O R D I F F E R E N T I A T I O N
4 8 . A s p a c e c u r v e i s d e f i n e d i n t e r m s o f t h e a r c l e n g t h p a r a m e t e r s b y t h e e q u a t i o n s
x = a r c t a n s , y = 2 V i l n ( s 2 + 1 ) , z = s - a r c t a n s
F i n d ( a ) T ,
( b ) N , ( c ) B , ( d ) K , ( e ) T , ( f ) p , ( g ) c r .
A n s .
a
T
i + y " 2 s j + s 2 k
( d ) K
Y " 2 -
S 2
)
+ 1
s 2 +
( b ) N = - V s i + ( 1 - s 2 ) j + / s k
( e )
r - _
y '
( g )
s 2 + 1
s 2 + 1
s 2 +
( c ) B = s
2
i - _ s j + k
( f ) p =
s 2 + 1
s 2 + 1
V r 2
4 9 . F i n d K a n d T f o r t h e s p a c e c u r v e x = t , y = t , z = t 3
c a l l e d t h e t w i s t e d c u b i c .
A n s . K
2 " 9
+ 9 t
+ 1
T
( 9 t 4 + 4 t 2 + 1 ) 3 / 2
3
9 t 4 + 9 t 2 + 1
5 5
5 0 . S h o w t h a t f o r a p l a n e c u r v e t h e t o r s i o n T = O .
5 1 . S h o w t h a t t h e r a d i u s o f c u r v a t u r e o f a p l a n e c u r v e w i t h e q u a t i o n s y = f ( x ) ,
z = 0 ,
i . e . a c u r v e i n t h e x y
p l a n e i s g i v e n b y p =
1 + ( y r ) 2 ] 3 l 2
I Y 1 1
5 2 . F i n d t h e c u r v a t u r e a n d r a d i u s o f c u r v a t u r e o f t h e c u r v e w i t h p o s i t i o n v e c t o r r = a c o s u i + b s i n u j , w h e r e
a a n d b a r e p o s i t i v e c o n s t a n t s . I n t e r p r e t t h e c a s e w h e r e a = b .
A n s . K =
a b
=
1
i f a = b , t h e g i v e n c u r v e w h i c h i s a n e l l i p s e , b e c o m e s a c i r -
( a
2
s i n
2
u + b 2 c o s 2 u ) 3 / 2
P
c l e o f r a d i u s a a n d i t s r a d i u s o f c u r v a t u r e p = a .
5 3 . S h o w t h a t t h e F r e n e t - S e r r e t f o r m u l a s c a n b e w r i t t e n i n t h e f o r m
d T = C x T , d N = w x N , d B = r v x B a n d
d e t e r m i n e a ) .
A n s .
= T T + K B
5 4 . P r o v e t h a t t h e c u r v a t u r e o f t h e s p a c e c u r v e r = r ( t )
i s g i v e n n u m e r i c a l l y b y K =
r x r 3 i
,
w h e r e d o t s d e -
n o t e d i f f e r e n t i a t i o n w i t h r e s p e c t t o t . I r
5 5 .
r . r x r
( a ) P r o v e t h a t
r =
f o r t h e s p a c e c u r v e r = r ( t ) .
I i x r , l 2
d r d 2 r
d 3 r
d
( b ) I f t h e p a r a m e t e r t i s t h e a r c l e n g t h s s h o w t h a t T =
s ' W s - 2 x
( d 2 r i d s )
2
2
5 6 .
I f Q = r x r , s h o w t h a t K
Q 3
, T =
Q + 2 '
r 1
Q
5 7 . F i n d K a n d T f o r t h e s p a c e c u r v e x = 8 - s i n 8 , y = 1 - c o s 8 , z = 4 s i n ( 6 / 2 ) .
1
( 3 + c o s 8 ) c o s 8 / 2 + 2 s i n 8 s i n 8 / 2
A n s . K =
8
6 - 2 c o s 8 ,
=
1 2 c o s 8 - 4
2
5 8 . F i n d t h e t o r s i o n o f t h e c u r v e x =
t t + 1
t
t
1
,
z = t + 2 .
E x p l a i n y o u r a n s w e r .
A n s . T = U . T h e c u r v e l i e s o n t h e p l a n e x - 3 y + 3 z = 5 .
5 9 . S h o w t h a t t h e e q u a t i o n s o f t h e t a n g e n t l i n e , p r i n c i p a l n o r m a l a n d b i n o r m a l t o t h e s p a c e c u r v e r = r ( t ) a t t h e
p o i n t t = t o c a n b e w r i t t e n r e s p e c t i v e l y r = r o + t T o ,
r = r o + t N o , r = r o + t B o , w h e r e t i s a p a r a m e t e r .
6 0 . F i n d e q u a t i o n s f o r t h e ( a ) t a n g e n t , ( b ) p r i n c i p a l n o r m a l a n d ( c ) b i n o r m a l t o t h e c u r v e x = 3 c o s t , y = 3 s i n t ,
z = 4 t
a t t h e p o i n t w h e r e t = R .
A n s . ( a ) T a n g e n t : r = - 3 i + 4 7 t k + t ( - 5 j + 5 k )
o r
x = - 3 , y
5 t ,
z = 4 7 L +
5
t .
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5 6
V E C T O R D I F F E R E N T I A T I O N
( b ) N o r m a l :
r = - 3 i + 4 1 t j + I i o r
x = - 3 + t , y = 4 T t , z = 0 .
( c ) B i n o r m a l : r = - 3 i + 4 7 L j + t ( 4 j +
5
k )
o r
x = - 3 , y = 4 1 t +
5
t , z =
5
t .
6 1 . F i n d e q u a t i o n s f o r t h e ( a ) o s c u l a t i n g p l a n e , ( b ) n o r m a l p l a n e a n d ( c ) r e c t i f y i n g p l a n e t o t h e c u r v e x = 3 t - t 3
Y = 3 t 2 ,
z = 3 t + t 3
a t t h e p o i n t w h e r e t = 1 .
A n s . ( a ) y - z + 1 = 0 ,
( b ) y + z - - 7 = 0 ,
( c ) x = 2
6 2 . ( a ) S h o w t h a t t h e d i f f e r e n t i a l o f a r c l e n g t h o n t h e s u r f a c e r = r ( u , v ) i s g i v e n b y
d s 2 = E d u e + 2 F d u d v + G d v 2
w h e r e E
a r a r
_
a r 2
a r
, ' 3 r
a r
=
a r 2
C u C u
( a u )
F
C u T V '
G
' 6 V
a v ( a v )
( b ) P r o v e t h a t a n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n t h a t t h e u , v c u r v i l i n e a r c o o r d i n a t e s y s t e m b e o r t h o g o n a l
i s F = O .
6 3 . F i n d a n e q u a t i o n f o r t h e t a n g e n t p l a n e t o t h e s u r f a c e z = x y a t t h e p o i n t ( 2 , 3 , 6 ) .
A n s .
3 x + 2 y - z = 6
6 4 . F i n d e q u a t i o n s f o r t h e t a n g e n t p l a n e a n d n o r m a l l i n e t o t h e s u r f a c e 4 z = x 2 - y 2 a t t h e p o i n t ( 3 , 1 , 2 ) .
A n s . 3 x - y - 2 z = 4 ; x = 3 t + 3 , y = 1 - t , z = 2 - 2 t
a r
x
a r
6 5 . P r o v e t h a t a u n i t n o r m a l t o t h e s u r f a c e r = r ( u , v ) i s n
a
,
w h e r e E , F , a n d G a r e d e f i n e d a s
i n P r o b l e m 6 2 .
G
M E C H A N I C S
6 6 . A p a r t i c l e m o v e s a l o n g t h e c u r v e r = ( t 3 - 4 t ) i + ( t 2 + 4 t ) J + ( 8 t 2 - 3 t 3 ) k ,
w h e r e t i s t h e t i m e . F i n d t h e
m a g n i t u d e s o f t h e t a n g e n t i a l a n d n o r m a l c o m p o n e n t s o f i t s a c c e l e r a t i o n w h e n t = 2 .
A n s . T a n g e n t i a l , 1 6 ;
n o r m a l , 2 V ' 7 3
6 7 .
I f a p a r t i c l e h a s v e l o c i t y v a n d a c c e l e r a t i o n a a l o n g a s p a c e c u r v e , p r o v e t h a t t h e r a d i u s o f c u r v a t u r e o f i t s
p a t h i s g i v e n n u m e r i c a l l y b y p =
v 3
I v x a I
6 8 . A n o b j e c t i s a t t r a c t e d t o a f i x e d p o i n t 0 w i t h a f o r c e F = f ( r ) r , c a l l e d a c e n t r a l f o r c e , w h e r e r
i s t h e p o s i -
t i o n v e c t o r o f t h e o b j e c t r e l a t i v e t o 0 . S h o w t h a t r x v = h w h e r e h i s a c o n s t a n t v e c t o r . P r o v e t h a t t h e
a n g u l a r m o m e n t u m i s c o n s t a n t .
6 9 . P r o v e t h a t t h e a c c e l e r a t i o n v e c t o r o f a p a r t i c l e m o v i n g a l o n g a s p a c e c u r v e a l w a y s l i e s i n t h e o s c u l a t i n g
p l a n e .
7 0 .
( a ) F i n d t h e a c c e l e r a t i o n o f a p a r t i c l e m o v i n g i n t h e x y p l a n e i n t e r m s o f p o l a r c o o r d i n a t e s ( p , c b )
.
( b ) W h a t a r e t h e c o m p o n e n t s o f t h e a c c e l e r a t i o n p a r a l l e l a n d p e r p e n d i c u l a r t o p ?
A n s . ( a ) r = [ ( P - p % 2 ) c o s
- ( p +
s i n q S ] i
+
s i n 0 + ( p c +
c o s
j
( b )
p * + 2 p
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T H E V E C T O R D I F F E R E N T I A L O P E R A T O R D E L , w r i t t e n V , i s d e f i n e d b y
z i + j +
a z k
=
i x
+ ' a
+ k a z
r
r
T h i s v e c t o r o p e r a t o r p o s s e s s e s p r o p e r t i e s a n a l o g o u s t o t h o s e o f o r d i n a r y v e c t o r s .
I t i s u s e f u l i n d e -
f i n i n g t h r e e q u a n t i t i e s w h i c h a r i s e i n p r a c t i c a l a p p l i c a t i o n s a n d a r e k n o w n a s t h e g r a d i e n t , t h e d i v e r -
g e n c e a n d t h e c u r l . T h e o p e r a t o r V i s a l s o k n o w n a s n a b l a .
T H E G R A D I E N T . L e t 4 ) ( x , y , z ) b e d e f i n e d a n d d i f f e r e n t i a b l e a t e a c h p o i n t ( x , y , z ) i n a c e r t a i n r e -
g i o n o f s p a c e ( i . e . 0 d e f i n e s a d i f f e r e n t i a b l e s c a l a r f i e l d ) . T h e n t h e g r a d i e n t o f 4 ) ,
w r i t t e n V 4 ) o r g r a d 0 , i s d e f i n e d b y
( a x l +
y
j
+ a k ) = a i +
j +
k
a
r
N o t e t h a t V 4 ) d e f i n e s a v e c t o r f i e l d .
T h e c o m p o n e n t o f V 4 ) i n t h e d i r e c t i o n o f a
n i t v e c t o r a s g i v e n b y V c . a a n d i s c a l l e d t h e d i -
r e c t i o n a l d e r i v a t i v e o f 4 ) i n t h e d i r e c t i o n a .
P h y s i c a l l y , t h i s i s t h e r a t e o f c h a n g e o f 0 a t ( x , y , z ) i n
u e c t i o n a .
T H E D I V E R G E N C E . L e t V ( x , y , z ) = V 1 i + 2 j + V k b e d e f i n e d a n d d i f f e r e n t i a b l e a t e a c h p o i n t
( x , y , z ) i n a c e r t a i n r e g i o n o f s p a c e ( i . e . V d e f i n e s a d i f f e r e n t i a b l e v e c t o r f i e l d ) .
T h e n t h e d i v e r g e n c e o f V , w r i t t e n V . V o r d i v V , i s d e f i n e d b y
( a x i
+
a y j
+
a
a
a V 3
a x
a y + a z
N o t e t h e a n a l o g y w i t h A - B = A l B 1 + A 2 B 2 + A 3 B 3
.
A l s o n o t e t h a t V V
V - V .
T H E C U R L . I f V ( z , y , z ) i s a d i f f e r e n t i a b l e v e c t o r f i e l d t h e n t h e c u r l o r r o t a t i o n o f V , w r i t t e n V x V ,
c u r l V o r r o t V , i s d e f i n e d b y
V x V =
( 3 - i + a y j + a z k ) x ( V l i + V 2 j + V 3 k )
5 7
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G R A D I E N T , D I V E R G E N C E a n d C U R L
5 9
z a x e s ( s e e P r o b l e m 3 8 ) .
I n c a s e t h e o r i g i n s o f t h e t w o c o o r d i n a t e s y s t e m s a r e n o t c o i n c i d e n t t h e
e q u a t i o n s o f t r a n s f o r m a t i o n b e c o m e
( 2 )
=
1 1 1 x ± 1 1 2 Y ± 1 1 3 2 + a 1
1 2 1 x + 1 2 2 Y + 1 2 3 z + a 2
1 3 1 x + 1 3 2 Y + 1 3 3 z
+ a 3
w h e r e o r i g i n 0 o f t h e x y z c o o r d i n a t e s y s t e m i s l o c a t e d a t ( a " , a ' , a 3 ) r e l a t i v e t o t h e x ' y ' z ' c o o r d i n a t e
s y s t e m .
T h e t r a n s f o r m a t i o n e q u a t i o n s ( 1 ) d e f i n e a p u r e r o t a t i o n , w h i l e e q u a t i o n s ( 2 ) d e f i n e a r o t a t i o n p l u s
t r a n s l a t i o n . A n y r i g i d b o d y m o t i o n h a s t h e e f f e c t o f a t r a n s l a t i o n f o l l o w e d b y a r o t a t i o n . T h e t r a n s -
f o r m a t i o n ( 1 ) i s a l s o c a l l e d a n o r t h o g o n a l t r a n s f o r m a t i o n . A g e n e r a l l i n e a r t r a n s f o r m a t i o n i s c a l l e d
a n a f f i n e t r a n s f o r m a t i o n .
P h y s i c a l l y a s c a l a r p o i n t f u n c t i o n o r s c a l a r f i e l d O ( x , y , z ) e v a l u a t e d a t a p a r t i c u l a r p o i n t s h o u l d
b e i n d e p e n d e n t o f t h e c o o r d i n a t e s o f t h e p o i n t . T h u s t h e t e m p e r a t u r e a t a p o i n t i s n o t d e p e n d e n t o n
w h e t h e r c o o r d i n a t e s ( x , y , z ) o r ( x ; y ; z ' ) a r e u s e d . T h e n i f O ( x , y , z ) i s t h e t e m p e r a t u r e a t p o i n t P w i t h
c o o r d i n a t e s ( x , y , z ) w h i l e 0 ' ( x , y , z ' ) i s t h e t e m p e r a t u r e a t t h e s a m e p o i n t P w i t h c o o r d i n a t e s ( x ; y ; z ' ) ,
w e m u s t h a v e 0 ( x , y , z ) = c ' ( x , y , z ' ) .
I f 0 ( x , y , z ) = = Y ' ( x ' , y ' z ' ) , w h e r e x , y , z a n d x , y ' z ' a r e r e l a t e d
b y t h e t r a n s f o r m a t i o n e q u a t i o n s ( 1 ) o r ( 2 ) , w e c a l l ( P ( x , y , z ) a n i n v a r i a n t w i t h r e s p e c t t o t h e t r a n s f o r -
m a t i o n . F o r e x a m p l e , x 2 + y 2 + z 2 i s i n v a r i a n t u n d e r t h e t r a n s f o r m a t i o n o f r o t a t i o n ( 1 ) , s i n c e x 2 + y 2 + z 2 =
1 2
x
+
y 1 2
+
Z 1 2 .
S i m i l a r l y , a v e c t o r p o i n t f u n c t i o n o r v e c t o r f i e l d A ( x , y , z ) i s c a l l e d a n i n v a r i a n t i f
A ( x , y , z ) _
A ' ( x , y , z ' ) .
T h i s w i l l b e t r u e i f
A 1 ( x , y , z ) i + A 2 ( x , y , z ) j + A 3 ( x , y , z ) k
A ' ( x , y ; z ' ) i ' + A 2 ( x y ' , z ) j ' + A 3 ( x , y , z ) k '
I n C h a p . 7 a n d 8 , m o r e g e n e r a l t r a n s f o r m a t i o n s a r e c o n s i d e r e d a n d t h e a b o v e c o n c e p t s a r e e x t e n d e d .
I t c a n b e s h o w n ( s e e P r o b l e m 4 1 ) t h a t t h e g r a d i e n t o f a n i n v a r i a n t s c a l a r f i e l d i s a n i n v a r i a n t
v e c t o r f i e l d w i t h r e s p e c t t o t h e t r a n s f o r m a t i o n s ( 1 ) o r ( 2 ) .
S i m i l a r l y , t h e d i v e r g e n c e a n d c u r l o f a n i n -
v a r i a n t v e c t o r f i e l d a r e i n v a r i a n t u n d e r t h i s t r a n s f o r m a t i o n .
S O L V E D P R O B L E M S
T H E G R A D I E N T
1 . I f 0 ( x , y , z ) = 3 x 2 y - y
3
z
2
,
f i n d V o ( o r g r a d q 5 ) a t t h e p o i n t ( 1 , - 2 , - 1 ) .
V
a x
i + a y i + a z k ) ( 3 x 2 Y - y 3 z 2 )
( 3 x 2 Y
- y 3 z 2 )
+
k a z ( 3 x 2 y - y 3 z 2 )
3 X
Y
+ i
- a y
=
6 x y i
+
( 3 x 2 - 3 y 2 z 2 ) j - 2 y 3 z k
6 ( 1 ) ( - 2 ) i
+ { 3 ( 1 ) 2 -
3 ( - 2 ) 2 ( - 1 ) 2 } j
- 1 2 i - 9 j - 1 6 k
-
2 ( - 2 ) 3 ( - 1 ) k
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6 0
G R A D I E N T , D I V E R G E N C E a n d C U R L
2 . P r o v e ( a ) V ( F + G ) = V F + V G , ( b ) V ( F G ) = F V G + G V F w h e r e F a n d G
l a r f u n c t i o n s o f x , y a n d z .
( a ) V ( F + G )
( b ) V ( F G )
= ( a i
k ) ( F + G )
+a
]
a z
x
Y
i
a x
( F + G )
+
j a ( F + G )
+
k a z ( F + G )
Y
i s +
i 3 x
+
=
i a F
+ j a F
+
a x
a y
j a F +
j a G
+
k a F +
k a G
a y
a y
a z a z
k a z + i a c
+ a
+ k a z
Y
a r e s c a -
( i a x + j a y + k a z ) F +
( i a + ; a y + k a ) G
= V F + V G
( a x i + a y j
+ a z k ) ( F G )
a x
( F G ) i
+ a ( F G ) ;
+
a z ( F G )
k
Y
( F a G + G a F ) i
+ ( F a G
+ G - ) j
+
( F a c + G a F ) k
a x
a x
a y
a y
a z
a z
F (
a G
i + a G
j +
a G
k )
+
G ( a F
i +
6 F
j +
a F
k )
=
F V G
a x
a y a z
a x a y a z
+ G V F
3 . F i n d V q 5 i f ( a ) = I n , r { , ( b ) 4 = r .
( a ) r = x i + y j + z k . T h e n
I r I
= x 2 + y 2 + z 2 a n d c = I n
` r f = 2 l n ( x 2 + y 2 + z 2 ) .
V c p
=
2 V l n ( x 2 + y 2 + z 2 )
=
2 { i a l n ( x 2 + y 2 + z 2 ) + j a l n ( x 2 , + y 2 + z 2 ) + k a z I n ( x 2 + y 2 + z 2 ) }
x
y
2 x
2 y
2 z
x i + y j + z k
r
2 { i x 2 + y 2 + z 2
+ j x 2 + y 2 + z 2
+
k x 2 + y 2 + Z 2 }
=
x 2 + y 2 + z 2
- r 2
( b ) V = V ( T )
_
V (
2
1 2 2 )
=
V { ( x 2 + y 2 + z 2 ) - 1 / }
x + y + z
i a ( x 2 + y 2 + z 2 ) - 1 / 2 +
j a ( x 2 + y 2 + z 2 ) -
1 / 2 + k a z ( x 2 + y 2 + z 2 ) - 1 / 2
Y
1 2 2
2 - 3 / 2
2
2
2 - 3 / 2
I
2 2
2 - 3 / 2
i { - Z ( x + y + z )
2 x }
+
j { - 2 ( x + y + z )
2 y }
+ k { _ 2 ( x + y + z )
2 z }
- x i - - y j - z k
_
r
( x 2 + y 2 + z 2 ) 3 / 2 -
r 3
4 . S h o w t h a t V r n = n r n - 2 r .
V r n =
V ( x 2 + Y 2 + z 2 )
V ( x 2 + y 2 + z 2 ) n / 2
=
i
{ ( x 2 + y 2 + z 2 ) n / 2 }
+ j
{ ( x 2 + y 2 + z 2 ) n / 2 } +
k
1 { ( x 2 + y 2 + z 2 ) n / 2 }
a x
a y
a z
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G R A D I E N T , D I V E R G E N C E a n d C U R L
{ 2 ( x 2 + y 2 + z 2 ) n / 2 - 1 2 x }
+ j { 2 ( x 2 + y 2 + z 2 ) n / 2 - 1 2 y } +
k
n ( x 2 + y 2 + z 2 ) n / 2 - 1 ( x i
+ y j + z k )
n ( r
2 ) n / 2
- i r =
n r n - 2 r
{ 2 ( x 2 + y 2 + z 2 ) n / 2 - 1
2 z }
N o t e t h a t i f r = r r 1 w h e r e r 1 i s a u n i t v e c t o r i n t h e d i r e c t i o n r , t h e n V r n = n r n - 1 r i .
5 . 1 S h o w t h a t V
i s a v e c t o r p e r p e n d i c u l a r t o t h e s u r f a c e O ( x , y , z ) = c w h e r e c i s a c o n s t a n t .
6 1
L e t
r = x i + y j + z k b e t h e p o s i t i o n v e c t o r t o a n y p o i n t P ( x , y , z ) o n t h e s u r f a c e . T h e n d
r = d x i +
d y j + d z k l i e s i n t h e t a n g e n t p l a n e t o t h e s u r f a c e a t P .
B u t d =
d x +
L O d y
+
a 4 d z
= 0
o r
( a - O i
+
a 0 i +
d y j + d z k ) = 0
a x
a y
a z
a x
a y
a z
i . e . 0 g b
d r = 0
s o t h a t V
i s p e r p e n d i c u l a r t o d r a n d t h e r e f o r e t o t h e s u r f a c e .
6 . F i n d a u n i t n o r m a l t o t h e s u r f a c e x 2 y + 2 x z = 4 a t t h e p o i n t ( 2 , - 2 , 3 ) .
V ( x 2 y + 2 x z ) = ( 2 x y + 2 z ) i
+ x 2 j + 2 x k = - 2 i + 4 j + 4 k
a t t h e p o i n t ( 2 , - 2 , 3 ) .
T h e n a u n i t n o r m a l t o t h e s u r f a c e =
- 2 1 + 4 j + 4 k 1 2 . 2
- - - t + - + - k
V ' ( - 2 ) 2 + ( 4 ) 2 + ( 4 ) 2 3 3 3
A n o t h e r u n i t n o r m a l i s 3 i -
2 j
- s k h a v i n g d i r e c t i o n o p p o s i t e t o t h a t a b o v e .
7 . F i n d a n e q u a t i o n f o r t h e t a n g e n t p l a n e t o t h e s u r f a c e 2 x z 2 - 3 x y - 4 x = 7 a t t h e p o i n t ( 1 , - 1 , 2 ) .
V ( 2 x z 2 - 3 x y - 4 x ) _ ( 2 z 2 - - 3 y - 4 ) i - 3 x j + 4 x z k
T h e n a n o r m a l t o t h e s u r f a c e a t t h e p o i n t ( 1 , - 1 , 2 ) i s
7 i - 3 j + 8 k .
T h e e q u a t i o n o f a p l a n e p a s s i n g t h r o u g h a p o i n t w h o s e p o s i t i o n v e c t o r i s r o a n d w h i c h i s p e r p e n d i c u l a r
t o t h e n o r m a l N i s ( r - r o ) N = 0 . ( S e e C h a p . 2 , P r o b . 1 8 . )
T h e n t h e r e q u i r e d e q u a t i o n i s
[ ( x i + y j + z k )
- ( i - i + 2 k ) ] ( 7 i - - 3 j + 8 k ) = 0
o r
7 ( x - 1 ) - 3 ( y + 1 ) + 8 ( z - 2 )
=
0 .
8 . L e t q S ( x , y , z ) a n d c ( x + A x , y + A y , z + A z ) b e t h e t e m p e r a t u r e s a t t w o n e i g h b o r i n g p o i n t s P ( x , y , z )
a n d Q ( x + A x , y + A y , z + A z ) o f a c e r t a i n r e g i o n .
( a ) I n t e r p r e t p h y s i c a l l y t h e q u a n t i t y
O =
O ( x + A x , y + A y ,
z
A z ) - 9 5 ( x , y , z )
d i s t a n c e b e t w e e n p o i n t s P a n d Q .
A s
A s
( b ) E v a l u a t e A l s m o 0 _
d o
a n d i n t e r p r e t p h y s i c a l l y .
S -
( c ) S h o w t h a t
d s
L O
= V q -
d s
w h e r e A s i s t h e
( a ) S i n c e A
i s t h e c h a n g e i n t e m p e r a t u r e b e t w e e n p o i n t s P a n d Q a n d A s i s t h e d i s t a n c e b e t w e e n t h e s e
p o i n t s ,
Q O r e p r e s e n t s t h e a v e r a g e r a t e o f c h a n g e i n t e m p e r a t u r e p e r u n i t d i s t a n c e i n t h e d i r e c t i o n f r o m
P t o Q .
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G R A D I E N T , D I V E R G E N C E a n d C U R L
( b ) F r o m t h e c a l c u l u s ,
A x +
o A y + a o A z + i n f i n i t e s i m a l s o f o r d e r h i g h e r t h a n L a x , A y a n d A z
T h e n
o r
x y z
a
d s m o A s
-
& 4
o a x A s
a y A s
+ a z A s
d o
a o d x
a 0 d y
a o d z
d s a x d s +
a y d s
+ a z d s
d
d s
r e p r e s e n t s t h e r a t e o f c h a n g e o f t e m p e r a t u r e w i t h r e s p e c t t o d i s t a n c e a t p o i n t P i n a d i r e c t i o n
t o w a r d Q . T h i s i s a l s o c a l l e d t h e d i r e c t i o n a l d e r i v a t i v e o f .
d o
a q d x
o d y
a o d z a q
. - 4
,
d x d y d z
( c ) d s
a x d s +
a a y
d s + a z d s ( a x 1 + a y J + a z
L o
k )
( d s i + d s ' + d s k )
d s
N o t e t h a t s i n c e d i s a u n i t v e c t o r , V c
d s
i s t h e c o m p o n e n t o f V 4 i n t h e d i r e c t i o n o f t h i s u n i t
v e c t o r .
9 . S h o w t h a t t h e g r e a t e s t r a t e o f c h a n g e o f 0 , i . e . t h e m a x i m u m d i r e c t i o n a l d e r i v a t i v e , t a k e s p l a c e
i n t h e d i r e c t i o n o f , a n d h a s t h e m a g n i t u d e o f , t h e v e c t o r V o .
B y P r o b l e m 8 ( c ) , d o = V c b
d s
i s t h e p r o j e c t i o n o f V V i n t h e d i r e c t i o n d s .
T h i s p r o j e c t i o n w i l l b e
a m a x i m u m w h e n V V a n d
h a v e t h e s a m e d i r e c t i o n . T h e n t h e m a x i m u m v a l u e o f d o t a k e s p l a c e i n t h e
d i r e c t i o n o f V o a n d i t s m a g n i t u d e i s
I V o I
.
1 0 . F i n d t h e d i r e c t i o n a l d e r i v a t i v e o f 0 = x 2 y z + 4 x z 2 a t ( 1 , - 2 , - - 1 ) i n t h e d i r e c t i o n 2 i
- j - 2 k .
0 =
V ( x 2 y z + 4 x z 2 )
_
( 2 x y z + 4 z 2 7 i + x 2 z j + ( x 2 y + 8 x z ) k
=
8 i - j - 1 0 k
a t
( 1 , - 2 , - 1 ) .
T h e u n i t v e c t o r i n t h e d i r e c t i o n o f 2 1 - j - 2 k i s
a 2 i - j - 2 k
=
( 2 ) 2 + ( - 1 ) 2 + ( - 2 ) 2
2 ,
- 3 J - 3 k
T h e n t h e r e q u i r e d d i r e c t i o n a l d e r i v a t i v e i s
V O - a = ( 8 i - j -
S i n c e t h i s i s p o s i t i v e , 0 i s i n c r e a s i n g i n t h i s d i r e c t i o n .
1 6 + 1 + 0 _ 3 7
3 3
3 3
1 1 . ( a ) I n w h a t d i r e c t i o n f r o m t h e p o i n t ( 2 , 1 , - 1 ) i s t h e d i r e c t i o n a l d e r i v a t i v e o f
= x y z 3 a m a x i m u m ?
( b ) W h a t i s t h e m a g n i t u d e o f t h i s m a x i m u m ?
V O = V ( x y z 3 )
=
2 x y z 3 i + x z 3 j
+ 3 x x y z 2 k
= - 4 1 - 4 j + 1 2 k
a t
( 2 , 1 , - 1 ) .
T h e n b y P r o b l e m 9 ,
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G R A D I E N T , D I V E R G E N C E a n d C U R L
6 3
( a ) t h e d i r e c t i o n a l d e r i v a t i v e i s a m a x i m u m i n t h e d i r e c t i o n V c b = - 4 i - 4 j + 1 2 k ,
( b ) t h e m a g n i t u d e o f t h i s m a x i m u m i s
I V O
( - 4 ) 2 + ( - 4 ) 2 + ( 1 2 ) 2
=
7 1 6 = 4 v " 1 1 .
1 2 . F i n d t h e a n g l e b e t w e e n t h e s u r f a c e s x 2 + y 2 + z 2 = 9 a n d z = x 2 + y 2 - 3 a t t h e p o i n t ( 2 , - 1 , 2 ) .
T h e a n g l e b e t w e e n t h e s u r f a c e s a t t h e p o i n t i s t h e a n g l e b e t w e e n t h e n o r m a l s t o t h e s u r f a c e s a t t h e
p o i n t .
A n o r m a l t o x 2 + y 2 + z 2 = 9
a t ( 2 , - 1 , 2 ) i s
V O 1 =
V ( x 2 + y 2 + z 2 )
=
2 x i
+ 2 y i + 2 z k
=
4 i - 2 j + 4 k
A n o r m a l t o z = x 2 + y 2 - 3 o r x 2 + y 2 - z = 3 a t ( 2 , - 1 , 2 ) i s
V q 5 2
= V ( x 2 + y 2 - z )
=
2 x i + 2 y j - k
= 4 i - 2 j - k
( V V 1 ) ( V q 5 2 ) =
I V ¢ I 1 I
I V 0 2 1 c o s 0 , w h e r e 8 i s t h e r e q u i r e d a n g l e . T h e n
I 4 i - 2 i + 4 k I I 4 i - 2 i - k I c o s 8
1 6 + 4 - 4 =
( 4 ) 2 + ( - 2 ) 2 + ( 4 ) 2
( 4 ) 2 + ( - 2 ) 2 + ( - 1 ) 2 c o s 8
a n d
c o s B =
1 6
= 8 6 3 = 0 . 5 8 1 9 ; t h u s t h e a c u t e a n g l e i s 8 = a r c c o s 0 . 5 8 1 9 = 5 4 ° 2 5 ' .
6 2 1
1 3 . L e t R b e t h e d i s t a n c e f r o m a f i x e d p o i n t A ( a , b , c ) t o a n y p o i n t P ( x , y , z ) . S h o w t h a t V R i s a u n i t
v e c t o r i n t h e d i r e c t i o n A P = R .
I f r A a n d r p a r e t h e p o s i t i o n v e c t o r s a i + b j + c k a n d x i + y j + z k
o f A a n d P r e s p e c t i v e l y , t h e n
R = r p - r A = ( x - a ) i + ( y - b ) j + ( z - c ) k ,
s o t h a t R =
( x - a ) 2 + ( y - b ) 2 + ( z - c ) 2
.
T h e n
V R
= V ( ( x - a ) 2 + ( y - b ) 2 + ( z - c ) 2 )
_
( x - a ) i + ( y - b ) j + ( z - c ) k
=
R
( x - a ) 2 + ( y - - b ) 2 + ( z - c ) 2
R
i s a u n i t v e c t o r i n t h e d i r e c t i o n R .
1 4 . L e t P b e a n y p o i n t o n a n e l l i p s e w h o s e f o c i a r e a t p o i n t s A a n d B , a s s h o w n i n t h e f i g u r e b e l o w .
P r o v e t h a t l i n e s A P a n d B P m a k e e q u a l a n g l e s w i t h t h e t a n g e n t t o t h e e l l i p s e a t P .
L e t R 1 = A P a n d R 2 = B P d e n o t e v e c t o r s d r a w n r e -
s p e c t i v e l y f r o m f o c i A a n d B t o p o i n t P o n t h e e l l i p s e , a n d
l e t T b e a u n i t t a n g e n t t o t h e e l l i p s e a t P .
S i n c e a n e l l i p s e i s t h e l o c u s o f a l l p o i n t s P t h e s u m
o f w h o s e d i s t a n c e s f r o m t w o f i x e d p o i n t s A a n d B i s a
c o n s t a n t p , i t i s s e e n t h a t t h e e q u a t i o n o f t h e e l l i p s e i s
R 1 + R 2 = p .
B y P r o b l e m 5 , V ( R 1 + R 2 ) i s a n o r m a l t o t h e e l l i p s e ;
h e n c e
[ V ( R 1 + R 2 ) ] T = 0 o r ( V R 2 ) T = - ( V R 1 ) . T
.
S i n c e V R 1 a n d V R 2 a r e u n i t v e c t o r s i n d i r e c t i o n R - 1
a n d R 2 r e s p e c t i v e l y ( P r o b l e m 1 3 ) , t h e c o s i n e o f t h e a n g l e
b e t w e e n V R 2 a n d T i s e q u a l t o t h e c o s i n e o f t h e a n g l e b e -
t w e e n V R 1 a n d - T ; h e n c e t h e a n g l e s t h e m s e l v e s a r e e q u a l .
T h e p r o b l e m h a s a p h y s i c a l i n t e r p r e t a t i o n .
L i g h t r a y s ( o r s o u n d w a v e s ) o r i g i n a t i n g a t f o c u s A , f o r
e x a m p l e , w i l l b e r e f l e c t e d f r o m t h e e l l i p s e t o f o c u s B .
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G R A D I E N T , D I V E R G E N C E a n d C U R L
1 2 y 2 - z 2 - x 22
/
a y 2 ( V x 2 + y 2 + z 2 )
T h e n b y a d d i t i o n ,
( x 2 + y 2 + z 2 ) 5 / 2
a n d
a - 2
1
2 z 2 - x 2 - y 2
a z 2 (
x 2 + y 2 + z 2 )
( x 2 + y 2 + z 2 ) 5 / 2
2 2 2
+ a
1
( a
V
+
a z 2 ) (
)
= 0 .
V X 2
T h e e q u a t i o n
0 2 0
= 0 i s c a l l e d L a p l a c e ' s e q u a t i o n . I t f o l l o w s t h a t 0 = 1 1 r i s a s o l u t i o n o f t h i s
e q u a t i o n .
1 8 . P r o v e :
( a ) V ( A + B ) = V A + V B
( b ) V ( O A ) = ( V O ) - A + 0 ( V ' A )
( a ) L e t A = A l i + A 2 j + A s k , B = B l i + B 2 j + B 3 k .
T h e n V . ( A + B ) =
( a a - i
+
j +
a k )
[ ( A 1 + B 1 ) i + ( A 2 + B 2 ) j + ( A 3 + B 3 ) k ]
x
Y
=
x
( A 1 + B 1 ) + a
Y
a
( A 2 + B 2 ) + a z
( A 3 + B 3 )
a A 1
+
a A 2
+
a A 3
+
- 6 B ,
a B 2
a B 3
= a x a y a z
a x + a y + a z
= ( a i
j + A
k )
+ A
) . ( A
+ a
3
zx
Y
+ ( a x i + a ] + a z k ) .
( B 1 1 + B 2 j + B 3 k )
Y
( b )
= V . A + V . B
V .
V . ( O A 1 i + O A 2 j + c A 3 k )
a x ( ( P A 1 )
+ a y a z
( O A 3 )
= a A 1 + 0 : + a 0 A 2 +
O x
O x
a
A l +
a O
A 2
+ a A 3
+
Y
a A 3
a z
_
( a i + a j
+
a o k )
( A 1 i + A 2 j + A 3 k ) + 0 ( a x 1
+ a j + a
z
k ) . ( A 1 1
y
_ ( V ) . A +
( V - A )
1 9 . P r o v e V ( 3 ) = 0
.
r
L e t
= r - 3 a n d A = r i n t h e r e s u l t o f P r o b l e m 1 8 ( b ) .
T h e n V ( r - 3 r )
= ( V r - 3 ) . r + ( r
s ) V .
r
= - 3 r - 5 r r + 3 r - S = 0 ,
u s i n g P r o b l e m 4 .
+ A 2 j + A s k )
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6 6
G R A D I E N T , D I V E R G E N C E a n d C U R L
2 0 . P r o v e
V - ( U V V - V V U ) = U V 2 V - V V 2 U .
F r o m P r o b l e m 1 8 ( b ) , w i t h < t = U a n d A = V V ,
V .
( U V V )
_
( V U ) .
( V V ) +
U
V
( V
( V V )
U
V
V
V
V
( V V )
( V U )
+ V
V 2 U
= U V 2 V -
V V 2 U
2 1 . A f l u i d m o v e s s o t h a t i t s v e l o c i t y a t a n y p o i n t i s v ( x , y , z ) .
S h o w t h a t t h e l o s s o f f l u i d p e r u n i t
v o l u m e p e r u n i t t i m e i n a s m a l l p a r a l l e l e p i p e d h a v i n g c e n t e r a t P ( x , y , z ) a n d e d g e s p a r a l l e l t o t h e
c o o r d i n a t e a x e s a n d h a v i n g m a g n i t u d e A x , A y , A z r e s p e c t i v e l y , i s g i v e n a p p r o x i m a t e l y b y d i v v =
V - V
.
z
R e f e r r i n g t o t h e f i g u r e a b o v e ,
x c o m p o n e n t o f v e l o c i t y v a t P
=
V 1
x c o m p o n e n t o f v a t c e n t e r o f f a c e A F E D
=
v i - i a x 1 A x
a p p r o x .
x c o m p o n e n t o f v a t c e n t e r o f f a c e G H C B
=
v i +
2
a z Q x
a p p r o x .
T h e n ( 1 ) v o l u m e o f f l u i d c r o s s i n g A F E D p e r u n i t t i m e
( 2 ) v o l u m e o f f l u i d c r o s s i n g G H C B p e r u n i t t i m e
L o s s i n v o l u m e p e r u n i t t i m e i n x d i r e c t i o n
( v 1
- 2 a x
A x ) A y A z
( v 1 +
I
a v i
A x ) A y A z
.
2
x
( 2 ) - ( 1 ) = a x
A x A y A z .
a v
S i m i l a r l y ,
l o s s i n v o l u m e p e r u n i t t i m e i n y d i r e c t i o n
=
2 A x A y A z
y
l o s s i n v o l u m e p e r u n i t t i m e i n z d i r e c t i o n
a v 3
A x 0 y A z .
a z
T h e n , t o t a l l o s s i n v o l u m e p e r u n i t v o l u m e p e r u n i t t i m e
a v 1
+ a v 2
+ a v 3
( a x
a y a z )
O x D y A z
=
d i v v
=
v
A x A y A z
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G R A D I E N T , D I V E R G E N C E a n d C U R L
6 7
T h i s i s t r u e e x a c t l y o n l y i n t h e l i m i t a s t h e p a r a l l e l e p i p e d s h r i n k s t o P , i . e . a s L x , L \ y a n d A z a p p r o a c h
z e r o .
I f t h e r e i s n o l o s s o f f l u i d a n y w h e r e , t h e n V v = 0 . T h i s i s c a l l e d t h e c o n t i n u i t y e q u a t i o n f o r a n i n -
c o m p r e s s i b l e f l u i d . S i n c e f l u i d i s n e i t h e r c r e a t e d n o r d e s t r o y e d a t a n y p o i n t , i t i s s a i d t o n a v e n o s o u r c e s
o r s i n k s . A v e c t o r s u c h a s v w h o s e d i v e r g e n c e i s z e r o i s s o m e t i m e s c a l l e d s o l e n o i d a l .
2 2 . D e t e r m i n e t h e c o n s t a n t a s o t h a t t h e v e c t o r V = ( x + 3 y ) i + ( y - 2 z ) j + ( x + a z ) k i s s o l e n o i d a l .
A v e c t o r V i s s o l e n o i d a l i f i t s d i v e r g e n c e i s z e r o ( P r o b l e m 2 1 ) .
' V . V
=
a
( x + 3 Y ) + a ( y - 2 z ) +
a z
( x + a z )
=
1
+
1
+ a
Y
T h e n
w h e n a = - 2 .
T H E C U R L
2 3 . I f A = x z ' i - 2 x 2 y z i + 2 y z 4 k ,
f i n d V x A ( o r c u r l A ) a t t h e p o i n t ( 1 , - 1 , 1 ) .
v x A = ( a i + a j + a k ) x ( x z ' i - 2 x 2 y z ] + 2 y z 4 k )
a x
a y
a z
i j k
a
a a
a x
a y a z
x z 3
- 2 x 2 y z
2 y z 4
=
[
a y
( 2 y z 4 )
a z
( - 2 x 2 y z ) ] i
+
[ a ( x z 3 ) - a ( 2 y Z 4 ) ] j
+ [ a X ( - 2 x 2 y z ) - a Y ( x z 3 ) ] k
=
( 2 z 4 + 2 x 2 y ) i + 3 x z 2 j - 4 x y z k
= 3 j
+ 4 k a t ( 1 , - 1 , 1 ) .
2 4 . I f A = x 2 y i - 2 x z j
+ 2 y z k , f i n d c u r l c u r l A .
c u r l c u r l A
= V x ( V x A )
= v x
i
j
a a
a
a x a y a Z
x 2 y - 2 x z
2 y z
= V x [ ( 2 x + 2 z ) i - ( x 2 + 2 z ) k ]
i j
k
a a a
a x
a y
a z
2 x + 2 z
0 - x 2 - 2 z
= ( 2 x + 2 ) j
2 5 . P r o v e :
( a ) V x ( A + B ) = V x A + V x B
( b ) V x ( V O ) x A + 0 ( v x A ) .
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G R A D I E N T , D I V E R G E N C E a n d C U R L
2 6 . E v a l u a t e V . ( A x r )
i f V x A = 0 .
L e t A = A 1 i + A 2 j + A 3 k , r = x i + y j + z k .
T h e n
A x r =
i j
k
A 1
A 2
A 3
x y
z
( z A 2 - y A 3 ) i
+
( x A 3 - z A 1 ) j
+
( 7 A 1 - x A 2 ) k
a n d V ( A x r )
=
a
( z A 2 - y A 3 )
+
a
( x A 3 - z A 1 )
+
( y A 1 - x A 2 )
O x
a y
a z
a A 2
a A 3 a A 3
a A 1
a A 1
O x
a x
z
- - y
+
X
a z
x ( d A 3
_
a A 2 )
+
y ( a A 1
-
a A 3 )
+
z ( a A 2
_
a A 1
a y
a z
a z
a x a x
a y
a A 3 a A 2
+
( a A 1 a A 3
+ (
a A 2
-
a A 1
) k
x i + y j + z k ] ' [ ( a
' 3 Z
a z
a x
a x
a
]
y y
r ( V x A )
=
r
c u r l A .
I f V x A = 0 t h i s r e d u c e s t o z e r o .
2 7 . P r o v e : ( a ) V x ( V O ) = 0
( c u r l g r a d 0 = 0 ) ,
( b ) V . ( V x A ) = 0
( d i v c u r l A = 0 ) .
( a ) V x ( V q )
= V x
( L O
i + a ( P i + a k )
y
i j
k
a a
a
a x
a y
a z
a q 5
a 0
a x a y
a z
[ a
a d ) -
z a
y
y
- a o
- 3 0
[ a x ( a
) -
( a ) ] k
y y
x
_ ( a 2 -
2
) i +
( a 2
-
a 2 0
) ]
+
a 2
-
a 2 -
) k
= 0
a y a z
a z a y a z a x a x a z
a x a y
a y a x
6 9
p r o v i d e d w e a s s u m e t h a t q b h a s c o n t i n u o u s s e c o n d p a r t i a l d e r i v a t i v e s s o t h a t t h e o r d e r o f d i f f e r e n t i a t i o n i s
i m m a t e r i a l .
( b ) V ( V x A ) = V
i
j
a
a
a x
a y
A l A 2
A
2 ) i
a y
a z
a a A 3
a A 2
) +a x ( a y - a z
a
a z
A 3
+
( a A 1
_
a A 3
) j
a z a x
a a A 1
a A
a y ( a z - a x
+ ( a A 2 _
a A 1 ) k ]
a x
a y
a
- a A 2
A l )
- 6 Z
(
a x
a y
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7 0
G R A D I E N T , D I V E R G E N C E a n d C U R L
a 2 A
3
a 2 A 2
a 2 A
1
a 2 A 3 a 2 A 2
a 2 A 1
=
0
a x a y a x a z
a y a z
a y a x a Z a x
a Z a y
a s s u m i n g t h a t A h a s c o n t i n u o u s s e c o n d p a r t i a l d e r i v a t i v e s .
N o t e t h e s i m i l a r i t y b e t w e e n t h e a b o v e r e s u l t s a n d t h e r e s u l t s ( C x C m ) = ( C x C ) m = 0 . w h e r e m i s a
s c a l a r a n d C ( C x A ) _ ( C x C ) A = 0 .
2 8 . F i n d c u r l ( r f ( r ) ) w h e r e f ( r ) i s d i f f e r e n t i a b l e .
c u r l
( r f ( r ) )
= V x ( r f ( r ) )
=
V x
( x f ( r ) i + y f ( r ) j
B u t
o f = a f ) c a a z
T h e n t h e r e s u l t
+ z f ( r ) k )
i
j
k
a a
a
a x
a y a z
x f ( r )
y f ( r )
z f ( r )
( z a y
y a z ) i
+ ( x a z -
Z a f ) j
2 9 . P r o v e V x ( V x A ) = - Q
A +
i j
V x ( V x A )
= V x
a
a z
=
o f a
(
x 2 + y 2 + z 2 ) =
f ' ( r ) x
=
f ' x
a r a x
x 2 + y 2 + z 2
r
' z
' z ' x
( z f r y - y f r ) i
+
( x
f f
- z
f T
) j +
A l A 2
A s
S i m i l a r l y ,
o f
=
L Y
a n d
o f
= ' z
a y r
a z
r
' x '
( y f T
-
x f Y ) k
=
0 .
a A 3 _ a A 2
) I
+
(
a A 1
a A 3
) '
+
(
a A 2
-
a A l
) k ]
x
(
a z a Z
-
- a x - a x
y
a y
i j
a
a x
a A 3
_
a A 2
a y a z
+
( y a f - x a a - f ) k
y
k
a
a z
a A 2 a A 1
a z
a x a x
a y
a (
a A 2 _
a A l )
_ a ( a A ,
_
a A 3 ) ]
i
a y a x
a y
a z
a z a x
+
a
a A 3
a A 2 )
_
a z a y
a z
+
a (
a A 1
a A 3
a x
a z
a x
a x ( a x e
_ a a A l ) ] j
y
2 ) ] k
-
( a a 3
a
y y
a
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G R A D I E N T , D I V E R G E N C E a n d C U R L
a - 2 A 1 a 2 A 1
a A 2
a 2 A 2 .
a A 3
a y 2
3
1
+ ( a y a x + a z a x ) i + ( a z a y + a x a y A +
- 5 z 2 ) 1 + ( -
' M
- M
) j
+
( -
a x 2
A
a 2 A a 2 A
2 A
a 2 A
a 2 A 3
} k
y e
c 2 A '
+ a 2
A '
A 2 ) k
a x a z
a y a z
7 1
2
2
2
2
2
2
2
2
a A 1
a A 1 - 3 ) . - a a A , 2
_
- a a A , 2
a A 3
3 A 3
-
a A 3
( -
a x 2
a y e
_
a z 2 + (
- 3 X 2
a y 2
a z 2
+
a x 2
-
a y 2
a z 2 )
k
2
+
( 9 A 2 + a A 2
+
a A 3
) i +
( a A 1 + a A 2
+
a A 3
) j
+ ( S A 1 +
a A 2 + a A s ) k
a x
a y a x
a z a x
a x a y
a y
a z a y a x a z a y a z a z 2
2
+
2
+
2
- ( ' 3 X 2
' 6 Y 2
a z 2 )
( A 1 i + A 2 j + A s k )
+ i
a - 3 A , + a A 2 + a A 3 ) +
j
a ( a A 1
+
a A 2
+
a A 3 )
+ k
a ( a A 1
+
a A 2
+
a A 3 )
a x ( a x a y
a z a y a x
a y
a z
a z a x
a y
a z
- v A + v (
a A 1
+ a A 2
+ a A 3 )
a x
a y
a z
_ - v A
+
I f d e s i r e d , t h e l a b o r o f w r i t i n g c a n b e s h o r t e n e d i n t h i s a s w e l l a s o t h e r d e r i v a t i o n s b y w r i t i n g o n l y t h e i
c o m p o n e n t s s i n c e t h e o t h e r s c a n b e o b t a i n e d b y s y m m e t r y .
T h e r e s u l t c a n a l s o b e e s t a b l i s h e d f o r m a l l y a s f o l l o w s . F r o m P r o b l e m 4 7 ( a ) , C h a p t e r 2 ,
( 1 )
P l a c i n g A = B = V a n d C = F ,
A x ( B x C )
= ( A - B ) C
V x ( V x F )
= V ( V - F ) - ( V - V ) F = V ( V . F ) -
V 2 F
N o t e t h a t t h e f o r m u l a ( 1 ) m u s t b e w r i t t e n s o t h a t t h e o p e r a t o r s A a n d B p r e c e d e t h e o p e r a n d C , o t h e r w i s e
t h e f o r m a l i s m f a i l s t o a p p l y .
3 0 . I f v = c o x r , p r o v e w = 2 c u r l v w h e r e w i s a c o n s t a n t v e c t o r .
i
j
c u r l y
= V x v
= V x ( c v x r )
= V x
N 1 6 0 2
W 3
x
y
z
= V x [ ( a 2 z - & s Y ) i + ( W 3 x - W 1 z ) j + ( w 1 Y - c v 2 x ) k ]
i
j
k
a a
a
I
=
2 ( c u 1 i + W O + c v 3 k )
=
2 c a
a x a y a z
w 2 z - c v 3 y W a x - W 1 Z
W 1 y - W 2 x
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7 2
G R A D I E N T , D I V E R G E N C E a n d C U R L
T h e n = 2 V x v = 2 c u r l v .
T h i s p r o b l e m i n d i c a t e s t h a t t h e c u r l o f a v e c t o r f i e l d h a s s o m e t h i n g t o d o w i t h r o t a t i o n a l p r o p e r t i e s o f
t h e f i e l d . T h i s i s c o n f i r m e d i n C h a p t e r 6 . I f t h e f i e l d F i s t h a t d u e t o a m o v i n g f l u i d , f o r e x a m p l e , t h e n a
p a d d l e w h e e l p l a c e d a t v a r i o u s p o i n t s i n t h e f i e l d w o u l d t e n d t o r o t a t e i n r e g i o n s w h e r e c u r l F # 0 , w h i l e i f
c u r l F = 0 i n t h e r e g i o n t h e r e w o u l d b e n o r o t a t i o n a n d t h e f i e l d F i s t h e n c a l l e d i r r o t a t i o n a l . A f i e l d w h i c h
i s n o t i r r o t a t i o n a l i s s o m e t i m e s c a l l e d a v o r t e x f i e l d .
2
3 1 . I f V E = 0 , V H = 0 , V X E
a H , V x H =
a t E
, s h o w t h a t E a n d H s a t i s f y
V 2 u = a i l
2
a H )
=
V x
a
x E
= V x v x H
_
a
a a E
(
( - -
(
)
-
( E )
- t
B y P r o b l e m 2 9 , V x ( V x E )
= -
V 2 E + V ( V - E )
_ - V E .
T h e n V E
S i m i l a r l y , V x ( V x H ) = V x
2
a E )
= a t ( v x E ) = a t ( -
a A )
_ -
H
B u t V x ( V X H )
=
- V 2 H +
V 2 H .
a 2
H
V 2h e n
H =
T h e g i v e n e q u a t i o n s a r e r e l a t e d t o M a x w e l l ' s e q u a t i o n s o f e l e c t r o m a g n e t i c t h e o r y .
T h e e q u a t i o n
2 2
2
2
i s c a l l e d t h e w a v e e q u a t i o n .
y 2
a x e + a + a z 2
a t e
M I S C E L L A N E O U S P R O B L E M S .
3 2 . ( a ) A v e c t o r V i s c a l l e d i r r o t a t i o n a l i f c u r l V = 0 ( s e e P r o b l e m 3 0 ) . F i n d c o n s t a n t s a , b , c s o t h a t
V
= ( x + 2 y + a z ) i
+ ( b x - 3 y - z ) j
+ ( 4 x + c y + 2 z ) k
i s i r r o t a t i o n a l .
( b ) S h o w t h a t V c a n b e e x p r e s s e d a s t h e g r a d i e n t o f a s c a l a r f u n c t i o n .
i j
( a ) c u r l V
= V x V
=
k
a a
a
a x
a y
a z
_
( c + l ) i + ( a - 4 ) j + ( b - 2 ) k
I x + 2 y + a z b x - 3 y - z
4 x + c y + 2 z '
T h i s e q u a l s z e r o w h e n a = 4 , b = 2 , c = - 1 a n d
V =
( x + 2 y + 4 z ) i + ( 2 x - 3 y - z ) j + ( 4 x - y + 2 z ) k
( b ) A s s u m e
V = V c = - i +
4 i
+ a O k
y
T h e n ( 1 )
a - = x + 2 y + 4 z ,
( 2 )
, a
= 2 x - 3 y - - z ,
( 3 ) a = 4 x - y + 2 z .
y
I n t e g r a t i n g ( 1 ) p a r t i a l l y w i t h r e s p e c t t o x , k e e p i n g y a n d z c o n s t a n t ,
2
( 4 )
c a
= 2 + 2 x y + 4 x z + f ( y , z )
w h e r e f ( y , z ) i s a n a r b i t r a r y f u n c t i o n o f y a n d z .
S i m i l a r l y f r o m ( 2 ) a n d ( 3 ) ,
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G R A D I E N T , D I V E R G E N C E a n d C U R L
( 5 )
( 6 )
0
2
2 x y -
3 y
- y z + g ( x , z )
=
4 x z - y z
+ z 2
+ h ( x , y ) .
C o m p a r i s o n o f ( 4 ) , ( 5 ) a n d ( 6 ) s h o w s t h a t t h e r e w i l l b e a c o m m o n v a l u e o f 0 i f w e c h o o s e
2
f ( Y , z ) _ -
3 y
+ z 2
2
s o t h a t
x 2
g ( x , z ) = 2 + z 2 ,
x 2
3 y 2
h ( x , y ) =
2 2
x 2
-
a
2
+ Z 2 + 2 x y + 4 x z - y z
7 3
N o t e t h a t w e c a n a l s o a d d a n y c o n s t a n t t o 0 . I n g e n e r a l i f V x V = 0 , t h e n w e c a n f i n d 4 s o t h a t
V = V 0 .
A v e c t o r f i e l d V w h i c h c a n b e d e r i v e d f r o m a s c a l a r f i e l d 0 s o t h a t V = V O i s c a l l e d a c o n s e r v a t i v e v e c t o r
f i e l d a n d 0 i s c a l l e d t h e s c a l a r p o t e n t i a l . N o t e t h a t c o n v e r s e l y i f V = V q 5 , t h e n V x V = 0 ( s e e P r o b . 2 7 a ) .
3 3 . S h o w t h a t i f O ( x , y , z ) i s a n y s o l u t i o n o f L a p l a c e ' s e q u a t i o n , t h e n V q b i s a v e c t o r w h i c h i s b o t h
s o l e n o i d a l a n d i r r o t a t i o n a l .
B y h y p o t h e s i s , 0 s a t i s f i e s L a p l a c e ' s e q u a t i o n
V 2 0
= 0 , i . e . V ( V 4 ) ) = 0 . T h e n V V i s s o l e n o i d a l ( s e e
P r o b l e m s 2 1 a n d 2 2 ) .
F r o m P r o b l e m 2 7 a , V x ( V V ) = 0 s o t h a t V V i s a l s o i r r o t a t i o n a l .
3 4 . G i v e a p o s s i b l e d e f i n i t i o n o f g r a d B .
A s s u m e B
a i i +
B 2 i j
+
a B 3
i k
+ a a l j i + a 2 j j + a a 3 j k
Y
Y
y
+
a B 1
k i +
a B 2
k J +
a B 3
k k
a z
a Z
a z
T h e q u a n t i t i e s i i , i j , e t c . , a r e c a l l e d u n i t d y a d s . ( N o t e t h a t i j , f o r e x a m p l e , i s n o t t h e s a m e a s j i . )
A q u a n t i t y o f t h e f o r m
a u i i + a 1 2 i j + a I s i k + a 2 2 j i + a 2 2 j j + a 2 3 j k + a 8 1 k i + a 3 2 k j + a 3 3 k k
i s c a l l e d a d y a d i c a n d t h e c o e f f i c i e n t s a l l , a 1 2 ,
. . .
a r e i t s c o m p o n e n t s . A n a r r a y o f t h e s e n i n e c o m p o -
n e n t s i n t h e f o r m
a l l a 1 2
a 1 3
a 2 1
a 2 2 a 2 3
a 3 1
C 3 2
a 3 3
= B 1 i + B 2 J + B 3 k . F o r m a l l y , w e c a n d e f i n e g r a d B a s
V B =
( a x
i +
j +
a z k ) ( B 1 i + B 2 j + B 3 k )
Y
i s c a l l e d a 3 b y 3 m a t r i x . A d y a d i c i s a g e n e r a l i z a t i o n o f a v e c t o r .
S t i l l f u r t h e r g e n e r a l i z a t i o n l e a d s t o
t r i a d i c s w h i c h a r e q u a n t i t i e s c o n s i s t i n g o f 2 7 t e r m s o f t h e f o r m a 1 1 1 i i i + a 2 1 1 j i i + . . . .
A s t u d y o f h o w
t h e c o m p o n e n t s o f a d y a d i c o r t r i a d i c t r a n s f o r m f r o m o n e s y s t e m o f c o o r d i n a t e s t o a n o t h e r l e a d s t o t h e s u b -
j e c t o f t e n s o r a n a l y s i s w h i c h i s t a k e n u p i n C h a p t e r 8 .
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7 4
G R A D I E N T , D I V E R G E N C E a n d C U R L
3 5 . L e t a v e c t o r A b e d e f i n e d b y A = A l i + A 2 j + A 3 k a n d a d y a d i c f i b y
0 =
a 1 1 i i + a 1 2 1 j + a 1 3 i k + a 2 1 j i + a 2 2 j j + a 2 3 j k + a 3 l k i + a 3 2 k j + a 3 3 k k
G i v e a p o s s i b l e d e f i n i t i o n o f A - 0 .
F o r m a l l y , a s s u m i n g t h e d i s t r i b u t i v e l a w t o h o l d ,
( A l i + A 2 j + A 3 k ) 4 > = A 1 i 4 i + A 2 j 4 i + A 3 k 4 b
A s a n e x a m p l e , c o n s i d e r i - 4 ' . T h i s p r o d u c t i s f o r m e d b y t a k i n g t h e d o t p r o d u c t o f i w i t h e a c h t e r m o f
4 ) a n d a d d i n g r e s u l t s . T y p i c a l e x a m p l e s a r e i
a l a i i , i
a 1 2 i j ,
i
a 2 1 j i ,
i
a 3 2 k j , e t c .
I f w e g i v e m e a n -
i n g t o t h e s e a s f o l l o w s
i
a s s i i
i a l 2 i j
i a 2 , j i
i
r i n k j
=
a l l ( i
1 ) 1
=
a 1 2 ( i
i ) j
= a 2 1 0 - h i
a 3 2 ( i k ) j
=
a l l i
s i n c e i
i
= 1
=
a i l j
s i n c e
i , 1 = 1
=
p
s i n c e
i j
= 0
= 0
s i n c e i k = 0
a n d g i v e a n a l o g o u s i n t e r p r e t a t i o n t o t h e t e r m s o f j
4 ) a n d k 4 0 , t h e n
A 4 )
=
A , ( a l l i + a 1 2 j + a 1 3 k ) + A 2 ( a 2 1 i + a 2 2 j + a 2 3 k ) + A 3 ( a 3 1 i + a 3 2 j +
a 3 3 k )
_ ( A l a s , + A 2 a 2 , + A 3 a 3 1 ) i + ( A 1 a 1 2 + A 2 a 2 2 + A 3 a 3 2 ) j + ( A l a i , + A 2 a 2 3 + A s a s s ) k
w h i c h i s a v e c t o r .
3 6 . ( a ) I n t e r p r e t t h e s y m b o l A - V . ( b ) G i v e a p o s s i b l e m e a n i n g t o ( A V ) B .
( c ) I s i t p o s s i b l e t o
w r i t e t h i s a s A V B w i t h o u t a m b i g u i t y ?
( a ) L e t A = A l i + A 2 j + A s k .
T h e n , f o r m a l l y ,
A - V =
( A s i + A 2 j + A 3 k ) ( a i
+ a
i
+ a z k )
a
a
A l
a
a x
+
A 2 a y
+
A s
a z
i s a n o p e r a t o r . F o r e x a m p l e ,
( A V )
( A 1 a x + A 2
a
+ A s
a ) =
A l
+ A 2
a
+
A s
a
y
N o t e t h a t t h i s i s t h e s a m e a s A V c .
( b ) F o r m a l l y , u s i n g ( a ) w i t h 0 r e p l a c e d b y B = B 1 i + B 2 j + B 3 k ,
( A 0 ) B
( A ,
a x
+
A 2 - a y
+
A 3 a z ) B
= A l
a B x + A 2 a B
+ A 3
a B
_
a B 1 a B 1 a B 1 a B 2 a B 2 a B 2 a B 3
a B 3
a
( A l
a x
+ A 2
a y
+ A 3
a z )
i
+
( A , - : a x
+ ( A l
a x
+ A 2
a y
+ A 3
- a z )
k
( c ) U s e t h e i n t e r p r e t a t i o n o f V B a s g i v e n i n P r o b l e m 3 4 . T h e n , a c c o r d i n g t o t h e s y m b o l i s m e s t a b l i s h e d
i n P r o b l e m 3 5 ,
A V B =
( A 1 i + A 2 j + A s k ) V B = A 1 i V B + A 2 j V B + A s k V B
a B 1 ,
- a a B . 2 a B 3
a B 1
a B 2
- 6 B 3
a B l
.
a B 2
B s
A , ( a x l + a x j +
a x
k ) + A 2 ( a y i + a y j + a y k ) + A s ( a z l + a z j + a z k )
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G R A D I E N T , D I V E R G E N C E a n d C U R L
7 5
w h i c h g i v e s t h e s a m e r e s u l t a s t h a t g i v e n i n p a r t ( b ) .
I t f o l l o w s t h a t ( A V ) B = A V B w i t h o u t a m b i -
g u i t y p r o v i d e d t h e c o n c e p t o f d y a d i c s i s i n t r o d u c e d w i t h p r o p e r t i e s a s i n d i c a t e d .
3 7 . I f A = 2 y z i - x 2 y j + x z 2 k , B = x 2 i + y z i - x y k a n d 0 = 2 x 2 y z 3 ,
( a ) ( b )
A x V 0 .
[ ( 2 y z i - x 2 y j + x z 2 k )
( i
i
+
i
+
k
- a y
) ] q 5
a z
( 2 y z
-
x 2 y +
a x
- a y
2 y z a ( 2 x 2 y z 3 )
-
x
x z 2 a z )
( 2 x 2 y z 3 )
x 2 y a ( 2 x y z 3 )
+
x z 2
a z
( 2 x 2 y z 3 )
Y
( 2 y z ) ( 4 x y z 3 ) -
( x 2 y ) ( 2 x 2 2 3 )
+
( x z 2 ) ( 6 x y z 2 )
8 x y 2 z 4 -
2 7 j 4 y z 3
+
6 x 3 y Z 4
( b )
( 2 y z i - x 2 y j + x z 2 k ) ( a ( p i
+
a 0 j
+
0 k )
8 x y 2 z 4 - 2 x 4 y z 3
+
6 x 3 y z 4
O x
a y
a z
( 2 y z i - x 2 y j + x z 2 k )
( 4 x y z 3 i + 2 x 2 z 3 j + f i x y z 2 k )
C o m p a r i s o n w i t h ( a ) i l l u s t r a t e s t h e r e s u l t ( A V ) 0 = A V 0 .
( c ) ( B - V ) A
=
[ ( x 2 i + y z j - x y k )
( a i
=
( X 2 - 1
a - ) A
Y z
-
x Y
a x
a
,
Y
+ a
j +
a z k ) ] A
Y
x 2 a A
+
z
a A
_ x
a A
a x
Y a y
Y
a z
x 2 ( - 2 x y j + z 2 k )
+
y z ( 2 z i - x 2 j ) - x y ( 2 y i + 2 x z k )
( 2 y z 2 - 2 x y 2 ) i - ( 2 x 3 y + x 2 Y z ) j
+
( x 2 2 2 - 2 x 2 y z ) k
F o r c o m p a r i s o n o f t h i s w i t h B - V A ,
s e e P r o b l e m 3 6 ( c ) .
( d ) ( A x V )
[ ( 2 y z i - x 2 y j + x z 2 k )
x ( a x l + a j +
a k ) 1
Y
i
i
k
2 y z
- x 2 y
x z 2
0
+
j ( x Z 2 a x
- 2 y z
a Z )
+
f i n d
k ( 2 y z
+
X 2 Y a x ) ] 0
Y
( x z 2
- - 2 y z : a
) j
+
( 2 y z a + x 2 y ) k
Y
a
- ( x y
a z
+
x z 2
a
) i
+
Y
a
a a
a x a y
a z
[ i ( - x 2 y -
x z 2 -
)
Y
a d )
a d )
= - (
6 X 4 Y 2
z
2
+ 2 x 3 z 5 ) i +
( 4 x 2 y z 5
- 1 2 x 2 y 2 z 3 ) j
+
( 4 x 2 y z 4 + 4 x 3 y 2 z 3 ) k
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G R A D I E N T , D I V E R G E N C E a n d C U R L
( e ) A x V O _
( 2 y z i - x 2 Y j + x z 2 k ) x
( a i
+
a j
+
_ a 4 ) a
k )
Y
i
j
k
2 y z
- x 2 y x z 2
a x
a y
a z
a a
a a
0
I N V A R I A N C E
2 2
2
_ ( - x Y
a z -
x z . a y ) i
+
( x z a x - 2 y z a z ) j
+
( 2 y z a y + x y
a x
) k
- ( 6 x y 2 z 2 + 2 x 3 z 5 ) i
+
( 4 x 2 y z 5 - 1 2 x 2 y 2 z 3 ) j
+
( 4 x 2 y z 4 + 4 x 3 y 2 z 3 ) k
C o m p a r i s o n w i t h ( d ) i l l u s t r a t e s t h e r e s u l t ( A x V ) = A x V q .
3 8 . T w o r e c t a n g u l a r x y z a n d x ' y ' z ' c o o r d i n a t e s y s t e m s h a v i n g t h e s a m e o r i g i n a r e r o t a t e d w i t h r e -
s p e c t t o e a c h o t h e r .
D e r i v e t h e t r a n s f o r m a t i o n e q u a t i o n s b e t w e e n t h e c o o r d i n a t e s o f a p o i n t i n
t h e t w o s y s t e m s .
L e t r a n d r ' b e t h e p o s i t i o n v e c t o r s o f a n y p o i n t P i n t h e t w o s y s t e m s ( s e e f i g u r e o n p a g e 5 8 ) . T h e n
s i n c e
r = r f ,
( 1 )
x + y ' j ' + z ' k '
=
x i + y j
N o w f o r a n y v e c t o r A w e h a v e ( P r o b l e m 2 0 , C h a p t e r 2 ) ,
A =
i '
+
j ' +
( A - k ' ) k '
T h e n l e t t i n g A = i , j , k i n s u c c e s s i o n ,
( 2 )
i
=
1 1 1 1 '
+
1 2 1 j f
+
1 3 1 k '
i
=
( j - k ' ) k '
=
1 1 2 1 '
+
1 2 2 i t
+
1 3 2 k '
k
=
( k . i ' ) i '
+
( k j ' ) j '
+
( k k ' ) k '
=
1 1 3 i '
+
1 2 3 j '
+
1 3 3 k '
S u b s t i t u t i n g e q u a t i o n s ( 2 ) i n ( 1 ) a n d e q u a t i n g c o e f f i c i e n t s o f i ' , j ' , k ' w e f i n d
( 3 )
x ' = 1 1 1 x + 1 1 2 Y
+ 1 1 3 z ,
t h e r e q u i r e d t r a n s f o r m a t i o n e q u a t i o n s .
3 9 . P r o v e i ' =
1 1 1 i + 1 1 2 j + 1 1 3 k
j f
=
1 2 1 i + 1 2 2 j + 1 2 3 k
k ' =
1 3 1 1 + 1 3 2 j + 1 3 3 k
y ' = 1 2 1 x + 1 2 2
Y + 1 2 3 z ,
z ' = 1 3 1 X + 1 3 2 Y + 1 3 3 Z
F o r a n y v e c t o r A w e h a v e
A = ( A i ) i
+ ( A . j ) j
+ ( A k ) k .
T h e n l e t t i n g A = i ' , j ' , k ' i n s u c c e s s i o n ,
1 '
_ ( i ' i ) i
+
( i ' j ) j
+
( i ' k ) k
=
j r
=
k ' _
( k ' i ) i + ( k ' j ) i
+
( k ' k ) k
=
1 1 1 i + 1 1 2 i + 1 1 3 k
1 2 1 i
+ 1 2 2 j
+ 1 2 3 k
1 3 1 1
+ 1 3 2 i
+ 1 3 3 k
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G R A D I E N T , D I V E R G E N C E a n d C U R L
S U P P L E M E N T A R Y P R O B L E M S
4 2 .
I f
= 2 x z 4 - x 2 y ,
f i n d V q a n d
I V ( f
a t t h e p o i n t ( 2 , - 2 , - 1 ) .
A n s .
1 0 i - 4 j - 1 6 k , 2 V 9 3
4 3 .
I f A = 2 x 2 i - 3 y z j + x z 2 k a n d q 5 = 2 z - x 3 y , f i n d A o g 5 a n d A x V c a t
t h e p o i n t ( 1 , - 1 , 1 ) .
A n s .
5 , 7 i - j - I l k
4 4 .
I f F = x 2 z + e
y / x
a n d G = 2 z 2 y - x y 2 , f i n d ( a ) V ( F + G ) a n d ( b ) V ( F G ) a t t h e p o i n t ( 1 , 0 , - 2 ) .
A n s .
( a ) - 4 i + 9 j + k , ( b ) - 8 j
4 5 . F i n d V I r
I 3 .
A n s .
3 r r
4 6 . P r o v e V f ( r ) =
f ( r ) r
r
4 7 . E v a l u a t e V ( 3 r 2 -
6
) .
A n s .
( 6 -
2 r - 3 / 2
-
2 r - 7 / 3 )
r
v c
4 8 .
I f V U = 2 r 4 r , f i n d U .
A n s . r e / 3 + c o n s t a n t
4 9 . F i n d 0 ( r ) s u c h t h a t V p = s a n d g ( 1 ) = 0 .
A n s . 0 ( r ) = 3 ( 1 -
r
r
2
2 2
5 0 . F i n d V q w h e r e q = ( x 2 + y 2 + Z 2 ) e
- , I X
+ y + z
A n s .
( 2 - r ) a - r r
5 1 .
I f V V = 2 x y z 3 i + x 2 z 3 j + 3 x 2 y z 2 k ,
f i n d O ( x , y , z ) i f
4 .
A n s .
= x 2 y z 3 + 2 0
5 2 .
I f V o _ ( y 2 _ 2 x y z 3 ) i + ( 3 + 2 x y _ x 2 z ° ) j + ( 6 z 3 - 3 x 2 y z 2 ) A , f i n d
A n s . 0 = x y 2 - x 2 y z 3 + 3 y + ( 3 / 2 ) z 4 + c o n s t a n t
5 3 . I f U i s a d i f f e r e n t i a b l e f u n c t i o n o f x , y , z ,
p r o v e W . d r = d U .
5 4 . I f F i s a d i f f e r e n t i a b l e f u n c t i o n o f x , y , z , t w h e r e x , y , z a r e d i f f e r e n t i a b l e f u n c t i o n s o f t , p r o v e t h a t
d F
_
a F
d r
+ O F
d t a t
d t
5 5 . I f A i s a c o n s t a n t v e c t o r , p r o v e V ( r A ) = A .
5 6 .
I f A ( x , y , z ) = A l i + A 2 j + A 3 k , s h o w t h a t d A =
5 7 . P r o v e v ( F ) =
G V F -
2
F V G
i f G 4 0 .
G G
5 8 . F i n d a u n i t v e c t o r w h i c h i s p e r p e n d i c u l a r t o t h e s u r f a c e o f t h e p a r a b o l o i d o f r e v o l u t i o n z = x 2 + y
2
a t t h e
p o i n t ( 1 , 2 , 5 ) .
A n s .
2 i + 4 ] - k
± 2 1
5 9 . F i n d t h e u n i t o u t w a r d d r a w n n o r m a l t o t h e s u r f a c e ( x - 1 ) 2 + y 2 + ( z + 2 )
2
= 9 a t t h e p o i n t ( 3 , 1 , - 4 ) .
A n s .
( 2 i + i - 2 k ) / 3
6 0 . F i n d a n e q u a t i o n f o r t h e t a n g e n t p l a n e t o t h e s u r f a c e x z 2 + x 2 y = z - 1 a t t h e p o i n t ( 1 , - 3 , 2 ) .
A n s .
2 x - y - 3 z + I = 0
6 1 . F i n d e q u a t i o n s f o r t h e t a n g e n t p l a n e a n d n o r m a l l i n e t o t h e s u r f a c e z = x 2 + y 2 a t t h e p o i n t ( 2 , - 1 , 5 ) .
A n s . 4 x - 2 y - z = 5 , x 4 2 = y
2 1 - z r 1 5
o r x = 4 t + 2 , y = - 2 t - 1 , z = - t + 5
6 2 . F i n d t h e d i r e c t i o n a l d e r i v a t i v e o f = 4 x z 3 - 3 x 2 y 2 z a t ( 2 , - 1 , 2 ) i n t h e d i r e c t i o n 2 i - 3 j + 6 k .
A n s . 3 7 6 / 7
6 3 . F i n d t h e d i r e c t i o n a l d e r i v a t i v e o f P = 4 e
2 x ^ y + Z
a t t h e p o i n t ( 1 , 1 , - 1 ) i n a d i r e c t i o n t o w a r d t h e p o i n t
( - 3 , 5 , 6 ) .
A n s . - 2 0 / 9
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G R A D I E N T , D I V E R G E N C E a n d C U R L
7 9
6 4 .
I n w h a t d i r e c t i o n f r o m t h e p o i n t ( 1 , 3 , 2 ) i s t h e d i r e c t i o n a l d e r i v a t i v e o f ( P = 2 x z - y 2 a m a x i m u m ' s W h a t i s
t h e m a g n i t u d e o f t h i s m a x i m u m ?
A n s .
I n t h e d i r e c t i o n o f t h e v e c t o r 4 i - 6 j + 2 k , 2 v " 1 4
6 5 . F i n d t h e v a l u e s o f t h e c o n s t a n t s a , b , c s o t h a t t h e d i r e c t i o n a l d e r i v a t i v e o f 0 = a x y 2 + b y z + c z 2 x 3 a t
( 1 , 2 , - 1 ) h a s a m a x i m u m o f m a g n i t u d e 6 4 i n a d i r e c t i o n p a r a l l e l t o t h e z a x i s .
A n s . a = 6 ,
b = 2 4 , c = - 8
6 6 . F i n d t h e a c u t e a n g l e b e t w e e n t h e s u r f a c e s x y 2 z = 3 x + z 2 a n d 3 x 2 . - y 2 + 2 z = 1
a t t h e p o i n t ( 1 , - 2 , 1 ) .
A n s . a r c c o s
1
= a r c c o s 1 4
= 7 9 ° 5 5 1
6 7 . F i n d t h e c o n s t a n t s a a n d b s o t h a t t h e s u r f a c e a x 2 - b y z = ( a + 2 ) x w i l l b e o r t h o g o n a l t o t h e s u r f a c e
4 x 2 y + z 3 = 4 a t t h e p o i n t ( 1 , - 1 , 2 ) .
A n s . a = 5 / 2 , b = 1
6 8 .
( a ) L e t u a n d v b e d i f f e r e n t i a b l e f u n c t i o n s o f x , y a n d z . S h o w t h a t a n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n
t h a t u a n d v a r e f u n c t i o n a l l y r e l a t e d b y t h e e q u a t i o n F ( u , v ) = 0 i s t h a t V u x V v = 0 .
( b ) D e t e r m i n e w h e t h e r u = a r e t a n x + a r e t a n y a n d v = 1 z y a r e f u n c t i o n a l l y r e l a t e d .
A n s . ( b ) Y e s ( v = t a n u )
6 9 .
( a ) S h o w t h a t a n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n t h a t u ( x , y , z ) , v ( x , y , z ) a n d w ( x , y , z ) b e f u n c t i o n a l l y r e -
l a t e d t h r o u g h t h e e q u a t i o n F ( u , v , w ) = 0 i s V u - V v X V w = 0 .
( b ) E x p r e s s V u - V v x V w i n d e t e r m i n a n t f o r m . T h i s d e t e r m i n a n t i s c a l l e d t h e J a c o b i a n o f u , v , w w i t h r e -
s p e c t t o x , y , z a n d i s w r i t t e n
u , v , w
o r J (
u , v , w
) .
a ( x , y , z )
x , y , z
( c ) D e t e r m i n e w h e t h e r u = x + y + z , v = x 2 + y 2 + z 2 a n d w = x y + y z + z x a r e f u n c t i o n a l l y r e l a t e d .
a u
a u a u
a x
a y
a z
A n s .
( b )
a v a v a v
a x
a y
a z
( c ) Y e s ( u 2 - v - 2 w = 0 )
a w a w
a w
a x a y
a z
7 0 .
I f A = 3 x y z 2 i + 2 x y 3 j - - x y z k a n d d ) = 3 x 2 - y z ,
f i n d
( a ) V A ,
( b ) A V O , ( c ) V - ( 0 A ) , ( d )
a t t h e p o i n t A n s .
( a ) 4 , ( b ) - 1 5 , ( c ) 1 , ( d ) 6
7 1 . E v a l u a t e d i v ( 2 x 2 z i - x y 2 z j + 3 y z 2 k ) .
A n s . 4 x z - - 2 x y z + b y z
7 2 .
I f = 3 x 2 z - y 2 z 3 + 4 x 3 y + 2 x - 3 y - 5 , f i n d
V 2 c b .
A n s .
6 z + 2 4 x y - 2 z 3 - - 6 y 2 z
2
7 3 . E v a l u a t e V ( l n r ) . A n s .
1 / r 2
7 4 . P r o v e
V 2 r n = n ( n + 1 ) r n - 2 w h e r e n i s a c o n s t a n t .
7 5 .
I f F = ( 3 x 2 y
4
3
y - z ) i + ( x z + y ) j - 2 2 z ' k ,
f i n d V ( V F ) a t t h e p o i n t ( 2 , - 1 , 0 ) .
A n s . - 6 i + 2 4 j - 3 2 k
7 6 .
I f w i s a c o n s t a n t v e c t o r a n d v = r v x r ,
p r o v e t h a t d i v v = 0 .
7 7 . P r o v e
V 2 ( 0 & )
= 0
V 2 q + 2 V O - V q + 0 V 2 0 .
7 8 .
I f U = 3 x 2 y , V = x z 2 - 2 y e v a l u a t e g r a d [ ( g r a d U ) ( g r a d V ) ] .
A n s . ( 6 y z 2 - - 1 2 x ) i + 6 x z 2 j + 1 2 x y z k
7 9 . E v a l u a t e V ( r 3 r ) .
A n s . 6 r 3
8 0 . E v a l u a t e V [ r V ( 1 / r 3 ) ] .
A n s .
3 r - 4
8 1 . E v a l u a t e
V 2 [ V _
( r / r 2 ) ] .
A n s .
2 r
- 4
8 2 .
I f A = r / r , ,
f i n d g r a d d i v A .
A n s . - 2 r _ 3 r
8 3 . ( a ) P r o v e V 2 f ( r ) =
d r f +
2
d f .
( b ) F i n d f ( r ) s u c h t h a t V 2 f ( r ) = 0 .
A n s . f ( r ) = A + B / r w h e r e A a n d B a r e a r b i t r a r y c o n s t a n t s .
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8 0
G R A D I E N T , D I V E R G E N C E a n d C U R L
4
z 2 i + 4 x 3 2 2 j - 3 x 2 y 2 k i s s o l e n o i d a l .
4 . P r o v e t h a t t h e v e c t o r A = 3 y
8 5 . S h o w t h a t A = ( 2 x 2 + 8 x y 2 z ) i + ( 3 x 3 y - 3 x y ) j - ( 4 y 2 z 2 + 2 x 3 2 ) k i s n o t s o l e n o i d a l b u t B = x y z 2 A i s
s o l e n o i d a l .
8 6 . F i n d t h e m o s t g e n e r a l d i f f e r e n t i a b l e f u n c t i o n f ( r ) s o t h a t f ( r ) r i s s o l e n o i d a l .
A n s . f ( r ) = C / r 3 w h e r e C i s a n a r b i t r a r y c o n s t a n t .
8 7 . S h o w t h a t t h e v e c t o r f i e l d V =
- x
1 - y j i s a 1 ' s i n k f i e l d " . P l o t a n d g i v e a p h y s i c a l i n t e r p r e t a t i o n .
x 2 + y 2
8 8 .
I f U a n d V a r e d i f f e r e n t i a b l e s c a l a r f i e l d s , p r o v e t h a t V U x V V i s s o l e n o i d a l .
8 9 .
I f A = 2 x z 2 i - y z j + 3 x z 3 k a n d c b = x 2 y z ,
f i n d
( a ) V x A , ( b ) c u r l ( O A ) ,
( c ) V x ( V x A ) , ( d ) V [ A - c u r l A ] , ( e ) c u r l g r a d
a t t h e p o i n t ( 1 , 1 , 1 ) .
A n s .
( a ) i + j ,
( b ) 5 i - 3 j - 4 k , ( c ) 5 i + 3 k ,
( d ) - 2 1 + j + 8 k ,
( e ) 0
9 0 .
I f F = x 2 y z , G = x y - 3 z 2 , f i n d
( a ) V
( b ) V - [ ( V F ) x ( V G ) ] ,
( c ) V x [ ( V F ) x ( V G ) ] .
A n s .
( a ) ( 2 y 2 z + 3 x 2 z - 1 2 x y z ) i + ( 4 x y z - 6 x 2 z ) j + ( 2 x y 2 + x 3 - 6 x 2 y ) k
( b ) 0
( c ) ( x 2 z - 2 4 x y z ) i - ( 1 2 x 2 2 + 2 x y z ) j + ( 2 x y 2 + 1 2 y z 2 + x 3 ) k
9 1 . E v a l u a t e V x ( r / r 2 ) .
A n s . 0
9 2 . F o r w h a t v a l u e o f t h e c o n s t a n t a w i l l t h e v e c t o r A = ( a x y - - - z 3 ) i
+ ( a - 2 ) x
2
j
+ ( 1 - a ) x z 2 k h a v e i t s
c u r l i d e n t i c a l l y e q u a l t o z e r o 9
A n s . a = 4
9 3 . P r o v e c u r l ( 0 g r a d j . ) = 0 .
9 4 . G r a p h t h e v e c t o r f i e l d s A = x i + y j
a n d B = y i - x j . C o m p u t e t h e d i v e r g e n c e a n d c u r l o f e a c h v e c t o r
f i e l d a n d e x p l a i n t h e p h y s i c a l s i g n i f i c a n c e o f t h e r e s u l t s o b t a i n e d .
9 5 .
I f A = x 2 z i + y z 3 j - 3 x y k ,
B ' = y 2 i - y z j + 2 x k a n d
= 2 x 2 + y z , f i n d
( a ) A - ( V ( P ) ,
( b ) ( A - V ) g b , ( c ) ( A . V ) B ,
( d )
( e )
A n s .
( a ) 4 x 3 z + y z 4 - 3 x y 2 ,
( b ) 4 x 3 z + y z 4 - 3 x y 2 ( s a m e a s ( a ) ) ,
( c ) 2 y 2 z 3 i + ( 3 x y 2 - y z 4 ) j + 2 x 2 z k ,
( d ) t h e o p e r a t o r ( x 2 y 2 z i - x 2 y z 2 j + 2 x 3 z k )
+ ( y 3 z 3 1 - y 2 z 4 j + 2 x y z 3 k )
a x
- a y
+ ( - 3 x y 3 i + 3 x y 2 z j - 6 x 2 y k ) a
( e ) ( 2 x y 2 z + y 2 z 3 ) i - ( 2 x y z 2 + y z 4 ) j + ( 4 x 2 2 + 2 x z 3 ) k
9 6 .
I f A = y z 2 i - 3 x z 2 j + 2 x y z k ,
B = 3 x i + 4 z j - x y k a n d O = x y z ,
f i n d
( a ) A x ( V f ) ,
( b ) ( A x V ) ( P ,
( c ) ( V x A ) x B ,
( d ) B . V x A .
A n s .
( a ) - 5 x 2 y z 2 i
+ x y 2 z 2 j
+ 4 x y z 3 k
( b ) - 5 x 2 y z 2 i
+ x y 2 z 2 j
+ 4 x y z 3 k ( s a m e a s ( a ) )
( c ) 1 6 x 3 i
+ ( 8 x 2 y z - 1 2 x z 2 ) j + 3 2 x z 2 k
( d ) 2 4 x 2 2 + 4 x y z 2
9 7 . F i n d A x ( V x B ) a n d ( A x V ) , ' c B a t t h e p o i n t ( 1 , - 1 , 2 ) , i f A = x z 2 i + 2 y j - 3 x z k a n d B = 3 x z i
+ 2 y z j - z 2 k .
A n s . A x ( V x B ) = 1 8 i -
1 2 j + 1 6 k , ( A x V ) x B = 4 j + 7 6 k
9 8 . P r o v e ( v - V ) v = 2 V v 2 - v x ( V x v ) .
9 9 . P r o v e V ( A x B ) = B - ( V x A ) - A - ( O x B ) .
1 0 0 . P r o v e V x ( A x B ) = B ( V . A ) -
A ( V - B ) .
1 0 1 . P r o v e V ( A B ) =
B x ( V x A ) + A x ( V x B ) .
1 0 2 . S h o w t h a t A = ( 6 x y + z 3 ) i + ( 3 x 2 - Z ) j + ( 3 x 2 2 - y ) k i s i r r o t a t i o n a l . F i n d c p s u c h t h a t A
= 0 .
A n s . c P = 3 x 2 y + x z 3 - y z + c o n s t a n t
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G R A D I E N T , D I V E R G E N C E a n d C U R L
8 1
1 0 3 . S h o w t h a t E = r / r 2 i s i r r o t a t i o n a l . F i n d 0 s u c h t h a t E
a n d s u c h t h a t 0 ( a ) = 0 w h e r e a > 0 .
A n s . g b = I n ( a 1 r )
1 0 4 . I f A a n d B a r e i r r o t a t i o n a l , p r o v e t h a t A I B i s s o l e n o i d a l .
1 0 5 .
I f f ( r ) i s a i f f e r e n t i a b l e , p r o v e t h a t f ( r ) r i s i r r o t a t i o n a l .
1 0 6 .
I s t h e r e a d i f f e r e n t i a b l e v e c t o r f u n c t i o n V s u c h t h a t ( a ) c u r l V = r ,
( b ) c u r l V = 2 i + j + 3 k ? I f s o , f i n d V .
A n s . ( a t N o , ( b ) V = 3 x j + ( 2 y - - x ) k + V O , w h e r e 0 i s a n a r b i t r a r y t w i c e d i f f e r e n t i a b l e f u n c t i o n .
1 0 7 . S h o w t h a t s o l u t i o n s t o M a x w e l l ' s e q u a t i o n s
V x H =
c
a E
V x E =
a H ,
V - H = 0 ,
V . E = 4 7 T p
w h e r e p i s a f u n c t i o n o f x , y , z a n d c i s t h e v e l o c i t y o f l i g h t , a s s u m e d c o n s t a n t , a r e g i v e n b y
E = - V V -
c
a ` A ,
H = V x A
w h e r e A a n d 0 , c a l l e d t h e v e c t o r a n d s c a l a r p o t e n t i a l s r e s p e c t i v e l y , s a t i s f y t h e e q u a t i o n s
, 2
1
2 2
2
1 ) V - A +
C
a
= 0 ,
( 2 ) V
c 2
a t _
( 3 ) V 2 A = c 2 a t
1 0 8 . ( a ) G i v e n t h e d y a d i c 4 = i i + j j + k k , e v a l u a t e
r ( 1 r ) a n d ( r - J ' ) r .
( b ) I s t h e r e a n y a m b i g u i t y i n
w r i t i n g r . 4 r ?
( c ) W h a t d o e s r 1
r = 1 r e p r e s e n t g e o m e t r i c a l l y ?
A n s . ( a ) r
( c l
r ) _ ( r . I ) r = x 2 + y 2 + z 2 ,
( b ) N o , ( c ) S p h e r e o f r a d i u s o n e w i t h c e n t e r a t t h e o r i g i n .
1 0 9 .
( a ) I f A = x z i - y 2 j + y z 2 k a n d B = 2 z 2 i - x y j + y 3 k , g i v e a
p o s s i b l e s i g n i f i c a n c e t o ( A x V ) B a t
t h e p o i n t
( b ) I s i t p o s s i b l e t o w r i t e t h e r e s u l t a s A I ( V B ) b y u s e o f d y a d i c s ?
A n s . ( a ) - - 4 1 i - i j + 3 i k - j j - 4 j i + 3 k k
( b ) Y e s , i f t h e o p e r a t i o n s a r e s u i t a b l y p e r f o r m e d .
1 1 0 . P r o v e t h a t c a ( x , y , z ) = x 2 + y 2 + z 2
i s a s c a l a r i n v a r i a n t u n d e r a r o t a t i o n o f a x e s .
1 1 1 .
I f A ( x , y , z ) i s a n i n v a r i a n t d i f f e r e n t i a b l e v e c t o r f i e l d w i t h r e s p e c t t o a r o t a t i o n o f a x e s , p r o v e t h a t ( a ) d i v A
a n d ( b ) c u r l A a r e i n v a r i a n t s c a l a r a n d v e c t o r f i e l d s r e s p e c t i v e l y u n d e r t h e t r a n s f o r m a t i o n .
1 1 2 . S o l v e e q u a t i o n s ( 3 ) o f S o l v e d P r o b l e m 3 8 f o r x , y , z i n t e r m s o f x ' , y ' , z ' .
A n s . x = 1 1 1 x ' + 1 2 1 Y ' + 1 3 1 Z ' ,
Y = 1 1 2 x ' + 1 2 2 Y I + 1 3 2 Z ' ,
Z = 1 1 3 x ' + 1 2 3 Y ' + 1 : 3 3 Z '
1 1 3 .
I f A a n d B a r e i n v a r i a n t u n d e r r o t a t i o n s h o w t h a t A B a n d A I B a r e a l s o i n v a r i a n t .
1 1 4 . S h o w t h a t u n d e r a r o t a t i o n
. a x
i
+ J a y + k a
z
1 1
a x ' +
j / a y I + k '
a z '
Q '
1 1 5 . S h o w t h a t t h e L a p l a c i a n o p e r a t o r i s i n v a r i a n t u n d e r a r o t a t i o n .
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O R D I N A R Y I N T E G R A L S O F V E C T O R S . L e t R ( u ) = R 1 ( u ) i + R 2 ( u ) j + R 3 ( u ) k b e a v e c t o r d e p e n d i n g
o n a s i n g l e s c a l a r v a r i a b l e u , w h e r e R 1 ( u ) , R 2 ( u ) , R 3 ( u ) a r e
s u p p o s e d c o n t i n u o u s i n a s p e c i f i e d i n t e r v a l . T h e n
f R ( u ) d u
=
i R 1 ( u ) d u +
i f
R 2 ( u ) d u + k
R s ( u ) d u
i s c a l l e d a n i n d e f i n i t e i n t e g r a l o f R ( u ) .
I f t h e r e e x i s t s a v e c t o r S ( u ) s u c h t h a t R ( u ) = d u ( S ( u ) ) , t h e n
f R ( u )
d u
=
f , / ( s ( u ) ) d u = S ( u ) + c
w h e r e c i s a n a r b i t r a r y c o n s t a n t v e c t o r i n d e p e n d e n t o f u . T h e d e f i n i t e i n t e g r a l b e t w e e n l i m i t s u = a
a n d u = b c a n i n s u c h c a s e b e w r i t t e n
f b
a
R ( u ) d u
=
f a
d
d u ( S ( u ) )
d u
S ( u ) + c I = S ( b ) - S ( a )
a
T h i s i n t e g r a l c a n a l s o b e d e f i n e d a s a l i m i t o f a s u m i n a m a n n e r a n a l o g o u s t o t h a t o f e l e m e n t a r y i n -
t e g r a l c a l c u l u s .
L I N E I N T E G R A L S . L e t r ( u ) = x ( u ) i + y ( u ) j + z ( u ) k ,
w h e r e r ( u ) i s t h e p o s i t i o n v e c t o r o f ( x , y , z ) ,
d e f i n e a c u r v e C j o i n i n g p o i n t s P 1 a n d P 2 , w h e r e u = u l a n d u = u 2 r e s p e c t i v e l y .
W e a s s u m e t h a t C i s c o m p o s e d o f a f i n i t e n u m b e r o f c u r v e s f o r e a c h o f w h i c h r ( u ) h a s a c o n t i n -
u o u s d e r i v a t i v e . L e t A ( x , y , z ) = A 1 i + A 2 j + A 3 k b e a v e c t o r f u n c t i o n o f p o s i t i o n d e f i n e d a n d c o n -
t i n u o u s a l o n g C . T h e n t h e i n t e g r a l o f t h e t a n g e n t i a l c o m p o n e n t o f A a l o n g C f r o m P 1 t o P 2
, w r i t t e n a s
P 2
P
A l d x + A 2 d y + A 3 d z
i s a n e x a m p l e o f a l i n e i n t e g r a l . I f A i s t h e f o r c e F o n a p a r t i c l e m o v i n g a l o n g C , t h i s l i n e i n t e g r a l
r e p r e s e n t s t h e w o r k d o n e b y t h e f o r c e .
I f C i s a c l o s e d c u r v e ( w h i c h w e s h a l l s u p p o s e i s a s i m p l e
c l o s e d c u r v e , i . e . a c u r v e w h i c h d o e s n o t i n t e r s e c t i t s e l f a n y w h e r e ) t h e i n t e g r a l a r o u n d C i s o f t e n
d e n o t e d b y
5
A l d x + A 2 d y + A 3 d z
I n a e r o d y n a m i c s a n d f l u i d m e c h a n i c s t h i s i n t e g r a l i s c a l l e d t h e c i r c u l a t i o n o f A a b o u t C , w h e r e A
r e p r e s e n t s t h e v e l o c i t y o f a f l u i d .
I n g e n e r a l , a n y i n t e g r a l w h i c h i s t o b e e v a l u a t e d a l o n g a c u r v e i s c a l l e d a l i n e i n t e g r a l . S u c h
i n t e g r a l s c a n b e d e f i n e d i n t e r m s o f l i m i t s o f s u m s a s a r e t h e i n t e g r a l s o f e l e m e n t a r y c a l c u l u s .
F o r m e t h o d s o f e v a l u a t i o n o f l i n e i n t e g r a l s , s e e t h e S o l v e d P r o b l e m s .
T h e f o l l o w i n g t h e o r e m i s i m p o r t a n t .
°
8 2
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V E C T O R I N T E G R A T I O N
8 3
T H E O R E M . I f A = V o e v e r y w h e r e i n a r e g i o n R o f s p a c e , d e f i n e d b y a 1 5 x
< a 2 , b 1 < y < b , 2 ,
C l
z C 2 ,
w h e r e c ( x , y , z )
i s s i n g l e - v a l u e d a n d h a s c o n t i n u o u s d e r i v a t i v e s i n R ,
t h e n
P 2
1 .
A - d r i s i n d e p e n d e n t o f t h e p a t h C i n R j o i n i n g P 1 a n d P 2 .
P 1
2 .
A - d r = 0 a r o u n d a n y c l o s e d c u r v e C i n R .
C
I n s u c h c a s e A i s c a l l e d a c o n s e r v a t i v e v e c t o r f i e l d a n d c b i s i t s s c a l a r p o t e n t i a l .
A v e c t o r f i e l d A i s c o n s e r v a t i v e i f a n d o n l y i f V x A = O , o r e q u i v a l e n t l y A = V c . I n s u c h c a s e
A . d r = A l d x + A 2 d y + A 3 d z = d o , a n e x a c t d i f f e r e n t i a l . S e e P r o b l e m s 1 0 - 1 4 .
S U R F A C E I N T E G R A L S . L e t S b e a t w o - s i d e d s u r f a c e , s u c h a s s h o w n i n t h e f i g u r e b e l o w . L e t o n e
s i d e o f S b e c o n s i d e r e d a r b i t r a r i l y a s t h e p o s i t i v e s i d e ( i f S i s a c l o s e d
s u r f a c e t h i s i s t a k e n a s t h e o u t e r s i d e ) .
A u n i t n o r m a l n t o a n y p o i n t o f t h e p o s i t i v e s i d e o f S i s
c a l l e d a p o s i t i v e o r o u t w a r d d r a w n u n i t n o r m a l .
A s s o c i a t e w i t h t h e d i f f e r e n t i a l o f s u r f a c e
a r e a d S a v e c t o r d S w h o s e m a g n i t u d e i s d S a n d
w h o s e d i r e c t i o n i s t h a t o f n . T h e n d S = n d S .
T h e i n t e g r a l
f f A . d S =
f f A . n d S
S S
i s a n e x a m p l e o f a s u r f a c e i n t e g r a l c a l l e d t h e
f l u x o f A o v e r S . O t h e r s u r f a c e i n t e g r a l s a r e
0 d S ,
f f c t
n d S ,
f J A
x d S
S S
w h e r e o i s a s c a l a r f u n c t i o n . S u c h i n t e g r a l s c a n
b e d e f i n e d i n t e r m s o f l i m i t s o f s u m s a s i n e l e -
m e n t a r y c a l c u l u s ( s e e P r o b l e m 1 7 ) .
z
T h e n o t a t i o n 9 j .
i s s o m e t i m e s u s e d t o i n d i c a t e i n t e g r a t i o n o v e r t h e c l o s e d s u r f a c e S . W h e r e
S
n o c o n f u s i o n c a n a r i s e t h e n o t a t i o n
m a y a l s o b e u s e d .
S
T o e v a l u a t e s u r f a c e i n t e g r a l s , i t i s c o n v e n i e n t t o e x p r e s s t h e m a s d o u b l e i n t e g r a l s t a k e n o v e r
t h e p r o j e c t e d a r e a o f t h e s u r f a c e S o n o n e o f t h e c o o r d i n a t e p l a n e s . T h i s i s p o s s i b l e i f a n y l i n e p e r -
p e n d i c u l a r t o t h e c o o r d i n a t e p l a n e c h o s e n m e e t s t h e s u r f a c e i n n o m o r e t h a n o n e p o i n t . H o w e v e r , t h i s
d o e s n o t p o s e a n y r e a l p r o b l e m s i n c e w e c a n g e n e r a l l y s u b d i v i d e S i n t o s u r f a c e s w h i c h d o s a t i s f y
t h i s r e s t r i c t i o n .
V O L U M E I N T E G R A L S . C o n s i d e r a c l o s e d s u r f a c e i n s p a c e e n c l o s i n g a v o l u m e V . T h e n
f f 5 . a v
a n d
5 f f d V
V
V
a r e e x a m p l e s o f v o l u m e i n t e g r a l s o r s p a c e i n t e g r a l s a s t h e y a r e s o m e t i m e s c a l l e d . F o r e v a l u a t i o n o f
s u c h i n t e g r a l s , s e e t h e S o l v e d P r o b l e m s .
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8 4
V E C T O R I N T E G R A T I O N
S O L V E D P R O B L E M S
2
1 . I f R ( u ) = ( u - u 2 ) i + 2 u 3 j - 3 k ,
f i n d ( a )
R ( u ) d u a n d ( b )
R ( u ) d u .
1
( a )
f R ( u ) d u
=
f
[ ( u - u 2 ) i + 2 u 3 j - 3 0 d u
i
f
( u - u 2 ) d u + j
, J
+
2 u 3 d u + k
J -
3 d u
J
2
3
i ( 2 - 3 + c 1 ) + j ( 3 + c 2 )
+
k ( - - 3 u + c 3 )
2
3
4
{ 2
3
) i
+ Z j
- 3 u k
+
c j i
+
c 2 j
+
c 3 k
2
3
4
( 2
3 ) i
+
2 j - 3 u k
+
c
w h e r e c i s t h e c o n s t a n t v e c t o r c l i + c 2 j + c 3 k .
2
( b ) F r o m ( a ) ,
f I
R ( u ) d u
A n o t h e r M e t h o d .
=
u 2
-
u 3
( 2
g ) i +
_ - 6 ' + 2 j
4
2
2 j
- 3 u k
+ c f 1
c i - 3 ( 2 ) k + c i - [ ( 2 -
4 - ) i + 2 j - 3 ( 1 ) k + c ]
- 3 k
2
2
2
2
. f
R ( u ) d u
=
i
f
1
( u - u 2 ) d u
+ i
f
1
2 u 3 d u +
k f I - 3 d u
2
3 2
4
2
2
i ( 3 - - 3 ) 1 1 + j ( 3 ) 1 1
+ k ( - 3 u ) I 1
=
- s 1 + 2 j
2 . T h e a c c e l e r a t i o n o f a p a r t i c l e a t a n y t i m e
t ? 0 i s g i v e n b y
a =
d t
=
1 2 c o s 2 t i - 8 s i n 2 t j
+ 1 6 t k
I f t h e v e l o c i t y v a n d d i s p l a c e m e n t r a r e z e r o a t t = 0 , f i n d v a n d r a t a n y t i m e .
I n t e g r a t i n g , v =
i
f
1 2 c o s 2 t d t
+
j f
-
8 s i n 2 t d t
+ k f
1 6 t d t
=
6 s i n 2 t i
+
4 c o s 2 t j
+
8 t 2 k + c 1
P u t t i n g v = 0 w h e n t = 0 , w e f i n d
0 = 0 i + 4 i + O k + c 1 a n d c 1 = - 4 j .
T h e n v =
6 s i n 2 t i
+ ( 4 c o s 2 t - 4 ) j
+ 8 t 2 k
s o t h a t
d a
=
6 s i n 2 t i +
( 4 c o s 2 t - 4 ) j
+
8 t 2 k .
I n t e g r a t i n g , r
=
i
f
6 s i n 2 t d t
+
J f ( 4 c o s 2 t - 4 ) d t +
k
f
8 t 2 d t
= - 3 c o s 2 t i +
( 2 s i n 2 t - 4 t ) j
+ 3 ? k +
c 2
P u t t i n g r = 0 w h e n t = 0 ,
0 = - 3 i + 0 j + 0 k + c 2 a n d c 2 = 3 i .
3 k
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V E C T O R I N T E G R A T I O N
8 5
T h e n
r = ( 3 - - 3 c o s 2 t ) i + ( 2 s i n 2 t - 4 t ) j + 8 t 3 k .
3
2
3 . E v a l u a t e
J A
x d t A d t
2
d t ( A x
d A )
= A X
d t 2 A + a A
a A
I n t e g r a t i n g ,
f
( A x
) d t
= A x
+ c
.
d t
d t
d t
4 . T h e e q u a t i o n o f m o t i o n o f a p a r t i c l e P o f m a s s m i s g i v e n b y
=
f ( r ) r 1
d 2
w h e r e r i s t h e p o s i t i o n v e c t o r o f P m e a s u r e d f r o m a n o r i g i n 0 , r 1 i s a u n i t v e c t o r i n t h e d i r e c t i o n r ,
a n d f ( r ) i s a f u n c t i o n o f t h e d i s t a n c e o f P f r o m 0 .
( a ) S h o w t h a t r x
= c w h e r e c i s a c o n s t a n t v e c t o r .
( b ) I n t e r p r e t p h y s i c a l l y t h e c a s e s f ( r ) < 0 a n d f ( r ) > 0 .
( c ) I n t e r p r e t t h e r e s u l t i n ( a ) g e o m e t r i c a l l y .
( d ) D e s c r i b e h o w t h e r e s u l t s o b t a i n e d r e l a t e t o t h e m o t i o n o f t h e p l a n e t s i n o u r s o l a r s y s t e m .
2
( a )
M u l t i p l y b o t h s i d e s o f i n d t 2 = f ( r ) r 1 b y r x .
T h e n
f i b I L x d
=
f ( r ) r x r 1
=
0
s i n c e r a n d r 1 a r e c o l l i n e a r a n d s o r x r 1 = 0 . T h u s
2
r x
a t e = 0
a n d
d t ( r x d t )
= 0
I n t e g r a t i n g ,
r x d = c , w h e r e c i s a c o n s t a n t v e c t o r . ( C o m p a r e w i t h P r o b l e m 3 ) .
2
( b )
I f f ( r ) < 0 t h e a c c e l e r a t i o n d t 2 h a s d i r e c t i o n o p p o s i t e t o r 1 ; h e n c e t h e f o r c e i s d i r e c t e d t o w a r d 0 a n d
t h e p a r t i c l e i s a l w a y s a t t r a c t e d t o w a r d 0 .
I f f ( r ) > 0 t h e f o r c e i s d i r e c t e d a w a y f r o m 0 a n d t h e p a r t i c l e i s u n d e r t h e i n f l u e n c e o f a r e p u l s i v e
f o r c e a t 0 .
A f o r c e d i r e c t e d t o w a r d o r a w a y f r o m a f i x e d p o i n t 0 a n d h a v i n g m a g n i t u d e d e p e n d i n g o n l y o n t h e
d i s t a n c e r f r o m 0 i s c a l l e d a c e n t r a l f o r c e .
( c )
I n t i m e A t t h e p a r t i c l e m o v e s f r o m M t o N ( s e e a d -
j o i n i n g f i g u r e ) . T h e a r e a s w e p t o u t b y t h e p o s i t i o n
v e c t o r i n t h i s t i m e i s a p p r o x i m a t e l y h a l f t h e a r e a o f
a p a r a l l e l o g r a m w i t h s i d e s r a n d A r , o r
2
r x A r .
T h e n t h e a p p r o x i m a t e a r e a s w e p t o u t b y t h e r a d i u s
v e c t o r p e r u n i t t i m e i s
I r x r ; h e n c e t h e i n s t a n -
t a n e o u s t i m e r a t e o f c h a n g e i n a r e a i s
l i m
2 r x
A r
=
A t 0 0
w h e r e v i s t h e i n s t a n t a n e o u s v e l o c i t y o f t h e p a r t i -
z
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8 6
V E C T O R I N T E G R A T I O N
c l e . T h e q u a n t i t y H =
Z r
x t = 2 r x v i s c a l l e d t h e a r e a l v e l o c i t y . F r o m p a r t ( a ) ,
A r e a l V e l o c i t y
=
f l
=
z r x
d t r
t
= c o n s t a n t
S i n c e r - H = 0 ,
t h e m o t i o n t a k e s p l a c e i n a p l a n e , w h i c h w e t a k e a s t h e x y p l a n e i n t h e f i g u r e a b o v e .
( d ) A p l a n e t ( s u c h a s t h e e a r t h ) i s a t t r a c t e d t o w a r d t h e s u n a c c o r d i n g t o N e w t o n ' s u n i v e r s a l l a w o f g r a v i t a -
t i o n , w h i c h s t a t e s t h a t a n y t w o o b j e c t s o f m a s s m a n d M r e s p e c t i v e l y a r e a t t r a c t e d t o w a r d e a c h o t h e r
w i t h a f o r c e o f m a g n i t u d e F =
G M 2
,
w h e r e r i s t h e d i s t a n c e b e t w e e n o b j e c t s a n d G i s a u n i v e r s a l
c o n s t a n t . L e t m a n d M b e t h e m a s s e s o f t h e p l a n e t a n d s u n r e s p e c t i v e l y a n d c h o o s e a s e t o f c o o r d i -
n a t e a x e s w i t h t h e o r i g i n 0 a t t h e s u n . T h e n t h e e q u a t i o n o f m o t i o n o f t h e p l a n e t i s
d 2 r
m d t 2
G M m
- r 2
r 1 o r
d 2 r
_ G M
d t 2
- r 2 r 1
a s s u m i n g t h e i n f l u e n c e o f t h e o t h e r p l a n e t s t o b e n e g l i g i b l e .
A c c o r d i n g t o p a r t ( c ) , a p l a n e t m o v e s a r o u n d t h e s u n s o t h a t i t s p o s i t i o n v e c t o r s w e e p s o u t e q u a l
a r e a s i n e q u a l t i m e s . T h i s r e s u l t a n d t h a t o f P r o b l e m 5 a r e t w o o f K e p l e r ' s f a m o u s t h r e e l a w s w h i c h h e
d e d u c e d e m p i r i c a l l y f r o m v o l u m e s o f d a t a c o m p i l e d b y t h e a s t r o n o m e r T y c h o B r a h e . T h e s e l a w s e n a -
b l e d N e w t o n t o f o r m u l a t e h i s u n i v e r s a l l a w o f g r a v i t a t i o n . F o r K e p l e r ' s t h i r d l a w s e e P r o b l e m 3 6 .
5 . S h o w t h a t t h e p a t h o f a p l a n e t a r o u n d t h e s u n i s a n e l l i p s e w i t h t h e s u n a t o n e f o c u s .
( 1 )
( 2 )
F r o m P r o b l e m s 4 ( c ) a n d 4 ( d ) ,
d v
G M
d t
=
- r 2 A l
r x v
=
2 H
= h
N o w r = r r 1 , d
r W t + d t r 1
s o t h a t
( 3 )
h r x v
r r 1 x ( r
d r l
+
d r
r l ) _
d t
d t
r 2 r 1 x
d t
F r o m ( 1 ) , d t x h
= G M r 1 x h
=
- - G M r 1 x ( r 1 x
d t l )
- G M [ ( r 1 .
d r 1 )
r 1 - -
{ r 1 r l ) d r l
= G M
d r 1
d t
d t
d t
u s i n g e q u a t i o n ( 3 ) a n d t h e f a c t t h a t r l . d a
= 0 ( P r o b l e m 9 , C h a p t e r 3 ) .
B u t s i n c e h i s a c o n s t a n t v e c t o r ,
d v
x h =
d
( v x h )
s o t h a t
d t d t
d t ( v
x
h )
=
G M d t
I n t e g r a t i n g ,
v x h =
G M r 1 + p
f r o m w h i c h
r . ( v x h )
= G M r r 1 + r p
= G M r + r r 1 . P
= G M r + r p c o s 8
w h e r e p i s a n a r b i t r a r y c o n s t a n t v e c t o r w i t h m a g n i t u d e p , a n d a i s t h e a n g l e b e t w e e n p a n d r 1 .
S i n c e r . ( v x h ) _ ( r x v ) . h = h h = h 2 ,
w e h a v e
h 2 = G M r + r p c o s B a n d
h 2
h 2 / G M
r
G M + p c o s B
1 + ( p / G M ) c o s 6
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V E C T O R I N T E G R A T I O N
F r o m a n a l y t i c g e o m e t r y , t h e p o l a r e q u a t i o n o f a c o n i c
s e c t i o n w i t h f o c u s a t t h e o r i g i n a n d e c c e n t r i c i t y E i s
r =
a w h e r e a i s a c o n s t a n t . C o m p a r i n g t h i s
1 + E C o s 9
w i t h t h e e q u a t i o n d e r i v e d , i t i s s e e n t h a t t h e r e q u i r e d
o r b i t i s a c o n i c s e c t i o n w i t h e c c e n t r i c i t y
E = p / G M .
T h e o r b i t i s a n e l l i p s e , p a r a b o l a o r h y p e r b o l a a c c o r d -
i n g a s E i s l e s s t h a n , e q u a l t o o r g r e a t e r t h a n o n e .
S i n c e o r b i t s o f p l a n e t s a r e c l o s e d c u r v e s i t f o l l o w s
t h a t t h e y m u s t b e e l l i p s e s .
E 1 1 '
-
a
a p s e r
i + E c 0 s 8
L I N E I N T E G R A L S
8 7
6 . I f A = ( 3 x 2 + 6 y ) i - 1 4 y z j + 2 0 x z 2 k , e v a l u a t e J A d r
f r o m ( 0 , 0 , 0 ) t o ( 1 , 1 , 1 ) a l o n g t h e f o l l o w -
i n g p a t h s C :
C
( a ) x = t , y = t 2 , z = t 3 .
( b ) t h e s t r a i g h t l i n e s f r o m ( 0 , 0 , 0 ) t o ( 1 , 0 , 0 ) , t h e n t o ( 1 , 1 , 0 ) , a n d t h e n t o ( 1 , 1 , 1 ) .
( c ) t h e s t r a i g h t l i n e j o i n i n g ( 0 , 0 , 0 ) a n d ( 1 , 1 , 1 ) .
f c
f c ,
[ ( 3 x 2 + 6 y ) i - 1 4 y z j + 2 0 x z 2 k ] ( d x i + d y j + d z k )
J ' ( 3 x 2 + 6 y ) d x
- 1 4 y z d y + 2 0 x z 2 d z
( a )
I f x = t , y = t 2 , Z = t 3 , p o i n t s ( 0 , 0 , 0 ) a n d ( 1 , 1 , 1 ) c o r r e s p o n d t o t = 0 a n d t = 1 r e s p e c t i v e l y . T h e n
1
f
A - d r
=
f
( 3 t 2 + 6 t 2 ) d t - 1 4 ( t 2 ) ( t 3 ) d ( t 2 )
t = 0
I '
t = o
9 t 2 d t - 2 8 t e d t + 6 0 t 9 d t
f ( 9 t _ 2 8 t 6 + 6 0 t 9 )
d t
t = o
+ 2 0 ( t ) ( t 3 ) 2 d ( t 3 )
1
3 t 2 - 4 t 7 + 6 t 1 0 1
= 5
0
A n o t h e r M e t h o d .
A l o n g C , A = 9 t 2 i - 1 4 t 5 j + 2 0 t 7 k a n d
r = x i + y j + z k = t i + t 2 j + t 3 k a n d d r = ( i + 2 t j + 3 t 2 k ) d t .
T h e n
J
A d r
C
I '
t = o
1
( 9 t 2 i - 1 4 t 5 j + 2 0 t 7 k ) . ( i + 2 t j + 3 t 2 k ) d t
( 9 t 2 - 2 8 t 6 + 6 0 t 9 ) d t
=
5
( b )
A l o n g t h e s t r a i g h t l i n e f r o m ( 0 , 0 , 0 ) t o ( 1 , 0 , 0 ) y = 0 , z = 0 , d y = 0 , d z = 0 w h i l e x v a r i e s f r o m 0 t o
1 . T h e n
t h e i n t e g r a l o v e r t h i s p a r t o f t h e p a t h i s
I '
1
3 x 2 + 6 ( 0 ) ) d x - 1 4 ( 0 ) ( 0 ) ( 0 ) + 2 0 x ( 0 ) 2 ( 0 ) =
f 3 x 2
d x
1
x
l
=
1
0
A l o n g t h e s t r a i g h t l i n e f r o m ( 1 , 0 , 0 ) t o ( 1 , 1 , 0 ) x = 1 , z = 0 , d x = 0 , d z = 0 w h i l e y v a r i e s f r o m 0
t o 1 .
T h e n t h e i n t e g r a l o v e r t h i s p a r t o f t h e p a t h i s
S I
( 3 ( 1 ) 2 + 6 y ) 0 - 1 4 y ( 0 ) d y + 2 0 ( 1 ) ( 0 ) 2 0
= 0
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8 8
V E C T O R I N T E G R A T I O N
A l o n g t h e s t r a i g h t l i n e f r o m ( 1 , 1 , 0 ) t o ( 1 , 1 , 1 ) x = 1 , y = 1 , d x = 0 , d y = 0 w h i l e z v a r i e s f r o m 0
t o 1 .
T h e n t h e i n t e g r a l o v e r t h i s p a r t o f t h e p a t h i s
f 1
z = 0
3 ( 1 ) 2 + 6 ( 1 ) ) 0 - 1 4 ( 1 ) z ( 0 ) + 2 0 ( 1 ) z 2 d z
= f 1 2 0 z 2 d z
=
2 0 3 z 3
z , = o
A d d i n g ,
J
A - d r
=
1
+ 0
C
+ 2 0
=
2 3
3
3
( c ) T h e s t r a i g h t l i n e j o i n i n g ( 0 , 0 , 0 ) a n d ( 1 , 1 , 1 ) i s g i v e n i n p a r a m e t r i c f o r m b y x = t , y = t , z = t .
T h e n
f
1
J A - d r
( 3 t 2 + 6 t ) d t - 1 4 ( t ) ( t ) d t + 2 0 ( t ) ( t ) 2 d t
C
t = 0
( `
t = 0
( 3 t 2 + 6 t - 1 4 t 2 + 2 0 t 3 ) d t
- f
1
( 6 t - 1 1 t 2 + 2 0 t 3 ) d t
t = 0
I '
7 . F i n d t h e t o t a l w o r k d o n e i n m o v i n g a p a r t i c l e i n a f o r c e f i e l d g i v e n b y F = 3 x y i - 5 z j + l O x k
a l o n g t h e c u r v e x = 1 2 + 1 , y = 2 t 2 , z = t 3 f r o m t = l t o t = 2 .
T o t a l w o r k
J F - d r
( 3 x y i - 5 z j +
+ d y j + d z k )
C
C
3 x y d x - 5 z d y + 1 0 x d z
2
3 ( t 2 + 1 ) ( 2 t 2 ) d ( t 2 + 1 ) - 5 ( t 3 ) d ( 2 t 2 )
+
1 0 ( t 2 + 1 ) d ( t 3 )
t = 1
2
( 1 2 t 5 + 1 0 t 4 + 1 2 t 2 + 3 0 t 2 ) d t = 3 0 3
8 . I f F = 3 x y i - y 2 j ,
e v a l u a t e
F - d r w h e r e C i s t h e c u r v e i n t h e x y p l a n e , y = 2 x 2 , f r o m ( 0 , 0 )
t o ( 1 , 2 ) . i c
S i n c e t h e i n t e g r a t i o n i s p e r f o r m e d i n t h e x y p l a n e ( z = 0 ) , w e c a n t a k e r = x i + y j .
T h e n
=
f ( 3 x y i _ y 2 j ) . ( d x i + d y j )
J F . d r
x y d x - y 2 d y
f e 3
F i r s t M e t h o d .
L e t x = t i n y = 2 x 2 . T h e n t h e p a r a m e t r i c e q u a t i o n s o f C a r e x = t , y = 2 t 2 .
P o i n t s ( 0 , 0 ) a n d
( 1 , 2 ) c o r r e s p o n d t o t = 0 a n d t = 1 r e s p e c t i v e l y . T h e n
f
F - d r
=
r 1
C
3 ( t ) ( 2 t 2 ) d t - ( 2 t 2 ) 2 d ( 2 t 2 )
t o
t = 0
( 6 t 3 - 1 6 1 5 ) d t
1 3
3
7
6
S e c o n d M e t h o d . S u b s t i t u t e y = 2 x 2 d i r e c t l y , w h e r e x g o e s f r o m 0 t o 1 .
T h e n
f
1
f F - d r
3 x ( 2 x 2 ) d x - ( 2 x 2 ) 2 d ( 2 x 2 )
C
x = o
( 6 x 3 - 1 6 x 5 ) d x
=
x = 0
N o t e t h a t i f t h e c u r v e w e r e t r a v e r s e d i n t h e o p p o s i t e s e n s e , i . e . f r o m ( 1 , 2 ) t o ( 0 , 0 ) , t h e v a l u e o f t h e i n t e g r a l
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V E C T O R I N T E G R A T I O N
8 9
9 . F i n d t h e w o r k d o n e i n m o v i n g a p a r t i c l e o n c e a r o u n d a c i r c l e C i n t h e x y p l a n e , i f t h e c i r c l e h a s
c e n t e r a t t h e o r i g i n a n d r a d i u s 3 a n d i f t h e f o r c e f i e l d i s g i v e n b y
F =
( 2 x - - y + z ) i
+ ( x + y - z 2 ) j
+ ( 3 x - 2 y + 4 z ) k
I n t h e p l a n e z = 0 , F = ( 2 x - y ) i + ( x + y ) j + ( 3 x - 2 y ) k a n d d r = d x i + d y j s o t h a t
t h e w o r k d o n e i s
[ ( 2 x - y ) i + ( x + y ) j + ( 3 x - 2 y ) k - [ d x i + d y j ]
C
J
f c
( 2 x - y ) d x + ( x + y ) d y
C h o o s e t h e p a r a m e t r i c e q u a t i o n s o f t h e c i r c l e a s x = 3 c o s t , y = 3 s i n t
w h e r e t v a r i e s f r o m 0 t o 2 n ( s e e a d j o i n i n g f i g u r e ) .
T h e n t h e l i n e i n t e g r a l
e q u a l s
2 7 1
[ 2 ( 3 c o s t ) - 3 s i n t ] [ - 3 s i n t ] d t + [ 3 c o s t + 3 s i n t ] [ 3 c o s . t ] d t
t = 0
I
( 9 - 9 s i n t c o s t ) d t
= 9 t -
2
s i n 2 t
2 7 T
1 2 7 T
0
=
1 8 7 L
I n t r a v e r s i n g C w e h a v e c h o s e n t h e c o u n t e r c l o c k w i s e d i r e c t i o n i n d i c a t e d
i n t h e a d j o i n i n g f i g u r e .
W e c a l l t h i s t h e p o s i t i v e d i r e c t i o n , o r s a y t h a t C
h a s b e e n t r a v e r s e d i n t h e p o s i t i v e s e n s e . I f C w e r e t r a v e r s e d i n t h e c l o c k -
w i s e ( n e g a t i v e ) d i r e c t i o n t h e v a l u e o f t h e i n t e g r a l w o u l d b e - 1 8 I T .
r = x i + y j
3 c o s t i + 3 s i n t j
1 0 . ( a ) I f F = V
,
w h e r e
i s s i n g l e - v a l u e d a n d h a s c o n t i n u o u s p a r t i a l d e r i v a t i v e s , s h o w t h a t t h e
w o r k d o n e i n m o v i n g a p a r t i c l e f r o m o n e p o i n t P 1 = ( x 1 , y 1 , z 1 ) i n t h i s f i e l d t o a n o t h e r p o i n t
P 2 = ( x 2 , y 2 , z 2 ) i s - i n d e p e n d e n t o f t h e p a t h j o i n i n g t h e t w o p o i n t s .
( b ) C o n v e r s e l y , i f
F . d r i s i n d e p e n d e n t o f t h e p a t h C j o i n i n g a n y t w o p o i n t s , s h o w t h a t t h e r e
C
e x i s t s a f u n c t i o n 0 s u c h t h a t F = V
.
P .
f P 1 2
( a ) W o r k d o n e
=
F - d r
=
V q b d r
I P 2
1
P
s
=
P 2
(
a i + j +
a o k )
( d x i + d y j + d z k )
f
Y z
i
P 2
d x +
a 0
d y +
a d z
P 1
Y
z
( P 2
, / ,
J
d Y =
O ( P 2 ) - O ( P i )
_ ( N x 2 , Y 2 , Z 2 ) - 0 ( x 1 , Y 1 , Z 1 )
P 1
T h e n t h e i n t e g r a l d e p e n d s o n l y o n p o i n t s P 1 a n d P 2 a n d n o t o n t h e p a t h j o i n i n g t h e m . T h i s i s t r u e
o f c o u r s e o n l y i f q 5 ( x , y , z ) i s s i n g l e - v a l u e d a t a l l p o i n t s P 1 a n d P 2 .
. d r i s i n d e p e n d e n t o f t h e p a t h C j o i n i n g a n y t w o
b ) L e t F = F 1 i + F 2 j + F 3 k . B y h y p o t h e s i s , f c F
p o i n t s , w h i c h w e t a k e a s ( x 1 , y 1 , z 1 ) a n d ( x , y , z ) r e s p e c t i v e l y . T h e n
( x , y , Z ) =
f ( x , y , Z )
c p ( x , y , z )
=
J
F 1 d x + F 2 d y + F 3 d z
f x 1 , Y 1 ,
z 1 )
z 1 )
i s i n d e p e n d e n t o f t h e p a t h j o i n i n g ( x 1 , y 1 i z 1 ) a n d ( x , y , z ) .
T h u s
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9 0
V E C T O R I N T E G R A T I O N
C / ( x + A x , y , z ) - c ( x , y , z )
f
( x + A x , y , z )
f ( x , y , z )
F . d r
- J
F
d r
x 1 , y i , z 1 )
( x 1 , Y 1 , z 1 )
( x 1 , y 1 , Z 1 ) ( x + I x , y , z )
r
, Y , z )
J ( x i , y j , z i )
( x + . x , Y . z )
- ' ( x + A x , y , z )
F 1 d x + F 2 d y + F 3 d z
( x , y , z ) ( x , y , z )
S i n c e t h e l a s t i n t e g r a l m u s t b e i n d e p e n d e n t o f t h e p a t h j o i n i n g ( x , y , z ) a n d ( x + A x , Y . z ) , w e m a y c h o o s e
t h e p a t h t o b e a s t r a i g h t l i n e j o i n i n g t h e s e p o i n t s s o t h a t d y a n d d z a r e z e r o . T h e n
O ( x + A x , y , z ) - O ( x , y , z )
A x
1
( x + A x , y , Z )
O x
J
( x , y , z )
T a k i n g t h e l i m i t o f b o t h s i d e s a s O x - , 0 , w e h a v e a 0 = F 1 .
x
S i m i l a r l y , w e c a n s h o w
t h a t a a y = F 2 a n d a 0 = F 3 .
T h e n F = F 1 i + F 2 j + F 3 k =
- i + a - j
+
a - k
= V 0 .
x y
F 1 d x
( ' P 2
I f J
F . d r i s i n d e p e n d e n t o f t h e p a t h C j o i n i n g P 1 a n d P 2 , t h e n F i s c a l l e d a c o n s e r v a t i v e f i e l d .
I t
P 1
f o l l o w s t h a t i f F = V O t h e n F i s c o n s e r v a t i v e , a n d c o n v e r s e l y .
P r o o f u s i n g v e c t o r s .
I f t h e l i n e i n t e g r a l i s i n d e p e n d e n t o f t h e p a t h , t h e n
( x , y , z )
f ( x , y , z )
( x , y , z )
=
F
d s
A r - d s
f x 1 , Y 1 ,
Z 1 )
( x 1 , Y i , z 1 )
B y d i f f e r e n t i a t i o n ,
d ( t =
F
d r
B u t
d 4
= 0
d r
s o t h a t ( V - F )
d r
= 0 .
d s
d s
d s
d s
d s
S i n c e t h i s m u s t h o l d i r r e s p e c t i v e o f d s
,
w e h a v e F = V g .
1 1 . ( a ) I f F i s a c o n s e r v a t i v e f i e l d , p r o v e t h a t
c u r l F = V x F = 0 ( i . e . F i s i r r o t a t i o n a l ) .
( b ) C o n v e r s e l y , i f V x F = 0 ( i . e . F i s i r r o t a t i o n a l ) , p r o v e t h a t F i s c o n s e r v a t i v e .
( a )
I f F i s a c o n s e r v a t i v e f i e l d , t h e n b y P r o b l e m 1 0 , F = V o
.
T h u s c u r l F = V x V V = 0 ( s e e P r o b l e m 2 7 ( a ) , C h a p t e r 4 ) .
i j
k
( b ) I f O x F = 0 , t h e n
W e
a
a a
a x
a y
a z
= 0
a n d t h u s
F 1 F 2 F 3
I
F 3
F 2
- a a F 1 .
F 3
F 2
F 1
a y
-
a z a z = a x
' a x
= a y
m u s t p r o v e t h a t F = V V f o l l o w s a s a c o n s e q u e n c e o f t h i s .
T h e w o r k d o n e i n m o v i n g a p a r t i c l e f r o m ( x 1 , y 1 , z 1 ) t o ( x , y , z ) i n t h e f o r c e f i e l d F i s
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V E C T O R I N T E G R A T I O N
9 1
f e
F j ( x , y , z ) d x
+
F 2 ( x , y , z ) d y +
F 3 ( x , y , z ) d z
w h e r e C i s a p a t h j o i n i n g ( x 1 , y 1 , z l ) a n d ( x , y , z ) .
L e t u s c h o o s e a s a p a r t i c u l a r p a t h t h e s t r a i g h t l i n e
s e g m e n t s f r o m ( x 1 , y 1 , z 1 ) t o ( x , y l , z 1 ) t o ( x , y , z 1 ) t o ( x , y , z ) a n d c a l l 0 ( x , y , z ) t h e w o r k d o n e a l o n g t h i s
p a r t i c u l a r p a t h . T h e n
x
Y
z
( x , y , z )
=
J
x
F 1 ( x , y 1 , z 1 ) d x
+
F 2 ( x , y , z l ) d y
+
F 3 ( x , y , z ) d z
1
9 . 0
Y 1
f
1
I t f o l l o w s t h a t
a z
F 3 ( x , y , z )
_
a y
f z F
F 2 ( x , y , z 1 ) +
3
( x , y , z ) d z
a
1
z
' 8 a F 2
( x , y , z ) d z
F 2 ( x , y , z 1 )
+
J 2 1
-
a x
z
F 2 ( x , y , Z 1 ) + F 2 ( x , y , Z ) I
F 2 ( x , y , Z 1 ) +
F 2 ( x , y , z ) - F 2 ( x , y , Z 1 )
= F 2 ( x , y , Z )
Z 1
z
=
F 1 ( x , y 1 , z 1 )
+ S
Y e ( x ,
y , z 1 ) d y
+ f a z ( x , y , z ) d z
1
. f
1
z
F , ( x , y 1 , z - 1 )
+ Y
a Y F l
( x , y , z 1 ) d y
+
f l
a
1 ( x , Y , z ) f d z
z
l
y z
=
F 1 ( x , Y 1 , z 1 )
+ F 1 ( x , y , z j ) I
+
F 1 ( x , y , z ) I
y 1
z 1
=
F 1 ( x , y 1 , z 1 )
+ F 1 ( x , y , z 1 ) - F 1 ( x , Y 1 , z 1 )
+ F 1 ( x , y , z ) -
F 1 ( x , y , z 1 )
=
F 1 ( x , y , z )
T h e n
F =
F 1 i + F 2 j + F 3 k
a - i +
a - j +
k
=
Y
T h u s a n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n t h a t a f i e l d F b e c o n s e r v a t i v e i s t h a t c u r l F = V x F = 0 .
1 2 . ( a ) S h o w t h a t F = ( 2 x y
+ z 3 ) i
+ x 2 j + 3 x z 2 k
i s a c o n s e r v a t i v e f o r c e f i e l d . ( b ) F i n d t h e s c a -
l a r p o t e n t i a l . ( c ) F i n d t h e w o r k d o n e i n m o v i n g a n o b j e c t i n t h i s f i e l d f r o m ( 1 , - 2 , 1 ) t o ( 3 , 1 , 4 ) .
( a ) F r o m P r o b l e m 1 1 , a n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n t h a t a f o r c e w i l l b e c o n s e r v a t i v e i s t h a t
c u r l F = V x F = 0 .
N o w
V x F =
i
j
k
a
a
a
a x
a y a Z
2 x y + z 3
x 2
3 x z 2
=
0 .
T h u s F i s a c o n s e r v a t i v e f o r c e f i e l d .
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9 2
( b ) F i r s t M e t h o d .
I n t e g r a t i n g , w e f i n d f r o m ( 1 ) , ( 2 ) a n d ( 3 ) r e s p e c t i v e l y ,
B y P r o b l e m 1 0 , F = V O
o r a
i +
L O
j + a o k = ( 2 x y + z 3 ) i +
X 2
j + 3 x z 2 k . T h e n
Y
( 1 ) a
= 2 x y + z 3
( 2 )
= x 2 ( 3 )
= 3 x z 2
- a z
Y
x 2 y
+
x z 3
+
f ( y , z )
=
' Y ' 2
I V
+
( x z )
Z 5
9
x z
3
+
h ( x , y )
T h e s e a g r e e i f w e c h o o s e f ( y , z ) = 0 , g ( x , z ) = x z 3 , h ( x , y ) = x 2 y s o t h a t = x 2 y + x z 3 t o w h i c h m a y
b e a d d e d a n y c o n s t a n t .
S e c o n d M e t h o d .
d r i s i n d e p e n d e n t o f t h e p a t h C j o i n i n g
i n c e F i s c o n s e r v a t i v e ,
f c F
U s i n g t h e m e t h o d o f P r o b l e m 1 1 ( b ) ,
x
f x l
V E C T O R I N T E G R A T I O N
( 2 x y 1 + z 1 ) d x
+
f
Y
Y 1
z
x 2 d y
+ f i 3 x z
2
d z
z
( y
( x 2 y 1 + x z 3 )
I x
x 1
+
x 2 Y I y 1
+
x z 3 1 z 1
( x 1 , Y 1 , Z 1 )
a n d ( x , y , z ) .
x 2 y 1 +
x z 3 - x i 2 y 1
-
X : 1 z 3
+
x 2 y - x 2 y 1
+
x z
- x z
=
x 2 y
+
x z
-
x i y i -
x 1 z 3
x 2 y
+ x Z 3
+
c o n s t a n t
T h i r d M e t h o d .
V o - d r
=
a 0 d x + L o
d y +
.
o
d z
=
d o
Y
T h e n
d o
=
( 2 x y + z 3 ) d x
+ x 2 d y
+
3 x z 2 d z
( 2 x y d x + x 2 d y )
+
( z 3 d x + 3 x z 2 d z )
= d ( x 2 y )
+
d ( x z 3 )
d ( x 2 y + x z 3 )
a n d 0 =
x 2 y + x z 3 + c o n s t a n t .
P 2
( c ) W o r k d o n e =
P
F . d r
1
P 1
f
p i
P 2
A n o t h e r M e t h o d .
( 2 x y + z 3 ) d x + x 2 d y + 3 x z 2 d z
d ( x 2 y + x z 3 ) = x 2 y + x z 3 I
P 2
= x 2 y + x z 3 I
( 3 , i t 4 )
= 2 0 2
P 1
F r o m p a r t ( b ) ,
c ( x , y , z )
=
x 2 y + x z 3 + c o n s t a n t .
( 1 , - 2 , 1 )
T h e n w o r k d o n e
=
0 ( 3 , 1 , 4 ) - 0 ( 1 , - 2 , 1 )
=
2 0 2 .
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V E C T O R I N T E G R A T I O N
9 3
P 2
1 3 . P r o v e t h a t i f
F . d r
i s i n d e p e n d e n t o f t h e p a t h j o i n i n g a n y t w o p o i n t s P 1 a n d P 2 i n a g i v e n
J
P 1
r e g i o n , t h e n
F d r = 0 f o r a l l c l o s e d p a t h s i n t h e r e g i o n a n d c o n v e r s e l y .
L e t P 1 A P 2 B P 1 ( s e e a d j a c e n t f i g u r e ) b e a c l o s e d c u r v e . T h e n
5
f
f
J
P 1 A P 2 8 P 1
P 1 A P 2
P 2 B P 1
s i n c e t h e i n t e g r a l f r o m P 1 t o P 2 a l o n g a p a t h t h r o u g h A i s t h e s a m e a s
t h a t a l o n g a p a t h t h r o u g h B , b y h y p o t h e s i s .
f F . d r =
0
P 1 A P 2
P 1 B P 2
s o t h a t ,
C o n v e r s e l y i f f F d r = 0 ,
t h e n
f F d r
f
f 0
P 1 A P 2 B P 1
P 1 A P 2
P 2 B P 1
P 1 A P 2
P 1 B P 2
J
P 1 A P 2
F d r =
f
F d r .
P 1 B P 2
1 4 . ( a ) S h o w t h a t a n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n t h a t F 1 d x + F 2 d y + F 3 d z b e a n e x a c t d i f f e r -
e n t i a l i s t h a t V x F = 0 w h e r e F = F 1 i + F 2 j + F 3 k .
( b ) S h o w t h a t ( y 2 z 3 c o s x - 4 x 3 z ) d x + 2 z 3 y s i n x d y + ( 3 y 2 z 2 s i n x - x 4 ) d z i s a n e x a c t d i f -
f e r e n t i a l o f a f u n c t i o n q 5 a n d f i n d 0 .
( a ) S u p p o s e
F 1 d x + F 2 d y + F 3 d z = d q 5 = a
d x + d y +
d z ,
a n e x a c t d i f f e r e n t i a l .
T h e n
s i n c e x , y a n d z a r e i n d e p e n d e n t v a r i a b l e s ,
x
y z
a
a
a
F 1 = .
,
F 2 = a
,
F 3 =
a
a n d s o F = F 1 i + F 2 J + F 3 k =
a
i +
j +
k = V 0 .
T h u s V x F = V x V q = 0 .
y
z
C o n v e r s e l y
i f O x F = 0 t h e n b y P r o b l e m 1 1 ,
F = V C a a n d s o F d r = V V V . d r = d O ,
i . e .
F 1 d x + F 2 d y + F 3 d z = d c / , a n e x a c t d i f f e r e n t i a l .
( b ) F = ( y 2 z 3 c o s x - 4 x 3 z ) i + 2 z 3 y s i n x j + ( 3 y 2 z 2 s i n x - x 4 ) k a n d O x F i s c o m p u t e d t o b e z e r o ,
s o t h a t b y p a r t ( a )
( y 2 z 3 c o s x - - 4 x 3 z ) d x
+
2 z 3 y s i n x d y
+
( 3 y 2 z 2 s i n x - x 4 ) d z
=
d o
B y a n y o f t h e m e t h o d s o f P r o b l e m 1 2 w e f i n d 0 = y 2 z 3 s i n x - x 4 z + c o n s t a n t .
1 5 . L e t F b e a c o n s e r v a t i v e f o r c e f i e l d s u c h t h a t F = - V O .
S u p p o s e a p a r t i c l e o f c o n s t a n t m a s s m
t o m o v e i n t h i s f i e l d .
I f A a n d B a r e a n y t w o p o i n t s i n s p a c e , p r o v e t h a t
( A ) + 2 m v A
f
0 ( B ) + 2 m v v
w h e r e v A a n d v B a r e t h e m a g n i t u d e s o f t h e v e l o c i t i e s o f t h e p a r t i c l e a t A a n d B r e s p e c t i v e l y .
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9 4
V E C T O R I N T E G R A T I O N
d 2 r
d r
=
d r d 2 r
_ i n
d
d r 2
F = m a = m d t 2
T h e n
F .
d t
' n d t d t 2
2
d t ( d t )
I n t e g r a t i n g ,
f A
B
J
B
I f F = - V o l
F d r
A
T h e n
0 ( A ) - 0 ( B )
=
2
v 2
B
- 2 m v A .
=
2 m V B
2
2
f B
2
2 m v B
A
B
d o
=
9 5 ( A ) - O ( B )
2 m A a n d t h e r e s u l t f o l l o w s .
O ( A ) i s c a l l e d t h e p o t e n t i a l e n e r g y a t A a n d 2 m v 2 i s t h e k i n e t i c e n e r g y a t A . T h e r e s u l t s t a t e s t h a t
t h e t o t a l e n e r g y a t A e q u a l s t h e t o t a l e n e r g y a t B
( c o n s e r v a t i o n o f e n e r g y ) . N o t e t h e u s e o f t h e m i n u s s i g n
i n F = - V V .
1 6 . I f 0 = 2 x y z 2 , F = x y i - z j + x 2 k a n d C I s t h e c u r v e x = t 2 , y = 2 t ,
z = t 3
f r o m t = 0 t o t = 1 ,
e v a l u a t e t h e l i n e i n t e g r a l s ( a )
C
d r ,
( b )
F x d r .
C
( a )
A l o n g C ,
d r
=
( 2 t i + 2 j + 3 t 2 k ) d t . T h e n
r
F - d r
2 x y z 2 =
2 ( t 2 ) ( 2 t ) ( t 3 ) 2
=
4 t 9 ,
=
x i + y j + z k
=
t 2 i + 2 t j + t 3 k ,
a n d
f 1
t = 0
I
4 t 9 ( 2 t i + 2 j + 3 t 2 k ) d t
1
1
1
=
i 8 t 1 O d t
+
j
8 t 9 d t
+
k
1 2 t 1 1 d t =
0
0
( b )
A l o n g C , F = x y t - z j + x 2 k = 2 t 3 i - t 3 1 + t 4 k .
T h e n F x d r =
( 2 t 3 i - t 3 j + t 4 k ) x ( 2 t i + 2 j + 3 t 2 k ) d t
d t
=
[ ( - 3 t
- 2 t 4 ) i + ( 2 t ' - 6 t 5 ) j
+ ( 4 t 3 + 2 t 4 ) k ] d t
( ' 1
1
1
- - 4 t 5 ) d t
+ k
( 4 t 3 + 2 t 4 ) d t
n d
F x d r
=
i J
( - 3 t 5 - 2 t 4 ) d t
+
j f o
C
o
S U R F A C E I N T E G R A L S .
1 7 . G i v e a d e f i n i t i o n o f
i
- 3 j + 5 k
A - n d S
o v e r a s u r f a c e S i n t e r m s o f l i m i t o f a s u m .
S
S u b d i v i d e t h e a r e a S i n t o M e l e m e n t s o f a r e a A S o w h e r e p = 1 , 2 , 3 , . . . , M .
C h o o s e a n y p o i n t P p w i t h i n
A S o w h o s e c o o r d i n a t e s a r e ( x p , y p , z , ) .
D e f i n e A ( x p , y p , z p ) = A . L e t n o b e t h e p o s i t i v e u n i t n o r m a l t o
A S a t P . F o r m t h e s u m
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V E C T O R I N T E G R A T I O N
M
A p n p A S p
w h e r e A p n p i s t h e n o r m a l c o m p o n e n t
o f A p a t P p .
N o w t a k e t h e l i m i t o f t h i s s u m a s
M - - a i n s u c h a w a y t h a t t h e l a r g e s t d i -
m e n s i o n o f e a c h A S p a p p r o a c h e s z e r o .
T h i s l i m i t , i f i t e x i s t s , i s c a l l e d t h e
s u r f a c e i n t e g r a l o f t h e n o r m a l c o m p o -
n e n t o f A o v e r S a n d i s d e n o t e d b y
f f A n d S
S
z
1 8 . S u p p o s e t h a t t h e s u r f a c e S h a s p r o j e c t i o n R o n t h e x y p l a n e ( s e e f i g u r e o f P r o b . 1 7 ) . S h o w t h a t
f f A . n d s
=
f f A . n %
S
R
B y P r o b l e m 1 7 , t h e s u r f a c e i n t e g r a l i s t h e l i m i t o f t h e s u m
M
( 1 )
A p . n p A s
p
= 1
9 5
T h e p r o j e c t i o n o f A S p o n t h e x y p l a n e i s
i ( n
p A S p ) k
i
o r
i n , - k i A S P w h i c h i s e q u a l t o
p y p
g y p
s o t h a t A S =
p
T h u s t h e s u m ( 1 ) b e c o m e s
p
( 2 )
M
A X P A n
p = 1
B y t h e f u n d a m e n t a l t h e o r e m o f i n t e g r a l c a l c u l u s t h e l i m i t o f t h i s s u m a s M - o o i n s u c h a m a n n e r t h a t
t h e l a r g e s t A x
P
a n d D y p a p p r o a c h z e r o i s
d x d y
A . n i n - k i
a n d s o t h e r e q u i r e d r e s u l t f o l l o w s .
R
S t r i c t l y s p e a k i n g , t h e r e s u l t A S p =
n p A y p
i s o n l y a p p r o x i m a t e l y t r u e b u t i t c a n b e s h o w n o n c l o s e r
p
k 1
e x a m i n a t i o n t h a t t h e y d i f f e r f r o m e a c h o t h e r b y i n f i n i t e s i m a l s o f o r d e r h i g h e r t h a n A x p A y p , a n d u s i n g t h i s
t h e l i m i t s o f ( 1 ) a n d ( 2 ) c a n i n f a c t b e s h o w n e q u a l .
1 9 . E v a l u a t e
A . n d S , w h e r e A = 1 8 z i - 1 2 j + 3 y k a n d S i s t h a t p a r t o f t h e p l a n e
S
2 x + 3 y + 6 z = 1 2 w h i c h i s l o c a t e d i n t h e f i r s t o c t a n t .
T h e s u r f a c e S a n d i t s p r o j e c t i o n R o n t h e x y p l a n e a r e s h o w n i n t h e f i g u r e b e l o w .
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9 6
V E C T O R I N T E G R A T I O N
z
F r o m P r o b l e m 1 7 ,
2 i + 3 j + 6 k
2 2 + 3 2 + 6 2
J ' f A n
R
d x d y
I n - k (
T o o b t a i n n n o t e t h a t a v e c t o r p e r p e n d i c u l a r t o t h e s u r f a c e 2 x + 3 y + 6 z = 1 2 i s g i v e n b y V ( 2 x + 3 y + 6 z ) _
2 i + 3 j + 6 k ( s e e P r o b l e m 5 o f C h a p t e r 4 ) .
T h e n a u n i t n o r m a l t o a n y p o i n t o f S ( s e e f i g u r e a b o v e ) i s
n
=
T h u s
n k
I I
7 i + 7 j + 7 k
( 7 i + 7 j + 7 k ) k =
7
a n d s o
1 d n - k y -
f i
d x d y .
2
3 i . 6
3 6 z - 3 6 + 1 8 y
3 6 - 1 2 x
A l s o
7 7
u s i n g t h e f a c t t h a t z =
1 2
- 6 -
3 y
f r o m t h e e q u a t i o n o f S .
T h e n
f f A . n d S
=
f f A - n
I n - k J
I f
S R R
( 3 6 7 1 2 x )
S
7
d x d y
=
f f
( 6 - 2 x )
d x d y
R
T o e v a l u a t e t h i s d o u b l e i n t e g r a l o v e r R , k e e p x f i x e d a n d i n t e g r a t e w i t h r e s p e c t t o y f r o m y = 0 ( P i n
t h e f i g u r e a b o v e ) t o y = 1 2 3
2 x
( Q i n t h e f i g u r e a b o v e ) ; t h e n i n t e g r a t e w i t h r e s p e c t t o x f r o m x = 0 t o
x = 6 . I n t h i s m a n n e r R i s c o m p l e t e l y c o v e r e d . T h e i n t e g r a l b e c o m e s
( 6 - 2 x ) d y d x
6
x = 0 y = O
x = 0
( 2 4 - 1 2 x +
3 2 ) d x
=
2 4
I f w e h a d c h o s e n t h e p o s i t i v e u n i t n o r m a l n o p p o s i t e t o t h a t i n t h e f i g u r e a b o v e , w e w o u l d h a v e o b t a i n e d
t h e r e s u l t - - 2 4 .
2 0 . E v a l u a t e
I
I
A - n d S ,
w h e r e A = z i + x j - 3 y 2 z k a n d S i s t h e s u r f a c e o f t h e c y l i n d e r
S
x 2 + y 2 = 1 6 i n c l u d e d i n t h e f i r s t o c t a n t b e t w e e n z = 0 a n d z = 5
.
6 5 ( 1 2 _ 2 x ) / 3
P r o j e c t S o n t h e x z p l a n e a s i n t h e f i g u r e b e l o w a n d c a l l t h e p r o j e c t i o n R . N o t e t h a t t h e p r o j e c t i o n o f
S o n t h e x y p l a n e c a n n o t b e u s e d h e r e . T h e n
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V E C T O R I N T E G R A T I O N
n
d z
f f
d S
=
f f
A
n o r m a l t o x 2 + y 2 = 1 6 i s V ( x 2 + y 2 ) = 2 x i + 2 y j .
T h u s t h e u n i t n o r m a l t o S a s s h o w n i n t h e a d j o i n i n g
f i g u r e , i s
n
2 x i + 2 y j
x i + y j
V ' r ( 2 x
+ ( 2 y ) 2
4
s i n c e x 2 + y 2 = 1 6 o n S .
( z i + x j - 3 y z k )
x i 4 + y j , j
= 4 .
T h e n t h e s u r f a c e i n t e g r a l e q u a l s
f f x z ; x Y d d z
=
Y
R
4 ( x z + x y )
x z
+ x ) d x d z
1 6 - - x 2
( x i + y j }
4
z = 0 x = 0
5
z = O
( 4 z + 8 ) d z
=
9 0
2 1 . E v a l u a t e
O n d S w h e r e 0 =
8
x y z a n d S i s t h e s u r f a c e o f P r o b l e m 2 0 .
S
W e h a v e
A d z
=
f f n
j J
n
R
f f c b n d S
S
U s i n g n = x i 4 y
n j =
4
a s i n P r o b l e m 2 0 , t h i s l a s t i n t e g r a l b e c o m e s
f f x z ( x i + Y i ) d x d z
8
R
z = O x = 0
3
8
( x 2 z i + x z 1 6 - x 2 j ) d x d z
( 3 4 z i + 3 4 z j ) d z =
1 0 0 1 + 1 0 0 j
9 7
2 2 . I f F = y i + ( x - g x z ) j - x y k ,
e v a l u a t e f f ( V x F ) n d S w h e r e S i s t h e s u r f a c e o f t h e s p h e r e
x 2 + y 2 + z 2 = a 2 a b o v e t h e x y p l a n e .
i j k
V x F =
a
a
a
a x
a y
a z
y
x - 2 x z
- - x y
S
= x i + y j - 2 z k
A n o r m a l t o x 2 + y 2 + z 2 = a 2 i s
V ( x 2 + y 2 + z 2 )
=
2 x 1 + 2 y j
+ 2 z k
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9 8
V E C T O R I N T E G R A T I O N
T h e n t h e u n i t n o r m a l n o f t h e f i g u r e a b o v e i s g i v e n b y
n =
s i n c e x 2 + y 2 + z 2 = a 2 .
2 x i + 2 y j + 2 z k
x i + y j + z k
4 x 2 + 4 y 2 + 4 z 2
a
T h e p r o j e c t i o n o f S o n t h e x y p l a n e i s t h e r e g i o n R b o u n d e d b y t h e c i r c l e x 2 + y 2 = a 2 , z = 0 ( s e e f i g -
u r e a b o v e ) . T h e n
r ( V x F ) . n
d S
=
f f ( V x F ) . n
f J
S
R
I f ( x i + y j - 2 z k ) . ( x i + y a j + z k )
d x d y
z 1 a
R
I r a
a
a 2 - x 2 - y 2
x = - a
d y d x
u s i n g t h e f a c t t h a t
z =
a 2 - x 2 - y 2 . T o e v a l u a t e t h e d o u b l e i n t e g r a l , t r a n s f o r m t o p o l a r c o o r d i n a t e s ( p , o )
w h e r e x = p c o s o , y = p s i n 0 a n d d y d x i s r e p l a c e d b y p d p d o .
T h e d o u b l e i n t e g r a l b e c o m e s
( ' 2 7
a
3 p
2 - 2 a 2
p
2 - 7 r
J
d p d
f
a 2 _ p 2
j a
0 = 0
p
= 0
= 0 p = 0
0 = 0
p = 0
f 2 7 7
J
( k = 0
3 ( p 2 - a 2 ) + a
2
d
d p
V / a 2 - p 2
2
( - 3 p / a 2 - p 2
+ a 2
p
p
2 ) d p d o
[ ( a 2 _ p 2 ) 3 / 2
- a 2 V a ` p `
; = i d o
5 2 1 7
_
( a 3 - a 3 ) d o
= 0
2 3 . I f
F = 4 x z i - y 2 j + y z k ,
e v a l u a t e
F - n d S
S
w h e r e S i s t h e s u r f a c e o f t h e c u b e b o u n d e d b y x = 0 ,
x = 1 , y = 0 , y = 1 , z = 0 , z = 1 .
F a c e D E F G : n = i , x = l . T h e n
1
1
J
F - n d S =
( 4 z 1 - y 2 j + y z k )
i d y d z
f j
E F G
_ / ' i
1
J J
4 z d y d z = 2
0 0
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V E C T O R I N T E G R A T I O N
F a c e A B C O : n = - i , x = 0 .
T h e n
f f
F n d S
f T
0
( - y 2
j + y z k ) . ( - i ) d y d z
=
0
A R C O
F a c e A B E F : n = j , y = 1 . T h e n
f f F . n d S
A B E F
( 4 x z i - j + z k ) . j d x d z
F a c e O G D C : n = - j , y = 0 .
T h e n
1
)
( - j ) d x d z
0
f F - n d S
= f f ( 4 x z i
O G D C
F a c e B C D E : n = k , z = 1 . T h e n
f f ( 4 x 1 _ y 2 i + Y k ) . k d x d Y
f F . n d S
=
o
B C D E
F a c e A F G O : n = - k , z = 0 .
T h e n
0
f f F . n d s
f f
J ( - y 2
A d d i n g ,
J ' J " F . n d S =
2 + 0 + ( - 1 ) + 0 +
S
2
+ 0
9 9
- d x d z =
- 1
Y
d x d y
2
2 4 . I n d e a l i n g w i t h s u r f a c e i n t e g r a l s w e h a v e r e s t r i c t e d o u r s e l v e s t o s u r f a c e s w h i c h a r e t w o - s i d e d .
G i v e a n e x a m p l e o f a s u r f a c e w h i c h i s n o t t w o - s i d e d .
T a k e a s t r i p o f p a p e r s u c h a s A B C D a s s h o w n i n
A
t h e a d j o i n i n g f i g u r e . T w i s t t h e s t r i p s o t h a t p o i n t s A a n d B
B f a l l o n D a n d C r e s p e c t i v e l y , a s i n t h e a d j o i n i n g f i g -
u r e . I f n i s t h e p o s i t i v e n o r m a l a t p o i n t P o f t h e s u r f a c e ,
w e f i n d t h a t a s n m o v e s a r o u n d t h e s u r f a c e i t r e v e r s e s
i t s o r i g i n a l d i r e c t i o n w h e n i t r e a c h e s P a g a i n .
I f w e
t r i e d t o c o l o r o n l y o n e s i d e o f t h e s u r f a c e w e w o u l d f i n d
t h e w h o l e t h i n g c o l o r e d . T h i s s u r f a c e , c a l l e d a M o e b i u s
s t r i p ,
i s a n e x a m p l e o f a o n e - s i d e d s u r f a c e . T h i s i s
s o m e t i m e s c a l l e d a n o n - o r i e n t a b l e s u r f a c e . A t w o - s i d e d
s u r f a c e i s o r i e n t a b l e .
V O L U M E I N T E G R A L S
C
A
D
D
2 5 . L e t q = 4 5 x 2 y a n d l e t V d e n o t e t h e c l o s e d r e g i o n b o u n d e d b y t h e p l a n e s 4 x + 2 y + z = 8 , x
= 0 ,
y = 0 , z = 0 . ( a ) E x p r e s s f f f 0 d V a s t h e l i m i t o f a s u m . ( b ) E v a l u a t e t h e i n t e g r a l i n ( a ) .
V
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1 0 0
V E C T O R I N T E G R A T I O N
( a ) S u b d i v i d e r e g i o n V i n t o M c u b e s h a v i n g v o l u m e
A V k
=
k A y k A z k k = 1 , 2 , . . . , M a s i n d i c a t e d
i n t h e a d j o i n i n g f i g u r e a n d l e t ( x k , y k , z k ) b e a
p o i n t w i t h i n t h i s c u b e . D e f i n e p ( x k , y k , z k )
q k . C o n s i d e r t h e s u m
M
( 1 ) O k A V k
k = 1
t a k e n o v e r a l l p o s s i b l e c u b e s i n t h e r e g i o n .
T h e l i m i t o f t h i s s u m , w h e n M - - c i n s u c h a
m a n n e r t h a t t h e l a r g e s t o f t h e q u a n t i t i e s A u k
w i l l a p p r o a c h z e r o , i f i t e x i s t s , i s d e n o t e d b y
f f f 0 d V .
I t c a n b e s h o w n t h a t t h i s l i m i t
V
i s i n d e p e n d e n t o f t h e m e t h o d o f s u b d i v i s i o n i f
i s c o n t i n u o u s t h r o u g h o u t V .
2
I n f o r m i n g t h e s u m ( 1 ) o v e r a l l p o s s i b l e c u b e s i n t h e r e g i o n , i t i s a d v i s a b l e t o p r o c e e d i n a n o r d e r -
l y f a s h i o n . O n e p o s s i b i l i t y i s t o a d d f i r s t a l l t e r m s i n ( 1 ) c o r r e s p o n d i n g t o v o l u m e e l e m e n t s c o n t a i n e d
i n a c o l u m n s u c h a s P Q i n t h e a b o v e f i g u r e . T h i s a m o u n t s t o k e e p i n g x k a n d y k f i x e d a n d a d d i n g o v e r
a l l z k ' s . N e x t , k e e p x k f i x e d b u t s u m o v e r a l l y k ' s . T h i s a m o u n t s t o a d d i n g a l l c o l u m n s s u c h a s P Q
c o n t a i n e d i n a s l a b R S , a n d c o n s e q u e n t l y a m o u n t s t o s u m m i n g o v e r a l l c u b e s c o n t a i n e d i n s u c h a s l a b .
F i n a l l y , v a r y x k . T h i s a m o u n t s t o a d d i t i o n o f a l l s l a b s s u c h a s R S .
I n t h e p r o c e s s o u t l i n e d t h e s u m m a t i o n i s t a k e n f i r s t o v e r z k ' s t h e n o v e r y k ' s a n d f i n a l l y o v e r x k ' s .
H o w e v e r , t h e s u m m a t i o n c a n c l e a r l y b e t a k e n i n a n y o t h e r o r d e r .
( b ) T h e i d e a s i n v o l v e d i n t h e m e t h o d o f s u m m a t i o n o u t l i n e d i n ( a ) c a n b e u s e d i n e v a l u a t i n g t h e i n t e g r a l .
K e e p i n g x a n d y c o n s t a n t , i n t e g r a t e f r o m
z = 0 ( b a s e o f c o l u m n P Q ) t o
z = 8 - 4 x - 2 y ( t o p o f c o l u m n
P Q ) . N e x t k e e p x c o n s t a n t a n d i n t e g r a t e w i t h r e s p e c t t o y . T h i s a m o u n t s t o a d d i t i o n o f c o l u m n s h a v i n g
b a s e s i n t h e x y p l a n e ( z = 0 ) l o c a t e d a n y w h e r e f r o m R ( w h e r e y = 0 ) t o S ( w h e r e 4 x + 2 y = 8 o r y =
4 - 2 x ) ,
a n d t h e i n t e g r a t i o n i s f r o m y = 0 t o y = 4 - 2 x . F i n a l l y , w e a d d a l l s l a b s p a r a l l e l t o t h e y z p l a n e , w h i c h
a m o u n t s t o i n t e g r a t i o n f r o m x = 0 t o x = 2 . T h e i n t e g r a t i o n c a n b e w r i t t e n
f 2
5
4 - 2 x
8 - 4 x - 2 y
2
4 - 2 x
f
4 5 x 2 y d z d y d x
4 5
r
J
X = 0 y = 0
z = 0
x = o y = o
x 2 y ( 8 - 4 x - 2 y ) d y d x
2
4 5 J 3 x 2 ( 4
- 2 x ) 3 d x =
1 2 8
x = 0
N o t e : P h y s i c a l l y t h e r e s u l t c a n b e i n t e r p r e t e d a s t h e m a s s o f t h e r e g i o n V i n w h i c h t h e d e n s i t y
v a r i e s a c c o r d i n g t o t h e f o r m u l a
= 4 5 z y .
2 6 . L e t F = 2 x z i - x j + y 2 k .
E v a l u a t e f f f F d V w h e r e V i s t h e r e g i o n b o u n d e d b y t h e s u r -
f a c e s
x = O , y = 0 , y = 6 , z = x 2 , z = 4 .
V
T h e r e g i o n V i s c o v e r e d ( a ) b y k e e p i n g x a n d y f i x e d a n d i n t e g r a t i n g f r o m z = x 2 t o z = 4 ( b a s e t o t o p
o f
c o l u m n P Q ) , ( b ) t h e n b y k e e p i n g x f i x e d a n d i n t e g r a t i n g f r o m y = 0 t o y = 6 ( R t o S i n t h e s l a b ) , ( c ) f i n a l l y
i n t e g r a t i n g f r o m x = 0 t o x = 2 ( w h e r e z = x 2 m e e t s z = 4 ) .
T h e n t h e r e q u i r e d i n t e g r a l i s
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V E C T O R I N T E G R A T I O N
f 2 5 6 1
x = 0 y = 0
z = x 2
6
4
2 ( ' 6
4
6 4
= i
2 x z d z d y d x -
i
2
f f f
2
x d z d y d x
+
k J J 2 y 2 d z d y d x
J J J
J
J
O x
0 O
x
0
O
x
= 1 2 8 i - 2 4 i
+ 3 8 4 k
1 0 1
. F i n d t h e v o l u m e o f t h e r e g i o n c o m m o n t o t h e i n t e r s e c t i n g c y l i n d e r s x 2 + y 2 = a
2 a n d x 2 + z 2 = a 2
.
1 1
x
R e q u i r e d v o l u m e =
8 t i m e s v o l u m e o f r e g i o n s h o w n i n a b o v e f i g u r e
8
( 2 x z i - x j + y 2 k ) d z d y d x
f a
a
v r a 2 - X 2
J
x = 0 y = 0
z = 0
f a
I
a 2 - x 2
d z d y d x
a 2 - x 2 d y d x
x = 0 Y = O
= 8
( a 2 - - x 2 ) d x
1 6 a 3
3
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V E C T O R I N T E G R A T I O N
1 0 3
4 1 . E v a l u a t e f F . d r w h e r e F = ( x - 3 y ) i + ( y - 2 x ) j a n d C i s t h e c l o s e d c u r v e i n t h e
x y p l a n e , x = 2 c o s t ,
C
y = 3 s i n t f r o m t = 0 t o t = 2 7 1 .
A n s .
6 7 1 , i f C i s t r a v e r s e d i n t h e p o s i t i v e ( c o u n t e r c l o c k w i s e ) d i r e c t i o n .
4 2 . I f T i s a u n i t t a n g e n t v e c t o r t o t h e c u r v e C , r = r ( u ) , s h o w t h a t t h e w o r k d o n e i n m o v i n g a p a r t i c l e i n a f o r c e
f i e l d F a l o n g C i s g i v e n b y
r
F . T d s w h e r e s i s t h e a r c l e n g t h .
4 3 . I f F = ( 2 x + y 2 ) i + ( 3 y - 4 x ) j ,
e v a l u a t e
F d r a r o u n d t h e t r i a n g l e C o f F i g u r e 1 , ( a ) i n t h e i n d i c a t e d
f c
d i r e c t i o n , ( b ) o p p o s i t e t o t h e i n d i c a t e d d i r e c t i o n .
A n s .
( a ) - 1 4 / 3
( b ) 1 4 / 3
( 2 , 1 )
0 1
( 2 , 0 )
x
F i g . 1
F i g . 2
d r a r o u n d t h e c l o s e d c u r v e C o f F i g . 2 a b o v e i f A = ( x - y ) i + ( x + y ) j .
A n s . 2 / 3
4 . E v a l u a t e
f r ' A
4 5 . I f A = ( y - 2 x ) i + ( 3 x + 2 y ) j ,
c o m p u t e t h e c i r c u l a t i o n o f A a b o u t a c i r c l e C i n t h e x y p l a n e w i t h c e n t e r a t
t h e o r i g i n a n d r a d i u s 2 , i f C i s t r a v e r s e d i n t h e p o s i t i v e d i r e c t i o n .
A n s . 8 7 1
- d r i s i n d e p e n d e n t o f t h e c u r v e C j o i n i n g
6 . ( a ) I f A = ( 4 x y - 3 x 2 z 2 ) i + 2 x 2 j - 2 x 3 z k , p r o v e t h a t
f c A
t w o g i v e n p o i n t s .
( b ) S h o w t h a t t h e r e i s a d i f f e r e n t i a b l e f u n c t i o n 4 s u c h t h a t A = V o a n d f i n d i t .
A n s . ( b ) 0 = 2 x 2 y - x 3 z 2 + c o n s t a n t
4 7 . ( a ) P r o v e t h a t F = ( y 2 c o s x + z 3 ) i + ( 2 y s i n x - 4 ) j + ( 3 x z 2 + 2 ) k i s
a c o n s e r v a t i v e f o r c e f i e l d .
( b ) F i n d t h e s c a l a r p o t e n t i a l f o r F .
( c ) F i n d t h e w o r k d o n e i n m o v i n g a n o b j e c t i n t h i s f i e l d f r o m ( 0 , 1 , - 1 ) t o ( 7 1 / 2 , - 1 , 2 ) .
A n s .
( b )
= y 2 s i n x + x z 3 - 4 y + 2 z + c o n s t a n t
( c ) 1 5 + 4 7 T
4
4 8 . P r o v e t h a t F = r 2 r i s c o n s e r v a t i v e a n d f i n d t h e s c a l a r p o t e n t i a l .
A n s .
= 4 + c o n s t a n t
4 9 . D e t e r m i n e w h e t h e r t h e f o r c e f i e l d F = 2 x z i + ( x 2 - y ) j + ( 2 z - x 2 ) k i s c o n s e r v a t i v e o r n o n - c o n s e r v a t i v e .
A n s . n o n - c o n s e r v a t i v e
5 0 . S h o w t h a t t h e w o r k d o n e o n a p a r t i c l e i n m o v i n g i t f r o m A t o B e q u a l s i t s c h a n g e i n k i n e t i c e n e r g i e s a t
t h e s e p o i n t s w h e t h e r t h e f o r c e f i e l d i s c o n s e r v a t i v e o r n o t .
A d r a l o n g t h e c u r v e x 2 + y 2 = 1 , z = 1 i n t h e p o s i t i v e d i r e c t i o n f r o m ( 0 , 1 , 1 ) t o ( 1 , 0 , 1 ) i f
1 . E v a l u a t e f c
A = ( y z + 2 x ) i + x z j + ( x y + 2 z ) k .
A n s . 1
5 2 . ( a ) I f E = r r , i s t h e r e a f u n c t i o n 0 s u c h t h a t E _ _ V ?
I f s o , f i n d i t .
( b ) E v a l u a t e 5 E d r i f C i s a n y
3
s i m p l e c l o s e d c u r v e . A n s .
( a )
+ c o n s t a n t
( b ) 0
C
2
3 . S h o w t h a t
( 2 x c o s y + z s i n y ) d x + ( x z c o s y - x s i n y ) d y + x s i n y d z
i s a n e x a c t d i f f e r e n t i a l .
H e n c e
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1 0 4
V E C T O R I N T E G R A T I O N
s o l v e t h e d i f f e r e n t i a l e q u a t i o n ( 2 x c o s y + z s i n y ) d x + ( x z c o s y - x 2 s i n y ) d y + x s i n y d z
= 0 .
A n s . x 2 c o s y + x z s i n y = c o n s t a n t
5 4 . S o l v e ( a )
( e - Y + 3 x 2 y 2 ) d x + ( 2 x 3 y - x e - Y ) d y = 0 ,
( b )
( z - e - x s i n y ) d x + ( I + e - x c o s y ) d y + ( x - 8 z ) d z = 0 .
A n s .
( a ) x e - y + x 3 y 2 = c o n s t a n t
( b ) x z +
e - x s i n y
+ y - 4 z 2 = c o n s t a n t
5 5 . I f
= 2 x y 2 z + x 2 y ,
e v a l u a t e f
4 ) d r w h e r e C
C
( a ) i s t h e c u r v e x = t , y = t 2 , z = t 3 f r o m t = 0 t o t = 1
( b ) c o n s i s t s o f t h e s t r a i g h t l i n e s f r o m ( 0 , 0 , 0 ) t o ( 1 , 0 , 0 ) , t h e n t o ( 1 , 1 , 0 ) , a n d t h e n t o ( 1 , 1 , 1 ) .
A n s .
( a ) 4 5 i
+ 1 5 j
+ 7 7 k
( b )
2
j
+ 2 k
5 6 . I f F = 2 y i - z j + x k ,
e v a l u a t e f
F x d r a l o n g t h e c u r v e x = c o s t , y = s i n t , z = 2 c o s t
f r o m t = 0
C
t o t = 7 T / 2 .
A n s . ( 2 - 4 ) i + ( 7 T - z ) j
5 7 . I f A = ( 3 x + y ) i - x j + ( y - 2 ) k a n d B = 2 i - 3 j + k ,
e v a l u a t e f c ( A x B ) x d r a r o u n d t h e c i r c l e i n t h e
x y p l a n e h a v i n g c e n t e r a t t h e o r i g i n a n d r a d i u s 2 t r a v e r s e d i n t h e p o s i t i v e d i r e c t i o n .
A n s . 4 7 T ( 7 i + 3 j )
5 8 . E v a l u a t e f J A - n d S f o r e a c h o f t h e f o l l o w i n g c a s e s .
S
( a ) A = y i + 2 x j - z k a n d S i s t h e s u r f a c e o f t h e p l a n e 2 x + y = 6 i n t h e f i r s t o c t a n t c u t o f f b y t h e p l a n e
z = 4 .
( b ) A = ( x + y 2 ) i - 2 x j + 2 y z k a n d S i s t h e s u r f a c e o f t h e p l a n e 2 x + y + 2 z
= 6 i n t h e f i r s t o c t a n t .
A n s . ( a ) 1 0 8
( b ) 8 1
5 9 . I f F = 2 y i - - z j + x 2 k a n d S i s t h e s u r f a c e o f t h e p a r a b o l i c c y l i n d e r y 2 = 8 x i n t h e f i r s t
o c t a n t b o u n d e d
b y t h e p l a n e s y = 4 a n d z = 6 , e v a l u a t e
f f F . n d S .
A n s .
1 3 2
S
6 0 . E v a l u a t e f f
d S o v e r t h e e n t i r e s u r f a c e S o f t h e r e g i o n b o u n d e d b y t h e c y l i n d e r x 2 + z 2 = 9 , x = 0 ,
3
Y = O , z = 0 a n d y = 8 , i f A = 6 z i + ( 2 x + y ) j - x k .
A n s .
1 8 7 T
6 1 . E v a l u a t e
J ' J r n d S o v e r : ( a ) t h e s u r f a c e S o f t h e u n i t c u b e b o u n d e d b y t h e c o o r d i n a t e p l a n e s a n d t h e
S
p l a n e s x = 1 , y = 1 , z = 1 ; ( b ) t h e s u r f a c e o f a s p h e r e o f r a d i u s a w i t h c e n t e r a t ( 0 , 0 , 0 ) .
A n s .
( a ) 3
( b ) 4 7 T a 3
6 2 . E v a l u a t e f f A . n d S o v e r t h e e n t i r e s u r f a c e o f t h e r e g i o n a b o v e t h e x y p l a n e b o u n d e d b y t h e c o n e
S
z 2 = x 2 + y 2 a n d t h e p l a n e z = 4 , i f A = 4 x z i + x y z 2 j + 3 z k
.
A n s .
3 2 0 7 T
6 3 . ( a ) L e t R b e t h e p r o j e c t i o n o f a s u r f a c e S o n t h e x y p l a n e .
P r o v e t h a t t h e s u r f a c e a r e a o f S i s g i v e n b y
i f
f
1 + ( ) + ( ) d x d y i f t h e e q u a t i o n f o r S i s z = f ( x , y ) .
v
Y
R
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V E C T O R I N T E G R A T I O N
1 0 5
( '
( a z 2
+ ( a F ) 2
+ ( F ) 2
( b ) W h a t i s t h e s u r f a c e a r e a i f S h a s t h e e q u a t i o n F ( x , y , z ) = 0 ?
A n s .
y
a F
d x d y
R
a z
6 4 . F i n d t h e s u r f a c e a r e a o f t h e p l a n e x + 2 y + 2 z = 1 2 c u t o f f b y :
( a ) x = 0 , y = 0 , x = 1 , y = 1 ; ( b ) x = 0 , y = 0 ,
a n d x 2 + y 2 = 1 6 .
A n s .
( a ) 3 / 2
( b ) 6 7 7
6 5 . F i n d t h e s u r f a c e a r e a o f t h e r e g i o n c o m m o n t o t h e i n t e r s e c t i n g c y l i n d e r s x 2 + Y 2 = a 2 a n d x 2 + z 2 =
a 2
.
A n s .
1 6 a 2
6 6 . E v a l u a t e ( a ) f f ( V x F ) . n d S a n d
( b ) f f 0 n d S
i f F = ( x + 2 y ) i - 3 z j + x k ,
= 4 x + 3 y - 2 z ,
S
S
a n d S i s t h e s u r f a c e o f 2 x + y + 2 z = 6 b o u n d e d b y x = 0 , x = 1 , y = 0 a n d y
= 2 .
A n s . ( a ) 1
( b ) 2 i + j + 2 k
6 7 . S o l v e t h e p r e c e d i n g p r o b l e m i f S i s t h e s u r f a c e o f 2 x + y + 2 z = 6 b o u n d e d b y x = 0 , y = 0 ,
a n d z = 0 .
A n s . ( a ) 9 / 2
( b ) 7 2 i + 3 6 j + 7 2 k
6 8 . E v a l u a t e
x 2 + y 2 d x d y o v e r t h e r e g i o n R i n t h e x y p l a n e b o u n d e d b y x 2 + y 2 = 3 6 .
A n s .
1 4 4 7 7
R
6 9 . E v a l u a t e
f f f ( 2 x + y )
d V , w h e r e V i s t h e c l o s e d r e g i o n b o u n d e d b y t h e c y l i n d e r
z = 4 - x 2 a n d t h e
V
p l a n e s x = 0 , y = 0 , y = 2 a n d z = 0 .
A n s . 8 0 / 3
7 0 . I f F = ( 2 x 2 - 3 z ) i - 2 x y j - 4 x k , e v a l u a t e ( a )
f f f V . F d V a n d ( b )
f f J ' v x F d V , w h e r e V i s
V
V
t h e c l o s e d r e g i o n b o u n d e d b y t h e p l a n e s x = 0 , y = 0 , z = 0 a n d 2 x + 2 y + z = 4 .
A n s .
( a ) 3
( b ) 3 ( j - k )
8/19/2019 M. R. Spiegel, Vector Analysis, Schaum's Series
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T H E D I V E R G E N C E T H E O R E M O F G A U S S s t a t e s t h a t i f V i s t h e v o l u m e b o u n d e d b y a c l o s e d s u r -
f a c e S a n d A i s a v e c t o r f u n c t i o n o f p o s i t i o n w i t h c o n -
t i n u o u s d e r i v a t i v e s , t h e n
f f f v .
A d V
=
Y
f f A . n d S
S
w h e r e n i s t h e p o s i t i v e ( o u t w a r d d r a w n ) n o r m a l t o S .
3
S T O K E S ' T H E O R E M s t a t e s t h a t i f S i s a n o p e n , t w o - s i d e d s u r f a c e b o u n d e d b y a c l o s e d , n o n - i n t e r -
s e c t i n g c u r v e C ( s i m p l e c l o s e d c u r v e ) t h e n i f A h a s c o n t i n u o u s d e r i v a t i v e s
J A . d r
=
f f ( v x A ) . n d s
c
f f v x
A ) . d s
S
w h e r e C i s t r a v e r s e d i n t h e p o s i t i v e d i r e c t i o n . T h e d i r e c t i o n o f C i s c a l l e d p o s i t i v e i f a n o b s e r v e r ,
w a l k i n g o n t h e b o u n d a r y o f S i n t h i s d i r e c t i o n , w i t h h i s h e a d p o i n t i n g i n t h e d i r e c t i o n o f t h e p o s i t i v e
n o r m a l t o S , h a s t h e s u r f a c e o n h i s l e f t .
G R E E N ' S T H E O R E M I N T H E P L A N E . I f R i s a c l o s e d r e g i o n o f t h e x y p l a n e b o u n d e d b y a s i m p l e
c l o s e d c u r v e C a n d i f M a n d N a r e c o n t i n u o u s f u n c t i o n s o f x
a n d y h a v i n g c o n t i n u o u s d e r i v a t i v e s i n R , t h e n
M d x + N d y
=
( a x
-
a t e ) d x d y
i t ,
w h e r e C i s t r a v e r s e d i n t h e p o s i t i v e ( c o u n t e r c l o c k w i s e d i r e c t i o n . U n l e s s o t h e r w i s e s t a t e d w e s h a l l
a l w a y s a s s u m e f t o m e a n t h a t t h e i n t e g r a l i s d e s c r i b e d i n t h e p o s i t i v e s e n s e .
G r e e n ' s t h e o r e m i n t h e p l a n e i s a s p e c i a l c a s e o f S t o k e s ' t h e o r e m ( s e e P r o b l e m 4 ) . A l s o , i t i s
o f i n t e r e s t t o n o t i c e t h a t G a u s s ' d i v e r g e n c e t h e o r e m i s a g e n e r a l i z a t i o n o f G r e e n ' s t h e o r e m i n t h e
p l a n e w h e r e t h e ( p l a n e ) r e g i o n R a n d i t s c l o s e d b o u n d a r y ( c u r v e ) C a r e r e p l a c e d b y a ( s p a c e ) r e g i o n
V a n d i t s c l o s e d b o u n d a r y ( s u r f a c e ) S . F o r t h i s r e a s o n t h e d i v e r g e n c e t h e o r e m i s o f t e n c a l l e d G r e e n ' s
t h e o r e m i n s p a c e ( s e e P r o b l e m 4 ) .
G r e e n ' s t h e o r e m i n t h e p l a n e a l s o h o l d s f o r r e g i o n s b o u n d e d b y a f i n i t e n u m b e r o f s i m p l e
c l o s e d c u r v e s w h i c h d o n o t i n t e r s e c t ( s e e P r o b l e m s 1 0 a n d 1 1 ) .
R
1 0 6
8/19/2019 M. R. Spiegel, Vector Analysis, Schaum's Series
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D I V E R G E N C E T H E O R E M , S T O K E S ' T H E O R E M , R E L A T E D I N T E G R A L T H E O R E M S
1 0 7
R E L A T E D I N T E G R A L T H E O R E M S .
1 .
+ ( v 0 ) ( v q ) ] d V
=
f f ( c 5 v ) . d s
Y
S
T h i s i s c a l l e d G r e e n ' s f i r s t i d e n t i t y o r t h e o r e m .
2 . f f f ( o
V ' O ) d V
Y
( O v a . - v t h ) . d s
3
T h i s i s c a l l e d G r e e n ' s s e c o n d i d e n t i t y o r s y m m e t r i c a l t h e o r e m . S e e P r o b l e m 2 1 .
3 .
f f f V x
A d V
=
f f ( n
x A ) d S =
f f d s
x A
V 3
S
N o t e t h a t h e r e t h e d o t p r o d u c t o f G a u s s ' d i v e r g e n c e t h e o r e m i s r e p l a c e d b y t h e c r o s s p r o d u c t .
S e e P r o b l e m 2 3 .
4 .
0
' c i r
C
f J ' ( n x V ) d S
=
3
f f d s
x V t h
S
5 .
L e t
r e p r e s e n t e i t h e r a v e c t o r o r s c a l a r f u n c t i o n a c c o r d i n g a s t h e s y m b o l o d e n o t e s a d o t o r
c r o s s , o r a n o r d i n a r y m u l t i p l i c a t i o n . T h e n
f f f v o q j d V
=
f n o q j d 5
= f f d s o q j
V
3
3
A o
J ' f ( n x V )
o i d S
=
J ' f c d S x V )
o
f
C
3 3
G a u s s ' d i v e r g e n c e t h e o r e m , S t o k e s ' t h e o r e m a n d t h e r e s u l t s 3 a n d 4 a r e s p e c i a l c a s e s o f t h e s e .
S e e P r o b l e m s 2 2 , 2 3 , a n d 3 4
.
I N T E G R A L O P E R A T O R F O R M F O R v . I t i s o f i n t e r e s t t h a t , u s i n g t h e t e r m i n o l o g y o f P r o b l e m 1 9 ,
t h e o p e r a t o r V c a n b e e x p r e s s e d s y m b o l i c a l l y i n t h e f o r m
v o
A v
1
o A V
d
0
' A S
w h e r e o d e n o t e s a d o t , c r o s s o r a n o r d i n a r y m u l t i p l i c a t i o n ( s e e P r o b l e m 2 5 ) . T h e r e s u l t p r o v e s u s e -
f u l i n e x t e n d i n g t h e c o n c e p t s o f g r a d i e n t , d i v e r g e n c e a n d c u r l t o c o o r d i n a t e s y s t e m s o t h e r t h a n r e c -
t a n g u l a r ( s e e P r o b l e m s 1 9 , 2 4 a n d a l s o C h a p t e r 7 ) .
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1 0 8
D I V E R G E N C E T H E O R E M , S T O K E S ' T H E O R E M , R E L A T E D I N T E G R A L T H E O R E M S
S O L V E D P R O B L E M S
G R E E N ' S T H E O R E M I N T H E P L A N E
1 . P r o v e G r e e n ' s t h e o r e m i n t h e p l a n e i f C i s a c l o s e d
c u r v e w h i c h h a s t h e p r o p e r t y t h a t a n y s t r a i g h t l i n e
p a r a l l e l t o t h e c o o r d i n a t e a x e s c u t s C i n a t m o s t t w o
p o i n t s .
L e t t h e e q u a t i o n s o f t h e c u r v e s A E B a n d A F B ( s e e
a d j o i n i n g f i g u r e ) b e y = Y j ( x ) a n d y = Y 2 ( x ) r e s p e c t i v e l y .
I f R i s t h e r e g i o n b o u n d e d b y C , w e h a v e
a M
d x d y =
Y
1 b [
f Y 2 ( x )
J
= a
= Y
D M M
d y
I
d x =
f b
x = a
Y
f
e
0
a
M ( x , y )
I Y 2 ( x )
d x
y = Y Y ( x )
b
f M ( Y )
- J
M ( x , Y Y ) d x
-
d x = - 5 M d x
a
b C
T h e n
( 1 ) 5 M d x
=
C
M d x d y
a , y
J a
R
f b
a
b
x
[ M ( x Y 2 )
- M ( x , Y i ) d x
S i m i l a r l y l e t t h e e q u a t i o n s o f c u r v e s E A F a n d E B F b e x = X 1 ( y ) a n d x = X 2 ( y ) r e s p e c t i v e l y . T h e n
J J
a x
d x d y
R
f
f
X 2 ( Y ) a N
( ' f
f
a x
d x d y
J
N ( X 2 , y ) N ( X l , Y )
d Y
y x = X 1 ( y )
e
f N ( x ) d y
+
f N ( X 2 . Y ) d 7
=
i f
N d y
f
e
C
T h e n
( 2 )
5 N d y
=
f f d x d y
f -
R
A d d i n g ( 1 ) a n d ( 2 ) ,
+ N d y
=
f f ( a N -
M ) d x d y .
M d x
a x
a
R Y
2 . V e r i f y G r e e n ' s t h e o r e m i n t h e p l a n e f o r
5 ( x y + y 2 ) d x + x 2 d y
w h e r e C i s t h e
c l o s e d c u r v e o f t h e r e g i o n b o u n d e d b y
y = x a n d y = x 2 .
y = x a n d y = x 2 i n t e r s e c t a t ( 0 , 0 ) a n d ( 1 , 1 ) .
T h e p o s i t i v e d i r e c t i o n i n t r a v e r s i n g C i s a s
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D I V E R G E N C E T H E O R E M , S T O K E S ' T H E O R E M , R E L A T E D I N T E G R A L T H E O R E M S
A l o n g y = x 2 , t h e l i n e i n t e g r a l e q u a l s
L I
( x ) ( x 2 ) + x 4 ) d x
+
( x 2 ) ( 2 x ) d x =
A l o n g y = x f r o m ( 1 , 1 ) t o ( 0 , 0 ) t h e l i n e i n t e g r a l e q u a l s
I
f t
0
( 3 x 3 + x 4 ) d x
1 9
2 0
0
( W ( x ) + x 2 ) d x
+
x 2 d x
f O
2 d x
=
- 1
x
T h e n t h e r e q u i r e d l i n e i n t e g r a l = 2 0 - 1
f f
a N a M
( a x - a M d x d y
=
I
f f ( x _ 2 7 ) d x d 7
f
f X _ Y X+
R x = = 0 y = x 2
I
f X
f
( x - 2 y ) d y I d x
2
t
x
( x Y - Y 2 ) 2
d x
0 x 0
f t
J
0
s o t h a t t h e t h e o r e m i s v e r i f i e d .
( x 4 - x 3 ) d x
1
2 0
3 . E x t e n d t h e p r o o f o f G r e e n ' s t h e o r e m i n t h e p l a n e
g i v e n i n P r o b l e m 1 t o t h e c u r v e s C f o r w h i c h l i n e s
p a r a l l e l t o t h e c o o r d i n a t e a x e s m a y c u t C i n m o r e
t h a n t w o p o i n t s .
-
C o n s i d e r a c l o s e d c u r v e C s u c h a s s h o w n i n t h e a d -
j o i n i n g f i g u r e , i n w h i c h l i n e s p a r a l l e l t o t h e a x e s m a y
m e e t C i n m o r e t h a n t w o p o i n t s . B y c o n s t r u c t i n g l i n e S T
t h e r e g i o n i s d i v i d e d i n t o t w o r e g i o n s R . a n d R 2 w h i c h a r e
o f t h e t y p e c o n s i d e r e d i n P r o b l e m 1 a n d f o r w h i c h G r e e n ' s
t h e o r e m a p p l i e s , i . e . ,
( 1 )
f M d x + N d y
S T U S
f f
( a x -
a M ) d x d y
Y
( 2 )
f
M d x + N d y
=
f f
( a N
-
E M - ) d x d y
Y
S V T S
R 2
0
A d d i n g t h e l e f t h a n d s i d e s o f ( 1 ) a n d ( 2 ) , w e h a v e , o m i t t i n g t h e i n t e g r a n d M d x + N d y i n e a c h c a s e ,
f + f + f + f
T U S
S V T S
S T
T U S
S V T
T S
u s i n g t h e f a c t t h a t
f
= - J
S T
T S
1
2 0
[
- x
( x 2 ) - - y ( x y + y 2 ) ] d x d y
R
R
= f + f f
U S
S V T
T U S V T
1 0 9
x
A d d i n g t h e r i g h t h a n d s i d e s o f ( 1 ) a n d ( 2 ) , o m i t t i n g t h e i n t e g r a n d ,
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1 1 0
D I V E R G E N C E T H E O R E M , S T O K E S ' T H E O R E M , R E L A T E D I N T E G R A L T H E O R E M S
f f
+
f f
R 1
R 2
w h e r e R c o n s i s t s o f r e g i o n s R 1 a n d R 2 .
f f
T h e n
M d x + N d y
=
f f ( i a z
a M ) d x d y a n d t h e t h e o r e m i s p r o v e d .
Y
T U S V T
R
A r e g i o n R s u c h a s c o n s i d e r e d h e r e a n d i n P r o b l e m 1 , f o r w h i c h a n y c l o s e d c u r v e l y i n g i n R c a n b e
c o n t i n u o u s l y s h r u n k t o a p o i n t w i t h o u t l e a v i n g R , i s c a l l e d a s i m p l y - c o n n e c t e d r e g i o n . A r e g i o n w h i c h i s
n o t s i m p l y - c o n n e c t e d i s c a l l e d m u l t i p l y - c o n n e c t e d . W e h a v e s h o w n h e r e t h a t G r e e n ' s t h e o r e m i n t h e p l a n e
a p p l i e s t o s i m p l y - c o n n e c t e d r e g i o n s b o u n d e d b y c l o s e d c u r v e s .
I n P r o b l e m 1 0 t h e t h e o r e m i s e x t e n d e d t o
m u l t i p l y - c o n n e c t e d r e g i o n s .
F o r m o r e c o m p l i c a t e d s i m p l y - c o n n e c t e d r e g i o n s i t m a y b e n e c e s s a r y t o c o n s t r u c t m o r e l i n e s , s u c h a s
S T , t o e s t a b l i s h t h e t h e o r e m .
4 . E x p r e s s G r e e n ' s t h e o r e m i n t h e p l a n e i n v e c t o r n o t a t i o n .
W e h a v e M d x + N d y = ( M i + N j ) . ( d x i + d y j ) = A = M i + N j a n d
t h a t
d r = d x i + d y j .
A l s o , i f A = M i + N j t h e n
V x A
i
j
k
a
a
a
a x
a y
a z
M
N
0
s o t h a t ( ` 7 x A ) k =
a N
_
a M
a x a y
` a z
i
+ a M ' +
( a x
a M ) k
Y
T h e n G r e e n ' s t h e o r e m i n t h e p l a n e c a n b e w r i t t e n
w h e r e d R = d x d y .
5 A . d r
=
C
f f ( V x A ) - k d R
R
r = x i + y j s o
A g e n e r a l i z a t i o n o f t h i s t o s u r f a c e s S i n s p a c e h a v i n g a c u r v e C a s b o u n d a r y l e a d s q u i t e n a t u r a l l y t o
S t o k e s ' t h e o r e m w h i c h i s p r o v e d i n P r o b l e m 3 1 .
A n o t h e r M e t h o d .
A s a b o v e , M d x f N d y = A - d r = A . L d s = A - T d s ,
w h e r e d = T = u n i t t a n g e n t v e c t o r t o C ( s e e a d j a c e n t f i g -
u r e ) .
I f n i s t h e o u t w a r d d r a w n u n i t n o r m a l t o C , t h e n T = k x n
s o t h a t
M d x + N d y = A - T d s = A ( k x n ) d s = ( A x k ) n d s
S i n c e A = M i + N j , B = A x k = ( M I + N j ) x k = N i - M j
a n d
a N
- a m
= V . B . T h e n G r e e n ' s t h e o r e m i n t h e p l a n e b e c o m e s
a x ` a y
f f
n
R
0
x
w h e r e d R = d x d y .
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D I V E R G E N C E T H E O R E M , S T O K E S ' T H E O R E M , R E L A T E D I N T E G R A L T H E O R E M S
1 1 1
G e n e r a l i z a t i o n o f t h i s t o t h e c a s e w h e r e t h e d i f f e r e n t i a l a r c l e n g t h d s o f a c l o s e d c u r v e C i s r e p l a c e d b y
t h e d i f f e r e n t i a l o f s u r f a c e a r e a d S o f a c l o s e d s u r f a c e S , a n d t h e c o r r e s p o n d i n g p l a n e r e g i o n R e n c l o s e d b y
C i s r e p l a c e d b y t h e v o l u m e V e n c l o s e d b y S , l e a d s t o G a u s s ' d i v e r g e n c e t h e o r e m o r G r e e n ' s t h e o r e m i n
s p a c e .
f f
f f f v . B d v
S V
5 . I n t e r p r e t p h y s i c a l l y t h e f i r s t r e s u l t o f P r o b l e m 4 .
I f A d e n o t e s t h e f o r c e f i e l d a c t i n g o n a p a r t i c l e , t h e n f e A d r i s t h e w o r k d o n e i n m o v i n g t h e p a r t i c l e
a r o u n d a c l o s e d p a t h C a n d i s d e t e r m i n e d b y t h e v a l u e o f V x A .
I t f o l l o w s i n p a r t i c u l a r t h a t i f V x A = 0 o r
e q u i v a l e n t l y i f A = V 0 , t h e n t h e i n t e g r a l a r o u n d a c l o s e d p a t h i s z e r o . T h i s a m o u n t s t o s a y i n g t h a t t h e w o r k
d o n e i n m o v i n g t h e p a r t i c l e f r o m o n e p o i n t i n t h e p l a n e t o a n o t h e r i s i n d e p e n d e n t o f t h e p a t h i n t h e p l a n e
j o i n i n g t h e p o i n t s o r t h a t t h e f o r c e f i e l d i s c o n s e r v a t i v e . T h e s e r e s u l t s h a v e a l r e a d y b e e n d e m o n s t r a t e d f o r
f o r c e f i e l d s a n d c u r v e s i n s p a c e ( s e e C h a p t e r 5 ) .
C o n v e r s e l y , i f t h e i n t e g r a l i s i n d e p e n d e n t o f t h e p a t h j o i n i n g a n y t w o p o i n t s o f a r e g i o n , i . e . i f t h e
i n t e g r a l a r o u n d a n y c l o s e d p a t h i s z e r o , t h e n V x A = 0 .
I n t h e p l a n e , t h e c o n d i t i o n V x A = 0 i s e q u i v a l e n t t o
t h e c o n d i t i o n a M = a N w h e r e A = M i + N j .
Y
( '
( 2 , 1 )
6 . E v a l u a t e
J
( 1 0 x 4 - 2 x y 3 ) d x - 3 x 2 y 2 d y a l o n g t h e p a t h x 4 - 6 x y 3 = 4 y 2 .
( 0 , 0 )
A d i r e c t e v a l u a t i o n i s d i f f i c u l t . H o w e v e r , n o t i n g t h a t M = l 0 x 4 - - 2 x y 3 , N = - 3 x 2 y 2 a n d a M = - 6 x y 2
Y
= a x ,
i t f o l l o w s t h a t t h e i n t e g r a l i s i n d e p e n d e n t o f t h e p a t h . T h e n w e c a n u s e a n y p a t h , f o r e x a m p l e t h e
p a t h c o n s i s t i n g o f s t r a i g h t l i n e s e g m e n t s f r o m ( 0 , 0 ) t o ( 2 , 0 ) a n d t h e n f r o m ( 2 , 0 ) t o ( 2 , 1 ) .
2
A l o n g t h e s t r a i g h t l i n e p a t h f r o m ( 0 , 0 ) t o ( 2 , 0 ) , y = 0 , d y = 0 a n d t h e i n t e g r a l e q u a l s
f
1 0 x 4 d x = 6 4 .
x = 0
A l o n g t h e s t r a i g h t l i n e p a t h f r o m ( 2 , 0 ) t o ( 2 , 1 ) , x = 2 , d x = 0 a n d t h e i n t e g r a l e q u a l s
T h e n t h e r e q u i r e d v a l u e o f t h e l i n e i n t e g r a l = 6 4 - 4 = 6 0 .
A n o t h e r M e t h o d .
a
a
I '
y = 0
- 1 2 y 2 d y = - 4 .
S i n c e
a M
= ,
( 1 0 x 4 - 2 x y 3 ) d x - 3 x 2 y 2 d y i s a n e x a c t d i f f e r e n t i a l ( o f 2 x 5 - x 2 y 3 ) .
T h e n
z
( 2 , 1 )
( 2 , 1 )
( 1 0 x 4
- 2 x y 3 ) d x
- 3 x 2 y 2
d y
J
d ( 2 x 5 - x 2 y 3 )
( 0 , 0 )
( 0 , 0 )
7 .
S h o w t h a t t h e a r e a b o u n d e d b y a s i m p l e c l o s e d c u r v e C i s g i v e n b y
I n G r e e n ' s t h e o r e m , p u t M = - y , N = x .
T h e n
f x d y - y d x
f f
( a ( x )
-
a y ( - y ) ) d x d y =
R
w h e r e A i s t h e r e q u i r e d a r e a . T h u s A = 2
x d y - y d x .
2 x 5 - x 2 y 3 I
( 2 , 1 )
=
6 0
( 0 . 0 )
2 J ' x d y - y d x .
. C
2 f
d x d y
2 A
R
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1 1 2
D I V E R G E N C E T H E O R E M , S T O K E S ' T H E O R E M , R E L A T E D I N T E G R A L T H E O R E M S
8 . F i n d t h e a r e a o f t h e e l l i p s e x = a c o s 6 , y = b s i n 6 .
A r e a
=
k f x d y _ y d x
=
r 2 7 7
` 2 2 J
( a c o s 6 ) ( b c o s O ) d O - ( b s i n 8 ) ( - a s i n 6 ) d O
0
2 7 r
2 7 7
f a b
( c o s 2 6 + s i n 2 8 ) d O
= Z
f a b
d O
=
9 . E v a l u a t e
( y - s i n x ) d x + c o s x d y , w h e r e C i s t h e
C
t r i a n g l e o f t h e a d j o i n i n g f i g u r e :
( a ) d i r e c t l y ,
( b ) b y u s i n g G r e e n ' s t h e o r e m i n t h e p l a n e .
( a ) A l o n g O A , y = 0 , d y = 0 a n d t h e i n t e g r a l e q u a l s
f o
0
c o s x
I T / 2
1 r / 2
( 0 - s i n x ) d x + ( c o s x ) ( 0 )
=
f
- s i n x d x
1 7 7 / 2
=
- 1
A l o n g A B , x = L T , d x = 0 a n d t h e i n t e g r a l e q u a l s
( y - 1 ) 0 + O d y
=
0
A l o n g B O , y = ,
f
0
( 2 x - s i n x ) d x
2
7 7
d y = 2 . d x a n d t h e i n t e g r a l e q u a l s
7 7 a b
+
7 T
c o s x d x
= ( - + c o s x +
7 7
s i n x )
1 0 / 2
=
1 - 4 - 2
T h e n t h e i n t e g r a l a l o n g C
= - 1 + 0 + 1 -
7 7
-
2 =
-
7 7
-
2
4
7 7
4
7 7
1
( b ) M = y - s i n x , N = c o s x , a z = - s i n x ,
' a m
= 1
a n d
y
M d x + N d y
= f f
( a x
-
a M ) d x d y
=
y
R
7 T / 2 [ 1 2 x / 1 7
X = 0
y = 0
R
f f ( - s i n x
( - s i n x - 1 ) d y
d x
1 7 / 2
=
J
( - k s i n x - 2 ) d x
=
0
2 x / n
( - y s i n x - y ) ' 0 d x
2
1 7 / 2
? ( - x c o s x + s i n x ) - x I
=
2
7 7
7 T
7 7
0
7 7
4
i n a g r e e m e n t w i t h p a r t ( a ) .
N o t e t h a t a l t h o u g h t h e r e e x i s t l i n e s p a r a l l e l t o t h e c o o r d i n a t e a x e s ( c o i n c i d e n t w i t h t h e c o o r d i -
n a t e a x e s i n t h i s c a s e ) w h i c h m e e t C i n a n i n f i n i t e n u m b e r o f p o i n t s , G r e e n ' s t h e o r e m i n t h e p l a n e s t i l l
h o l d s . I n g e n e r a l t h e t h e o r e m i s v a l i d w h e n C i s c o m p o s e d o f a f i n i t e n u m b e r o f s t r a i g h t l i n e s e g m e n t s .
1 0 . S h o w t h a t G r e e n ' s t h e o r e m i n t h e p l a n e i s a l s o v a l i d f o r a m u l t i p l y - c o n n e c t e d r e g i o n R s u c h a s
s h o w n i n t h e f i g u r e b e l o w .
- 1 ) d y d x
T h e s h a d e d r e g i o n R , s h o w n i n t h e f i g u r e b e l o w , i s m u l t i p l y - c o n n e c t e d s i n c e n o t e v e r y c l o s e d c u r v e
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I V E R G E N C E T H E O R E M , S T O K E S ' T H E O R E M , R E L A T E D I N T E G R A L T H E O R E M S
1 1 3
l y i n g i n R c a n b e s h r u n k t o a p o i n t w i t h o u t l e a v i n g
R . a s i s o b s e r v e d b y c o n s i d e r i n g a c u r v e s u r r o u n d i n g
D E F G D f o r e x a m p l e . T h e b o u n d a r y o f R , w h i c h c o n -
s i s t s o f t h e e x t e r i o r b o u n d a r y A H J K L A a n d t h e i n t e -
r i o r b o u n d a r y D E F G D , i s t o b e t r a v e r s e d i n t h e p o s -
i t i v e d i r e c t i o n , s o t h a t a p e r s o n t r a v e l i n g i n t h i s d i -
r e c t i o n a l w a y s h a s t h e r e g i o n o n h i s l e f t .
I t i s s e e n
t h a t t h e p o s i t i v e d i r e c t i o n s a r e t h o s e i n d i c a t e d i n t h e
a d j o i n i n g f i g u r e .
I n o r d e r t o e s t a b l i s h t h e t h e o r e m , c o n s t r u c t a
l i n e , s u c h a s A D , c a l l e d a c r o s s - c u t , c o n n e c t i n g t h e
e x t e r i o r a n d i n t e r i o r b o u n d a r i e s . T h e r e g i o n b o u n d e d
b y A D E F G D A L K J H A i s s i m p l y - c o n n e c t e d , a n d s o
G r e e n ' s t h e o r e m i s v a l i d . T h e n
M d x + N d y
I
( a x
a M ) d x d y
y
A D E F G D A L K J H A
R
B u t t h e i n t e g r a l o n t h e l e f t , l e a v i n g o u t t h e i n t e g r a n d , i s e q u a l t o
x
f +
I
+ f
+
- I
+
I
A D
D E F G D
D A
A L K J H A
D E F G D A L K J H A
s i n c e f z D = - £ A .
T h u s i f C 1 i s t h e c u r v e A L K J H A , C 2 i s t h e c u r v e D E F G D
c o n s i s t i n g o f C 1 a n d C 2 ( t r a v e r s e d i n t h e p o s i t i v e d i r e c t i o n s ) , t h e n f C +
f 2 =
1 C
M d x + N d y
_ 3 N
-
a x
a M ) d x d y
y
R
I I
a n d C i s t h e b o u n d a r y o f R
f c
a n d s o
1 1 . S h o w t h a t G r e e n ' s t h e o r e m i n t h e p l a n e h o l d s f o r t h e r e g i o n R , o f t h e f i g u r e b e l o w , b o u n d e d b y
t h e s i m p l e c l o s e d c u r v e s C 1 ( A B D E F G A )
,
C 2 ( H K L P H ) , C s ( Q S T U Q ) a n d C 4 ( V W X Y V )
.
C o n s t r u c t t h e c r o s s - c u t s A H , L Q a n d T V . T h e n t h e r e g i o n b o u n d e d b y A H K L Q S T V W X Y V T U Q L P H A -
B D E F G A i s s i m p l y - c o n n e c t e d a n d G r e e n ' s t h e o r e m a p p l i e s . T h e i n t e g r a l o v e r t h i s b o u n d a r y i s e q u a l t o
1 1 1 1 1 f + f + f + f + f + f + I
H H K L
L Q Q S T
T V
V W X Y V
V T
T ( I Q Q L
L P H
H A
A B D E F G A
S i n c e t h e i n t e g r a l s a l o n g A H a n d H A , L Q a n d Q L , T V a n d V T c a n c e l o u t i n p a i r s , t h i s b e c o m e s
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1 1 4
D I V E R G E N C E T H E O R E M , S T O K E S ' T H E O R E M , R E L A T E D I N T E G R A L T H E O R E M S
f + f
H K L Q S T
U L
L P H
Q S T
T U Q
= f + f + f +
f
H K L P H
Q S T U Q
V W X Y V
A B D E F G A
f + f + f
3
C 4
C 1
w h e r e C i s t h e b o u n d a r y c o n s i s t i n g o f C 1 , C 2 , C 3 a n d C 4 . T h e n
( S
M d x + N d y
=
C
a s r e q u i r e d .
+ f
+
f + f + f
V W X Y V
T U Q L P H
A B D E F G A
f +
f
+
f + r
+
f
+
f
J . 1
J
V W X Y V
A B D E F G A
I I
R
- -
a x
a M ) d x d y
y
1 2 . P r o v e t h a t
M d x + N d y = 0 a r o u n d e v e r y c l o s e d c u r v e C i n a s i m p l y - c o n n e c t e d r e g i o n i f a n d
o n l y i f
7 Y -
M = a N e v e r y w h e r e i n t h e r e g i o n .
A s s u m e t h a t M a n d N a r e c o n t i n u o u s a n d h a v e c o n t i n u o u s p a r t i a l d e r i v a t i v e s e v e r y w h e r e i n t h e r e g i o n
R b o u n d e d b y C , s o t h a t G r e e n ' s t h e o r e m i s a p p l i c a b l e . T h e n
f M d x + N d y
C
I f
a M
a x
I f
( a x -
a M ) d x d y
R
y
i n R , t h e n c l e a r l y f M d x + N d y = 0 .
C
C o n v e r s e l y , s u p p o s e
M d x + N d y = 0 f o r a l l c u r v e s C .
I f
a x
- a y
> 0
a t a p o i n t P , t h e n
C y
- A U
a N
-
r o m t h e c o n t i n u i t
o f t h e d e r i v a t i v e s i t f o l l o w s t h a t
.
, a > 0 i n s o m e r e g i o n A s u r r o u n d i n g P .
I f
x
y
I ' i s t h e b o u n d a r y o f A t h e n
M d x + N d y
f f
( a x
a M ) d x d y > 0
y
A
w h i c h c o n t r a d i c t s t h e a s s u m p t i o n t h a t t h e l i n e i n t e g r a l i s z e r o a r o u n d e v e r y c l o s e d c u r v e . S i m i l a r l y t h e
a s s u m p t i o n
a x _
a M < 0 l e a d s t o a c o n t r a d i c t i o n . T h u s
a x
- - = 0 a t a l l p o i n t s .
y
a M
_ a N
a y
a x
i s e q u i v a l e n t t o t h e c o n d i t i o n V x A = 0 w h e r e A = M i + N j
( s e e P r o b l e m s 1 0 a n d 1 1 , C h a p t e r 5 ) . F o r a g e n e r a l i z a t i o n t o s p a c e c u r v e s , s e e P r o b l e m 3 1 .
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I V E R G E N C E T H E O R E M , S T O K E S ' T H E O R E M , R E L A T E D I N T E G R A L T H E O R E M S
1 1 5
1 3 . L e t F
- y i + x j
=
x 2 + y 2
.
( a ) C a l c u l a t e V x F .
( b ) E v a l u a t e
F A a r o u n d a n y c l o s e d p a t h a n d
e x p l a i n t h e r e s u l t s .
i j
( a ) V x F =
a
a a
a x
a y
a z
- y
x
x 2 + Y 2
x 2 + y 2
0
= 0
i n a n y r e g i o n e x c l u d i n g ( 0 , 0 ) .
x 2 + 2
L e t x = p c o s 0 , y = p s i n 0 , w h e r e ( p , o ) a r e p o l a r c o o r d i n a t e s .
b )
F d r
5 _ y d x + x d y
T h e n
d x
a n d s o
- y d x + x d y
y
x 2 + y 2
= d o =
d ( a r c t a n x
F o r a c l o s e d c u r v e A B C D A ( s e e F i g u r e ( a ) b e l o w ) s u r r o u n d i n g t h e o r i g i n ,
= 0 a t A a n d = 2 7 T
f 2 d
f t e r a c o m p l e t e c c u i t b a c k t o A . I n t h i s c a s e t h e l i n e i n t e g r a l e q u a l s
2 .
F i g . ( a )
F i g . ( b )
F o r a c l o s e d c u r v e P Q R S P ( s e e F i g u r e ( b ) a b o v e ) n o t s u r r o u n d i n g t h e o r i g i n , o _ o o a t P a n d
0 0
o _ 0 o a f t e r a c o m p l e t e c i r c u i t b a c k t o P . I n t h i s c a s e t h e l i n e i n t e g r a l e q u a l s
f
d o = 0
.
S i n c e F = M i + N j , V x F = 0 i s e q u i v a l e n t t o
M
= a N a n d t h e r e s u l t s w o u l d s e e m t o c o n t r a -
a y
d i c t t h o s e o f P r o b l e m 1 2 .
H o w e v e r , n o c o n t r a d i c t i o n e x i s t s s i n c e M =
a n d N = x 2 + y 2 d o
X T ?
n o t h a v e c o n t i n u o u s d e r i v a t i v e s t h r o u g h o u t a n y r e g i o n i n c l u d i n g ( 0 , 0 ) , a n d t h i s w a s a s s u m e d i n P r o b . 1 2 .
T H E D I V E R G E N C E T H E O R E M
1 4 . ( a ) E x p r e s s t h e d i v e r g e n c e t h e o r e m i n w o r d s a n d ( b ) w r i t e i t i n r e c t a n g u l a r f o r m .
- p s i n 0 d o + d p c o s
0 ,
d y =
p c o s 0 d o + d p s i n 0
( a ) T h e s u r f a c e i n t e g r a l o f t h e n o r m a l c o m p o n e n t o f a v e c t o r A t a k e n o v e r a c l o s e d s u r f a c e i s e q u a l t o t h e
i n t e g r a l o f t h e d i v e r g e n c e o f A t a k e n o v e r t h e v o l u m e e n c l o s e d b y t h e s u r f a c e .
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1 1 6 D I V E R G E N C E T H E O R E M , S T O K E S ' T H E O R E M , R E L A T E D I N T E G R A L T H E O R E M S
( b ) L e t A = A l i + A 2 j + A s k . T h e n d i v A = V . A = 2 x 1 + a
2 + a s 3
y
T h e u n i t n o r m a l t o S i s n = n 1 i + n 2 j + n 3 k . T h e n n 1 = n i = c o s a , n 2 = n j = c o s 8 a n d
n 3 = n k = c o s y , w h e r e a , , 8 , y a r e t h e a n g l e s w h i c h n m a k e s w i t h t h e p o s i t i v e x , y , z a x e s o r i ,
j , k
d i r e c t i o n s r e s p e c t i v e l y . T h e q u a n t i t i e s c o s a , c o s / 3 , c o s y a r e t h e d i r e c t i o n c o s i n e s o f n . T h e n
A - n
= ( A 1 i + A 2 j + A 3 k )
( c o s a i + c o s , 8 j + c o s y k )
= A l c o s a + A 2 c o s / 3 + A s c o s y
a n d t h e d i v e r g e n c e t h e o r e m c a n b e w r i t t e n
f f f ( a A 1
+
d A 2
+
- A s )
d x d y d z
f f ( A i c o s a + A 2 c o s , 8 + A s c o s y ) d S
V
a x
a y
a z
S
1 5 . D e m o n s t r a t e t h e d i v e r g e n c e t h e o r e m p h y s i c a l l y .
L e t A = v e l o c i t y v a t a n y p o i n t o f a m o v i n g f l u i d . F r o m F i g u r e ( a ) b e l o w :
V o l u m e o f f l u i d c r o s s i n g d S i n A t s e c o n d s
= v o l u m e c o n t a i n e d i n c y l i n d e r o f b a s e d S a n d s l a n t h e i g h t v A t
=
T h e n , v o l u m e p e r s e c o n d o f f l u i d c r o s s i n g d S = v n d S
F i g . ( a )
F r o m F i g u r e ( b ) a b o v e :
F i g . ( b )
T o t a l v o l u m e p e r s e c o n d o f f l u i d e m e r g i n g f r o m c l o s e d s u r f a c e S
f f
S
F r o m P r o b l e m 2 1 o f C h a p t e r 4 ,
d V i s t h e v o l u m e p e r s e c o n d o f f l u i d e m e r g i n g f r o m a v o l u m e e l e
m e n t d V . T h e n
T o t a l v o l u m e p e r s e c o n d o f f l u i d e m e r g i n g f r o m a l l v o l u m e e l e m e n t s i n S
=
f f f v . v d v
V
T h u s
J ' f
f f f
V
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D I V E R G E N C E T H E O R E M , S T O K E S ' T H E O R E M , R E L A T E D I N T E G R A L T H E O R E M S
1 1 7
1 6 . P r o v e t h e d i v e r g e n c e t h e o r e m .
L e t S b e a c l o s e d s u r f a c e w h i c h i s s u c h t h a t a n y l i n e p a r a l l e l t o t h e c o o r d i n a t e a x e s c u t s S i n a t
m o s t t w o p o i n t s . A s s u m e t h e e q u a t i o n s o f t h e l o w e r a n d u p p e r p o r t i o n s , S 1 a n d S 2 , t o b e z = f 1 ( x , y )
a n d
z = f 2 ( x , y ) r e s p e c t i v e l y . D e n o t e t h e p r o j e c t i o n o f t h e s u r f a c e o n t h e x y p l a n e b y R . C o n s i d e r
f f f d v
V
f f f a A 3 d z d y d x
a z
V
f A 3 ( x , y , z )
I f z f d y d x
R
f f
[ A 3 ( x , y , f 2 ) - A 3 ( x , y , f 1 ) ] d y d x
R
F o r t h e u p p e r p o r t i o n S 2 , d y d x = c o s y 2 d S 2 = k . n 2 d S 2 s i n c e t h e n o r m a l n 2 t o S 2 m a k e s a n a c u t e
a n g l e y 2 w i t h k .
F o r t h e l o w e r p o r t i o n S 1 ,
d y d x = - c o s y 1 d S 1 = - k n 1 d S 1
t u s e a n g l e y i w i t h k .
T h e n
a n d
f f A o ( x y r 2 )
d y d x
R
f f
A 3 ( x , y , f 1 ) d y d x
R
f 2 ( x , y ) a A 3
f f
f a z
R
L z = f 1 ( x , y )
s i n c e t h e n o r m a l n 1 t o S 1 m a k e s a n o b -
f A s k n 2
d S 2
S 2
f f A s k . n 1 d S 1
S i
f f A n k . n 2 d s 2
+
f f A s k . n i d S i
S 2
3 1
f f A 2 k . f l d s
S
s o t h a t
( 1 )
f f f d v
=
f f A k . f l d s
V
S
f f A 3 ( x , y , f 2 ) d y d x
R
f f
A s ( x , y , f 1 ) d y d x
=
R
S i m i l a r l y , b y p r o j e c t i n g S o n t h e o t h e r c o o r d i n a t e p l a n e s ,
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1 1 8
D I V E R G E N C E T H E O R E M , S T O K E S ' T H E O R E M , R E L A T E D I N T E G R A L T H E O R E M S
( 2 )
f f f - 1 d v
=
f f A i i . n d s
V
S
( 3 )
f j ' f 4 _ a d v
=
f f A 2 J . n d s
V S
A d d i n g ( 1 ) , ( 2 ) a n d ( 3 ) ,
I
f f A 1
+
a A 2
+
a A 3 ) d V
a x
a y
a z
V
o r
f f f v . A d v
f f ( A i i
+ A 2 j
d S
f f A . n d s
V
S
T h e t h e o r e m c a n b e e x t e n d e d t o s u r f a c e s w h i c h a r e s u c h t h a t l i n e s p a r a l l e l t o t h e c o o r d i n a t e a x e s
m e e t t h e m i n m o r e t h a n t w o p o i n t s . T o e s t a b l i s h t h i s e x t e n s i o n , s u b d i v i d e t h e r e g i o n b o u n d e d b y S i n t o
s u b r e g i o n s w h o s e s u r f a c e s d o s a t i s f y t h i s c o n d i t i o n . T h e p r o c e d u r e i s a n a l o g o u s t o t h a t u s e d i n G r e e n ' s
t h e o r e m f o r t h e p l a n e .
1 7 . E v a l u a t e f f F . n d S ,
w h e r e F = 4 x z i - y 2 j + y z k a n d S i s t h e s u r f a c e o f t h e c u b e b o u n d e d
S
b y x = 0 , x = 1 , y = 0 , y = 1 , z = 0 , z = 1 .
B y t h e d i v e r g e n c e t h e o r e m , t h e r e q u i r e d i n t e g r a l i s e q u a l t o
f f f v . F d v
=
1 f f a x ( 4 x z ) +
a y ( - y 2 )
+ a z ( y z )
d V
V V
=
f f f 4 Z _ Y ) d v
V
=
f
r J
2 z 2 - y z
I
z _ o d y d x
x = o y = o
z = o
( 4 z - y ) d z d y d x
f , f
= o y = o
x = o
y = o
( 2 - y ) d y d x = 3
2
T h e s u r f a c e i n t e g r a l m a y a l s o b e e v a l u a t e d d i r e c t l y a s i n P r o b l e m 2 3 , C h a p t e r 5 .
1 8 . V e r i f y t h e d i v e r g e n c e t h e o r e m f o r A = 4 x i - 2 y 2 j + z 2 k t a k e n o v e r t h e r e g i o n b o u n d e d b y
x 2 + y 2 = 4 , z = 0 a n d z = 3 . '
f f f
o l u
m e i n t e g r a l
f J J V . A d V
a x
( 4 x ) + ( - 2 y 2 ) +
( z 2 )
d V
1 1 1 ( 4 - 4 y + 2 z ) d V
V
f 2
x = - - 2
4 - x 2
3
f ( 4 - 4 y + 2 z ) d z d y d x = 8 4 7 r
y
4 - x 2 z = 0
T h e s u r f a c e S o f t h e c y l i n d e r c o n s i s t s o f a b a s e S 1 ( z = 0 ) , t h e t o p S 2 ( z = 3 ) a n d t h e c o n v e x p o r t i o n
S 3 ( x 2 + y 2 = 4 ) . T h e n
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1 2 0
D I V E R G E N C E T H E O R E M , S T O K E S ' T H E O R E M , R E L A T E D I N T E G R A L T H E O R E M S
B y t h e d i v e r g e n c e t h e o r e m ,
V
( a x i + a y j
f f
f f
d S
A S
B y t h e m e a n - v a l u e t h e o r e m f o r i n t e g r a l s , t h e l e f t s i d e c a n b e w r i t t e n
d i v A 5 f f d V
=
7 1 v - A A V
A V
w h e r e d i v A i s s o m e v a l u e i n t e r m e d i a t e b e t w e e n t h e m a x i m u m a n d m i n i m u m o f d i v A t h r o u g h o u t A V . T h e n
f f
d i v A
A V
T a k i n g t h e l i m i t a s A V - . 0 s u c h t h a t P i s a l w a y s i n t e r i o r t o A V , d i v A a p p r o a c h e s t h e v a l u e d i v A a t
p o i n t P ; h e n c e
f f f d i v A d V
=
A V
f f
A
=
l i m
A S
A V
T h i s r e s u l t c a n b e t a k e n a s a s t a r t i n g p o i n t f o r d e f i n i n g t h e d i v e r g e n c e o f A , a n d f r o m i t a l l t h e p r o p -
e r t i e s m a y b e d e r i v e d i n c l u d i n g p r o o f o f t h e d i v e r g e n c e t h e o r e m .
I n C h a p t e r 7 w e u s e t h i s d e f i n i t i o n t o
e x t e n d t h e c o n c e p t o f d i v e r g e n c e o f a v e c t o r t o c o o r d i n a t e s y s t e m s o t h e r t h a n r e c t a n g u l a r . P h y s i c a l l y ,
f f
A S
A Y
r e p r e s e n t s t h e f l u x o r n e t o u t f l o w p e r u n i t v o l u m e o f t h e v e c t o r A f r o m t h e s u r f a c e A S . I f d i v A i s p o s i t i v e
i n t h e n e i g h b o r h o o d o f a p o i n t P i t m e a n s t h a t t h e o u t f l o w f r o m P i s p o s i t i v e a n d w e c a l l P a s o u r c e . S i m -
i l a r l y , i f d i v A i s n e g a t i v e i n t h e n e i g h b o r h o o d o f P t h e o u t f l o w i s r e a l l y a n i n f l o w a n d P i s c a l l e d a s i n k .
I f i n a r e g i o n t h e r e a r e n o s o u r c e s o r s i n k s , t h e n d i v A = 0 a n d w e c a l l A a s o l e n o i d a l v e c t o r f i e l d .
2 0 . E v a l u a t e j f r . n
d S , w h e r e S i s a c l o s e d s u r f a c e .
S
B y t h e d i v e r g e n c e t h e o r e m ,
f f r - n d S
S
V
w h e r e V i s t h e v o l u m e e n c l o s e d b y S .
3 f f f
d V
=
3 V
V
2 1 . P r o v e
f / f ( q 5 V 2
q - q V 2 ( S ) d V =
f f ( v
q - & V o ) d S .
V
S
V
f f f
( a x + a
+ a z )
d V
=
Y
L e t A = q V b i n t h e d i v e r g e n c e t h e o r e m . T h e n
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D I V E R G E N C E T H E O R E M , S T O K E S ' T H E O R E M , R E L A T E D I N T E G R A L T H E O R E M S
1 2 1
f f f
V . ( v l j ) d V
V
B u t
T h u s
o r
f f c q v i i . n
d S
S
V . ( g v q )
_
O ( V . V ) + ( V ) . ( v b )
_
f f f v . ( v ) d v
=
V
( 1 )
f f f
« v ,
V
f f ( v ) . d s
S
O V 2 0 + ( V O ) ( V
)
( V O ) ( V / ) J d V
f f f [ O v 2
+ ( v O )
( v o ) ) d V f f ( v ) . d S
V
S
w h i c h p r o v e s G r e e n ' s f i r s t i d e n t i t y .
I n t e r c h a n g i n g 0 a n d / i i n ( 1 ) ,
( 2 )
f f f [ & v 2 ( + ( V
d V
=
f f ( v ) . d s
V S
S u b t r a c t i n g ( 2 ) f r o m ( 1 ) , w e h a v e
( 3 )
f f f ( q 5 v 2 b - - V 2 O ) d V =
f f ( v a
-
V
S
w h i c h i s G r e e n ' s s e c o n d i d e n t i t y o r s y m m e t r i c a l t h e o r e m .
I n t h e p r o o f w e h a v e a s s u m e d t h a t 0 a n d
s c a l a r f u n c t i o n s o f p o s i t i o n w i t h c o n t i n u o u s d e r i v a t i v e s o f t h e s e c o n d o r d e r a t l e a s t .
2 2 . P r o v e
j f f v c )
d V
= f f 0 n d S .
V
S
I n t h e d i v e r g e n c e t h e o r e m , l e t A = O C w h e r e C i s a c o n s t a n t v e c t o r . T h e n
f f f v . ( C ) d v =
f f c . n d s
V
S
S i n c e
a n d
f / f
C . V
d V
=
f f C . n d s
V
S
T a k i n g C o u t s i d e t h e i n t e g r a l s ,
'
C . J + f
v O d V
c o J f
O n d S
V
S
a n d s i n c e C i s a n a r b i t r a r y c o n s t a n t v e c t o r ,
f f f v c
d V
f f n d s
V
S
q
a r e
2 3 . P r o v e f f f v x B d V
=
n
x B d S .
V
3
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1 2 2
D I V E R G E N C E T H E O R E M , S T O K E S ' T H E O R E M , R E L A T E D I N T E G R A L T H E O R E M S
I n t h e d i v e r g e n c e t h e o r e m , l e t A = B x C w h e r e C i s a c o n s t a n t v e c t o r . T h e n
f f f v . B x c d v
=
V
f f B x c . n
d S
S
( C x n ) _ ( C x n ) B
i n c e
V . ( B x C ) = C ( V x B )
a n d
( B x C ) n = B f f c . n x B d s
f f f c . v x B d v
=
V
S
T a k i n g C o u t s i d e t h e i n t e g r a l s , '
C
f
J f
O x B d V
= C J J n x B d S
V
S
a n d s i n c e C i s a n a r b i t r a r y c o n s t a n t v e c t o r ,
2 4 . S h o w t h a t a t a n y p o i n t P
f f f v x B d v
=
V
f f n x
B d S
S
C ( n x B ) ,
f f n d s
f f n x
A d S
( a ) V
l i m
A S
a n d
( b ) V x A =
l i m
A S
A 7 - 0
A V
A 7 - 0
A V
w h e r e A V i s t h e v o l u m e e n c l o s e d b y t h e s u r f a c e A S , a n d t h e l i m i t i s o b t a i n e d b y s h r i n k i n g A V
t o t h e p o i n t P .
( a ) F r o m P r o b l e m 2 2 ,
f f f v c i
d V =
f f n
d S .
T h e n
f f f v c
i d V =
f f c t ' n . i
d S .
A V
A S
A V
A S
U s i n g t h e s a m e p r i n c i p l e e m p l o y e d i n P r o b l e m 1 9 , w e h a v e
f f q n . i d S
L A S
A V
w h e r e V V i
i s s o m e v a l u e i n t e r m e d i a t e b e t w e e n t h e m a x i m u m a n d m i n i m u m o f V ( k
i t h r o u g h o u t A V .
T a k i n g t h e l i m i t a s
i n s u c h a w a y t h a t P i s a l w a y s i n t e r i o r t o A V , V V i a p p r o a c h e s t h e v a l u e
f f c t n . i d s
( 1 )
V
t
o v
S
H
A 7 - 0
A V
S i m i l a r l y w e f i n d
( 2 )
( 3 )
7 o . j
f f
d S
l i m
S
A Y - 0 A V
f f c b n . k d s
l i m
S
A V
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D I V E R G E N C E T H E O R E M , S T O K E S ' T H E O R E M , R E L A T E D I N T E G R A L T H E O R E M S
1 2 3
M u l t i p l y i n g ( 1 ) , ( 2 ) , ( 3 ) b y i , j , k r e s p e c t i v e l y , a n d a d d i n g , u s i n g
O c p =
( V ( t . j ) j +
n =
( s e e P r o b l e m 2 0 , C h a p t e r 2 ) t h e r e s u l t f o l l o w s .
( b ) F r o m P r o b l e m 2 3 , r e p l a c i n g B b y A , f f f V x A d V =
A V
T h e n a s i n p a r t ( a ) , w e c a n s h o w t h a t
( O x
l i m
A V m
I
o A V
a n d s i m i l a r r e s u l t s w i t h j a n d k r e p l a c i n g i .
M u l t i p l y i n g b y i , j , k a n d a d d i n g , t h e r e s u l t f o l l o w s .
T h e r e s u l t s o b t a i n e d c a n b e t a k e n a s s t a r t i n g p o i n t s f o r d e f i n i t i o n o f g r a d i e n t a n d c u r l . U s i n g
t h e s e d e f i n i t i o n s , e x t e n s i o n s c a n b e m a d e t o c o o r d i n a t e s y s t e m s o t h e r t h a n r e c t a n g u l a r .
2 5 . E s t a b l i s h t h e o p e r a t o r e q u i v a l e n c e
V O A
=
l i m
L J J d S 0 A
A Y - o A V
d S o
w h e r e o i n d i c a t e s a d o t p r o d u c t , c r o s s p r o d u c t o r o r d i n a r y p r o d u c t .
T o e s t a b l i s h t h e e q u i v a l e n c e , t h e r e s u l t s o f t h e o p e r a t i o n o n a v e c t o r o r s c a l a r f i e l d m u s t b e c o n s i s t -
e n t w i t h a l r e a d y e s t a b l i s h e d r e s u l t s .
I f o i s t h e d o t p r o d u c t , t h e n f o r a v e c t o r A ,
o r
A S
=
l i m 1
f J
d S
A V
A S
e s t a b l i s h e d i n P r o b l e m 1 9 .
S i m i l a r l y i f o i s t h e c r o s s p r o d u c t ,
c u r l A
=
O x A
A V l i m
_ - L
- 0 A V
i i d s x A
A S
l i m 1
f f n x A d S
A V - ' o
A V
A S
e s t a b l i s h e d i n P r o b l e m 2 4 ( b ) .
A l s o i f o i s o r d i n a r y m u l t i p l i c a t i o n , t h e n f o r a s c a l a r 0 ,
V o
l i m
o r
o
l i m
f
d S
f d s o q
A V - 0 A V
A S
A S
I
n x A d S .
f f
d S
A S
A S
A S
d i v A
= l i m 1 j J d S A
A V - o
A V
e s t a b l i s h e d i n P r o b l e m 2 4 ( a ) .
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1 2 4
D I V E R G E N C E T H E O R E M , S T O K E S ' T H E O R E M , R E L A T E D I N T E G R A L T H E O R E M S
2 6 . L e t S b e a c l o s e d s u r f a c e a n d l e t r d e n o t e t h e p o s i t i o n v e c t o r o f a n y p o i n t ( x , y , z ) m e a s u r e d f r o m
a n o r i g i n 0 . P r o v e t h a t
f f n . r
d S
T 3
S
i s e q u a l t o ( a ) z e r o i f 0 l i e s o u t s i d e S ; ( b ) 4 7 7 i f 0 l i e s i n s i d e S . T h i s r e s u l t i s k n o w n a s G a u s s '
t h e o r e m .
( a ) B y t h e d i v e r g e n c e t h e o r e m , J J
n n 3 r d l
= j j J
V .
r
d V .
S
V
B u t V .
3
= 0 ( P r o b l e m 1 9 , C h a p t e r 4 ) e v e r y w h e r e w i t h i n V p r o v i d e d r
0 i n V . i . e . p r o v i d e d 0
r 3
i s o u t s i d e o f V a n d t h u s o u t s i d e o f S .
T h e n f f n n
r
d S = 0 .
r 3
S
( b )
I f 0 i s i n s i d e S , s u r r o u n d 0 b y a s m a l l s p h e r e s o f r a d i u s a .
L e t ' r d e n o t e t h e r e g i o n b o u n d e d b y S a n d
s . T h e n b y t h e d i v e r g e n c e t h e o r e m
f f
n r
d S
r 3
f f r d s +
r 3
S + S
S
s i n c e r / 0 i n - r . T h u s
I f
n * - r
d S
r 3
f f f v .
d V
S
T
f f r r d s
=
-
f f r d 5
S
N o w o n s , r = a , n = -
f I
d S
T 3
S
d S
=
I f
d S
=
2
f f d s
a
S
S
2 7 . I n t e r p r e t G a u s s ' t h e o r e m ( P r o b l e m 2 6 ) g e o m e t r i c a l l y .
L e t d S d e n o t e a n e l e m e n t o f s u r f a c e a r e a a n d
c o n n e c t a l l p o i n t s o n t h e b o u n d a r y o f d S t o 0 ( s e e
a d j o i n i n g f i g u r e ) , t h e r e b y f o r m i n g a c o n e . L e t d O b e
t h e a r e a o f t h a t p o r t i o n o f a s p h e r e w i t h 0 a s c e n t e r
a n d r a d i u s r w h i c h i s c u t o u t b y t h i s c o n e ; t h e n t h e
s o l i d a n g l e s u b t e n d e d b y d S a t 0 i s d e f i n e d a s d w =
r
a n d i s n u m e r i c a l l y e q u a l t o t h e a r e a o f t h a t p o r -
2
t i o n o f a s p h e r e w i t h c e n t e r 0 a n d u n i t r a d i u s c u t o u t
b y t h e c o n e . L e t n b e t h e p o s i t i v e u n i t n o r m a l t o d S
a n d c a l l 0 t h e a n g l e b e t w e e n n a n d r ;
t h e n c o s 0 =
n T r r
.
A l s o , d O _ ± d S c o s 6 = ± n r r d S s o t h a t
d w
n n 3 r d S
,
t h e + o r - b e i n g c h o s e n a c c o r d i n g
a s n a n d r f o r m a n a c u t e o r a n o b t u s e a n g l e 0 w i t h
e a c h o t h e r .
r s o t h a t
n . r = - r / a . r
_ -
r . r
-
a 2
_ .
1
a n d
a
r 3
a 3 a 4
a 4 -
0
= 0
4 7 T a 2
a 2
4 7 T
L e t S b e a s u r f a c e , a s i n F i g u r e ( a ) b e l o w , s u c h t h a t a n y l i n e m e e t s S i n n o t m o r e t h a n t w o p o i n t s .
I f 0 l i e s o u t s i d e S , t h e n a t a p o s i t i o n s u c h a s 1 ,
1 1 3 r d S = d w ; w h e r e a s a t t h e c o r r e s p o n d i n g p o s i t i o n 2 ,
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D I V E R G E N C E T H E O R E M , S T O K E S ' T H E O R E M , R E L A T E D I N T E G R A L T H E O R E M S
1 2 5
n 3 r
d S = - d o ) . A n i n t e g r a t i o n o v e r t h e s e t w o r e g i o n s g i v e s z e r o , s i n c e t h e c o n t r i b u t i o n s t o t h e s o l i d
r
a n g l e c a n c e l o u t . W h e n t h e i n t e g r a t i o n i s p e r f o r m e d o v e r S i t t h u s f o l l o w s t h a t
f f - _ r
d S = 0 , s i n c e f o r
e v e r y p o s i t i v e c o n t r i b u t i o n t h e r e i s a n e g a t i v e o n e .
S
I n c a s e 0 i s i n s i d e S . h o w e v e r , t h e n a t a p o s i t i o n s u c h a s 3 ,
n n 3 r d S = d a ) a n d a t 4 , n 3 r d S = d c v
, s o t h a t t h e c o n t r i b u t i o n s a d d i n s t e a d o f c a n c e l . T h e t o t a l s o l i d a n g l e i n t h i s c a s e i s e q u a l t o t h e a r e a o f a
u n i t s p h e r e w h i c h i s 4 7 T , s o t h a t
F i g . ( a )
F i g . ( b )
F o r s u r f a c e s S , s u c h t h a t a l i n e m a y m e e t S i n m o r e t h a n t w o p o i n t s , a n e x a c t l y s i m i l a r s i t u a t i o n
h o l d s a s i s s e e n b y r e f e r e n c e t o F i g u r e ( b ) a b o v e .
I f 0 i s o u t s i d e S , f o r e x a m p l e , t h e n a c o n e w i t h v e r t e x
a t 0 i n t e r s e c t s S a t a n e v e n n u m b e r o f p l a c e s a n d t h e c o n t r i b u t i o n t o t h e s u r f a c e i n t e g r a l i s z e r o s i n c e t h e
s o l i d a n g l e s s u b t e n d e d a t 0 c a n c e l o u t i n p a i r s .
I f 0 i s i n s i d e S , h o w e v e r , a c o n e h a v i n g v e r t e x a t 0 i n -
t e r s e c t s S a t a n o d d n u m b e r o f p l a c e s a n d s i n c e c a n c e l l a t i o n o c c u r s o n l y f o r a n e v e n n u m b e r o f t h e s e ,
t h e r e w i l l a l w a y s b e a c o n t r i b u t i o n o f 4 7 T f o r t h e e n t i r e s u r f a c e S .
2 8 . A f l u i d o f d e n s i t y p ( x , y , z , t ) m o v e s w i t h v e l o c i t y v ( x , y , z , t ) . I f t h e r e a r e n o s o u r c e s o r s i n k s ,
p r o v e t h a t
O J + a p = 0
w h e r e J = p v
V i s
C o n s i d e r a n a r b i t r a r y s u r f a c e e n c l o s i n g a v o l u m e V o f t h e f l u i d . A t a n y t i m e t h e m a s s o f f l u i d w i t h i n
M
=
f f f p d v
V
T h e t i m e r a t e o f i n c r e a s e o f t h i s m a s s i s
' a m
a t
a
f f f p d v
=
f f f d v
t
V V
T h e m a s s o f f l u i d p e r u n i t t i m e l e a v i n g V i s
f f p v n d S
f f n . r
d S = 4 T r .
r
S
S
( s e e P r o b l e m 1 5 ) a n d t h e t i m e r a t e o f i n c r e a s e i n m a s s i s t h e r e f o r e
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1 2 6
D I V E R G E N C E T H E O R E M , S T O K E S ' T H E O R E M , R E L A T E D I N T E G R A L T H E O R E M S
- f f p v . n d S
S
b y t h e d i v e r g e n c e t h e o r e m . T h e n
5 f f a p d v
t
V
o r
f f f v . p v ) +
a p ) d V
a t
V
= 0
S i n c e V i s a r b i t r a r y , t h e i n t e g r a n d , a s s u m e d c o n t i n u o u s , m u s t b e i d e n t i c a l l y z e r o , b y r e a s o n i n g s i m i -
l a r t o t h a t u s e d i n P r o b l e m 1 2 . T h e n
V J + L P
=
0 w h e r e J = p v
a t
T h e e q u a t i o n i s c a l l e d t h e c o n t i n u i t y e q u a t i o n .
I f p i s a c o n s t a n t , t h e f l u i d i s i n c o m p r e s s i b l e a n d V . v =
0 , i . e . v i s s o l e n o i d a l .
T h e c o n t i n u i t y e q u a t i o n a l s o a r i s e s i n e l e c t r o m a g n e t i c t h e o r y , w h e r e p i s t h e c h a r g e d e n s i t y a n d
J = p v i s t h e c u r r e n t d e n s i t y .
2 9 . I f t h e t e m p e r a t u r e a t a n y p o i n t ( x , y , z ) o f a s o l i d a t t i m e t i s U ( x , y , z , t ) a n d i f K , p
a n d c a r e r e -
s p e c t i v e l y t h e t h e r m a l c o n d u c t i v i t y , d e n s i t y a n d s p e c i f i c h e a t o f t h e s o l i d , a s s u m e d c o n s t a n t ,
s h o w t h a t
a t -
k V 2 U
w h e r e k = K / p c
L e t V b e a n a r b i t r a r y v o l u m e l y i n g w i t h i n t h e s o l i d , a n d l e t S d e n o t e i t s s u r f a c e . T h e t o t a l f l u x o f
h e a t a c r o s s S , o r t h e q u a n t i t y o f h e a t l e a v i n g S p e r u n i t t i m e , i s
f f K v u ) . f l d S
S
T h u s t h e q u a n t i t y o f h e a t e n t e r i n g S p e r u n i t t i m e i s
( 1 )
f f ( K V u ) . n d S
=
f f f v . ( K v u )
d V
S
V
b y t h e d i v e r g e n c e t h e o r e m . T h e h e a t c o n t a i n e d i n a v o l u m e V i s g i v e n b y
f f f c p U d V
V
T h e n t h e t i m e r a t e o f i n c r e a s e o f h e a t i s
( 2 )
a
J J J
c p U d V
=
f f f c p d v
a t
t
V V
E q u a t i n g t h e r i g h t h a n d s i d e s o f ( 1 ) a n d ( 2 ) ,
f f f
[ c p a U - -
V ( K V U ) ] d V
=
0
V
a n d s i n c e V i s a r b i t r a r y , t h e i n t e g r a n d , a s s u m e d c o n t i n u o u s , m u s t b e i d e n t i c a l l y z e r o s o t h a t
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D I V E R G E N C E T H E O R E M , S T O K E S ' T H E O R E M , R E L A T E D I N T E G R A L T H E O R E M S
1 2 7
o r i f K , c , ) o a r e c o n s t a n t s ,
c p a c
=
V ( K D U )
T h e q u a n t i t y k i s c a l l e d t h e d i f f u s i v i t y .
F o r s t e a d y - s t a t e h e a t f l o w
t i m e ) t h e e q u a t i o n r e d u c e s t o L a p l a c e ' s e q u a t i o n V U = 0 .
S T O K E S ' T H E O R E M
( i . e .
a U
= 0
o r U i s i n d e p e n d e n t o f
3 0 . ( a ) E x p r e s s S t o k e s ' t h e o r e m i n w o r d s a n d ( b ) w r i t e i t i n r e c t a n g u l a r f o r m .
( a ) T h e l i n e i n t e g r a l o f t h e t a n g e n t i a l c o m p o n e n t o f a v e c t o r A t a k e n a r o u n d a s i m p l e c l o s e d c u r v e C i s
e q u a l t o t h e s u r f a c e i n t e g r a l o f t h e n o r m a l c o m p o n e n t o f t h e c u r l o f A t a k e n o v e r a n y s u r f a c e S h a v i n g
C a s i t s b o u n d a r y .
( b ) A s i n P r o b l e m 1 4 ( b ) ,
A = A 1 i + A 2 j + A 3 k , n = c o s a i + c o s / 3 j + c o s y k
T h e n
V X A =
A d r
i j
a a
a
a x a y a z
A l
A 2 A 3
- a U
k 0 2 U
a t
c
A 3
_
a A 2 )
i
+
a
_ a A 3
+
A 2
-
a A 1
k
a y
a z ( a z
a x
a x a y
( a A 3
_
a A 2 )
c o s a +
( A l a _ a A 3
a A e
_
a A 1
) c o s y
z a z a x )
c o s , C 3
+ (
a x
a y
a n d S t o k e s ' t h e o r e m b e c o m e s
f a A 3 a A 2
c
a A 1
_ 3 A 3
_ 3 A 2
a A 1
[ ( a y
a z ) o s
a +
( a z - a x ) c o s
+ ( a x
a
) c o s y ] d S =
f A i d x + A 2 d y + A s d z
y
C
S
L e t S b e a s u r f a c e w h i c h i s s u c h t h a t i t s p r o j e c t i o n s
o n t h e x y , y z a n d x z p l a n e s a r e r e g i o n s b o u n d e d b y s i m p l e
c l o s e d c u r v e s , a s i n d i c a t e d i n t h e a d j o i n i n g f i g u r e . A s -
s u m e S t o h a v e r e p r e s e n t a t i o n
z = f ( x , y ) o r x = g ( y , z ) o r
y = , h ( x , z ) ,
w h e r e f , g , h a r e s i n g l e - v a l u e d , c o n t i n u o u s a n d
d i f f e r e n t i a b l e f u n c t i o n s . W e m u s t s h o w t h a t
f f ( v x A ) . n d s
=
f f [ v x ( A i i + 1 4 2 i + A s k ) ] . n d s
S
S
n _
f
C
w h e r e C i s t h e b o u n d a r y o f S .
z
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1 2 8
D I V E R G E N C E T H E O R E M , S T O K E S ' T H E O R E M , R E L A T E D I N T E G R A L T H E O R E M S
C o n s i d e r f i r s t
f f
d S .
S
i
j
k
S i n c e
V x ( A 1 i )
_
a a a
a x
a y a z
A 1
0 0
a A I
a 4 1
a z a y
( 1 )
- a A ' n - k )
y
I f z = f ( x , y ) i s t a k e n a s t h e e q u a t i o n o f S , t h e n t h e p o s i t i o n v e c t o r t o a n y p o i n t o f S i s r = x i
+ y j + z k =
x i + y j + f ( x , y ) k s o t h a t
a r = j +
a z
k = j +
o f
k . B u t
a r
i s a v e c t o r t a n g e n t t o S ( s e e P r o b l e m 2 5 ,
C h a p t e r 3 ) a n d t h u s
p e r p e n d i c u l a r t o n , s o t h a t
a y a y
S u b s t i t u t e i n ( 1 ) t o o b t a i n
a A 1
a
d S
( a z
.
n
a r
=
a z
0
o r
- a z
n . k
a y
a y
a y
o r
( 2 ) d S
a a A 1
a z
a a 1
y
a A 1 a A 1 a z )
n . k d S
a y + a z a y
N o w o n S , A 1 ( x , y , z ) = A 1 ( x , y , f ( x , y ) ) = F ( x , y ) ; h e n c e
T h e n
a A 1
a A 1 a z
=
a F
a y + a z a y
a y
x ( A 1 i ) ] n d S
=
-
a F
n k d S
=
- -
a F
d x d y
a y
a y
f f [ V x ( A 1 i ) ] n d S
=
S
d x w h e r e F i s t h e b o u n d a r y o f R .
S i n c e a t e a c h p o i n t ( x , y ) o f F t h e v a l u e o f F i s t h e s a m e a s t h e
I T ,
F
f f 4
a n d ( 2 ) b e c o m e s
w h e r e R i s t h e p r o j e c t i o n o f S o n t h e x y p l a n e . B y G r e e n ' s t h e o r e m f o r t h e p l a n e t h e l a s t i n t e g r a l e q u a l s
v a l u e o f A 1 a t e a c h p o i n t ( x , y , z ) o f C , a n d s i n c e d x i s t h e s a m e f o r b o t h c u r v e s , w e m u s t h a v e
o r
f T F d x
f
A 1 d x
0
f f [ V x ( A 1 i ) ]
n d S
A , d x
S
S i m i l a r l y , b y p r o j e c t i o n s o n t h e o t h e r c o o r d i n a t e p l a n e s ,
f f
d S
S
A 2 d y
A s d z
f f
9
S
C
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I V E R G E N C E T H E O R E M , S T O K E S ' T H E O R E M , R E L A T E D I N T E G R A L T H E O R E M S
1 2 9
T h u s b y a d d i t i o n ,
f f ( v x A . n d s
S
T h e t h e o r e m i s a l s o v a l i d f o r s u r f a c e s S w h i c h m a y n o t s a t i s f y t h e r e s t r i c t i o n s i m p o s e d a b o v e . F o r
a s s u m e t h a t S c a n b e s u b d i v i d e d i n t o s u r f a c e s S 1 , S 2 , . . . S k w i t h b o u n d a r i e s C 1 , C 2 , . . . C k w h i c h d o s a t i s f y
t h e r e s t r i c t i o n s . T h e n S t o k e s ' t h e o r e m h o l d s f o r e a c h s u c h s u r f a c e . A d d i n g t h e s e s u r f a c e i n t e g r a l s , t h e
t o t a l s u r f a c e i n t e g r a l o v e r S i s o b t a i n e d . A d d i n g t h e c o r r e s p o n d i n g l i n e i n t e g r a l s o v e r C 1 , C 2 , . . . C k , t h e
l i n e i n t e g r a l o v e r C i s o b t a i n e d .
3 2 . V e r i f y S t o k e s ' t h e o r e m f o r A = ( 2 x - y ) i - y z 2 j - y 2 z k ,
w h e r e S i s t h e u p p e r h a l f s u r f a c e o f
t h e s p h e r e x 2 + y 2 + z 2 = 1
a n d C i s i t s b o u n d a r y .
T h e b o u n d a r y C o f S i s a c i r c l e i n t h e x y p l a n e o f r a d i u s o n e a n d c e n t e r a t t h e o r i g i n . L e t x = c o s t ,
y = s i n t , z = 0 , 0 < t < 2 7 7 b e p a r a m e t r i c e q u a t i o n s o f C . T h e n
f ( 2 x
- y ) d x - y z 2 d y - y 2 z d z
C
r 2 n
=
J
( 2 c o s t - s i n t ) ( - s i n t ) d t
=
? r
0
i j
k
A l s o ,
V x A
=
a a
a
a x a y a z
2 x - y
- y Z 2 - y 2 z
k
T h e n
f f
( V x
A ) n d S
=
f f k . n d S
=
J ' f d x d y
S S
R
s i n c e n k d S = d x d y a n d R i s t h e p r o j e c t i o n o f S o n t h e x y p l a n e . T h i s l a s t i n t e g r a l e q u a l s
x - = -
( i
= x 2
t
, / , = x 2
i
d x
=
7
f d y d x
=
4 f
( '
J
d y d x
4 I -
r
- x 2
- V i
a n d S t o k e s ' t h e o r e m i s v e r i f i e d .
3 3 . P r o v e t h a t a n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n t h a t A A . d r = 0 f o r e v e r y c l o s e d c u r v e C i s
t h a t V x A = 0 i d e n t i c a l l y .
S u f f i c i e n c y . S u p p o s e O x A = 0 . T h e n b y S t o k e s ' t h e o r e m
f
C
f f ( V X A ) . n d S
=
S
0
N e c e s s i t y . S u p p o s e
f
A - d r = 0 a r o u n d e v e r y c l o s e d p a t h C , a n d a s s u m e O x A
0 a t s o m e p o i n t
C
P . T h e n a s s u m i n g O x A i s c o n t i n u o u s t h e r e w i l l b e a r e g i o n w i t h P a s a n i n t e r i o r p o i n t , w h e r e O x A # 0 .
L e t S b e a s u r f a c e c o n t a i n e d i n t h i s r e g i o n w h o s e n o r m a l n a t e a c h p o i n t h a s t h e s a m e d i r e c t i o n a s O x A ,
i . e . O x A = a n w h e r e 0 6 i s a p o s i t i v e c o n s t a n t . L e t C b e t h e b o u n d a r y o f S . T h e n b y S t o k e s ' t h e o r e m
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1 3 0
D I V E R G E N C E T H E O R E M , S T O K E S ' T H E O R E M , R E L A T E D I N T E G R A L T H E O R E M S
f A - d r
C
J ' f ( n x v )
x B d S .
S
f f ( V x A ) . n
d S =
S
w h i c h c o n t r a d i c t s t h e h y p o t h e s i s t h a t 5 A d r = 0 a n d s h o w s t h a t V x A = 0 .
P 2
I t f o l l o w s t h a t V x A = 0 i s a l s o a n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n f o r a l i n e I n t e g r a l
A . d r
t o b e i n d e p e n d e n t o f t h e p a t h j o i n i n g p o i n t s P 1 a n d P 2 .
( S e e P r o b l e m s 1 0 a n d 1 1 , C h a p t e r 5 . )
3 4 . P r o v e
j I
d r x B
I n S t o k e s ' t h e o r e m , l e t A = B x C w h e r e C i s a c o n s t a n t v e c t o r , T h e n
f f [ V x ( B x C ) ] n d S
S
f f
[ ( C V ) B - C ( V . B ) ] n d S
S
f f
n d S
S
c t f f n . n
d S
>
0
S
I I
[ C ( V B ) ] n d S
=
f f c . [ V ( B n ) ] d S
S
- - -
f f c . [ n ( V B ) ] d S
S
C f f [ V ( B n ) - n V . B ) ] d S
=
C f f ( n x V ) x B d S
S S
S i n c e C i s a n a r b i t r a r y c o n s t a n t v e c t o r 5 d r x B
=
f f ( n x v )
x B d S
S
P 1
3 5 . I f A S i s a s u r f a c e b o u n d e d b y a s i m p l e c l o s e d c u r v e C , P i s a n y p o i n t o f A S n o t o n C a n d n i s
a u n i t n o r m a l t o A S a t P , s h o w t h a t a t P
A A
( c u r l A ) . n
=
l i r a C
A S
w h e r e t h e l i m i t i s t a k e n i n s u c h a w a y t h a t A S s h r i n k s t o P .
B y S t o k e s ' t h e o r e m ,
f f ( c u r l
A ) n d S
=
5
A A . d r .
A S
C
U s i n g t h e m e a n v a l u e t h e o r e m f o r i n t e g r a l s a s i n P r o b l e m s 1 9 a n d 2 4 , t h i s c a n b e w r i t t e n
( c u r l A ) n
f A d r
C
A s
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D I V E R G E N C E T H E O R E M , S T O K E S ' T H E O R E M , R E L A T E D I N T E G R A L T H E O R E M S
1 3 1
a n d t h e r e q u i r e d r e s u l t f o l l o w s u p o n t a k i n g t h e l i m i t a s A S - 0 .
T h i s c a n b e u s e d a s a s t a r t i n g p o i n t f o r d e f i n i n g c u r l A ( s e e P r o b l e m 3 6 ) a n d i s u s e f u l i n o b t a i n i n g
c u r l A i n c o o r d i n a t e s y s t e m s o t h e r t h a n r e c t a n g u l a r . S i n c e
,
A A . d r i s c a l l e d t h e c i r c u l a t i o n o f A a b o u t
C
C , t h e n o r m a l c o m p o n e n t o f t h e c u r l c a n b e i n t e r p r e t e d p h y s i c a l l y a s t h e l i m i t o f t h e c i r c u l a t i o n p e r u n i t
a r e a , t h u s a c c o u n t i n g f o r t h e s y n o n y m r o t a t i o n o f A ( r o t A ) i n s t e a d o f c u r l o f A .
3 6 . I f
c u r l A i s d e f i n e d a c c o r d i n g t o t h e l i m i t i n g p r o c e s s o f P r o b l e m 3 5 , f i n d t h e z c o m p o n e n t o f
c u r l A .
z
L e t E F G H b e a r e c t a n g l e p a r a l l e l t o t h e x y p l a n e w i t h i n t e r i o r p o i n t P ( x , y , z ) t a k e n a s m i d p o i n t , a s
s h o w n i n t h e f i g u r e a b o v e . L e t A l a n d A 2 b e t h e c o m p o n e n t s o f A a t P i n t h e p o s i t i v e x a n d y d i r e c t i o n s
r e s p e c t i v e l y .
I f C i s t h e b o u n d a r y o f t h e r e c t a n g l e , t h e n
J
J
f J
A d r
= ( A 1
1 a A 1
A Y ) A X
2 a y
F G
G H
H E
J
A ' d r
=
- ( A 1 +
I
a A l
A Y ) A x
Y
E F
G H
J
A . d r
=
( A 2 + 1 a s 2 A X ) A y
J A A
= - ( A 2
3 A 2
A x ) A y
t a x
F G
H E
e x c e p t f o r i n f i n i t e s i m a l s o f h i g h e r o r d e r t h a n A x A y .
A d d i n g , w e h a v e a p p r o x i m a t e l y 5 A . d r
=
( a x e
-
a A 1 ) A x A y .
C
Y
T h e n , s i n c e A S = A x A y ,
f A - d r
z c o m p o n e n t o f c u r l A
=
( c u r l A ) k
=
l i m
A 1 - o
A s
( a z 2
-
a 1 ) A x A y
l i m
y
A x A y
a A 2
a A 1
a x
a y
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1 3 2
D I V E R G E N C E T H E O R E M , S T O K E S ' T H E O R E M , R E L A T E D I N T E G R A L T H E O R E M S
S U P P L E M E N T A R Y P R O B L E M S
3 7 . V e r i f y G r e e n ' s t h e o r e m i n t h e p l a n e f o r f ( 3 x 2 - 8 y 2 ) d x + ( 4 y - 6 x y ) d y , w h e r e C i s t h e b o u n d a r y o f t h e
C
r e g i o n d e f i n e d b y : ( a ) y = f x , y = x 2 ;
( b ) x = 0 , y = 0 , x + y = 1 .
A n s .
( a ) c o m m o n v a l u e = 3 / 2 ( b ) c o m m o n v a l u e = 5 / 3
3 8 . E v a l u a t e f ( 3 x + 4 y ) d x + ( 2 x - - 3 y ) d y w h e r e C , a c i r c l e o f r a d i u s t w o w i t h c e n t e r a t t h e o r i g i n o f t h e x y
C
p l a n e , i s t r a v e r s e d i n t h e p o s i t i v e s e n s e . A n s . - 8 7 T
3 9 . W o r k t h e p r e v i o u s p r o b l e m f o r t h e l i n e i n t e g r a l f ( x 2 + y 2 ) d x + 3 x y 2 d y .
A n s . 1 2 7 T
C
4 0 . E v a l u a t e f ( x 2 - 2 x y ) d x + ( x 2 y + 3 ) d y a r o u n d t h e b o u n d a r y o f t h e r e g i o n d e f i n e d b y y 2 = 8 x a n d x = 2
( a ) d i r e c t l y , ( b ) b y u s i n g G r e e n ' s t h e o r e m .
A n s . 1 2 8 / 5
( T T . 2 )
4 1 . E v a l u a t e
f
( 6 x y - y 2 ) d x + ( 3 x 2 - - - 2 x y ) d y a l o n g t h e c y c l o i d x = 6 - s i n 6 , y = 1 - c o s 6 .
( o , o )
A n s . 6 7 7 2 - 4 7 T
4 2 . E v a l u a t e ' ( 3 x 2 + 2 y ) d x - - ( x + 3 c o s y ) d y a r o u n d t h e p a r a l l e l o g r a m h a v i n g v e r t i c e s a t ( 0 , 0 ) , ( 2 , 0 ) , ( 3 , 1 )
a n d ( 1 , 1 ) .
A n s . - 6
4 3 . F i n d t h e a r e a b o u n d e d b y o n e a r c h o f t h e c y c l o i d x = a ( 6 - s i n 6 ) , y = a ( l - c o s 6 ) , a > 0 ,
a n d t h e x a x i s .
A n s . 3 7 T a 2
4 4 . F i n d t h e a r e a b o u n d e d b y t h e h y p o c y c l o i d x
2 / 3
+ y 2 / 3 = a
2 / 3 ,
a > 0 .
H i n t : P a r a m e t r i c e q u a t i o n s a r e x = a c o s 3 6 , y = a s i n 3 6 . A n s . 3 7 7 a 2 / 8
4 5 . S h o w t h a t i n p o l a r c o o r d i n a t e s ( p , 0 ) t h e e x p r e s s i o n x d y - y d x = p 2 d c .
I n t e r p r e t
4 6 . F i n d t h e a r e a o f a l o o p o f t h e f o u r - l e a f e d r o s e p = 3 s i n 2 0 .
A n s . 9 7 T / 8
4 7 . F i n d t h e a r e a o f b o t h l o o p s o f t h e l e m n i s c a t e p 2 = a 2 c o s
A n s . a 2
4 8 . F i n d t h e a r e a o f t h e l o o p o f t h e f o l i u m o f D e s c a r t e s
x 3 + y 3 = 3 a x y , a > 0 ( s e e a d j o i n i n g f i g u r e ) .
H i n t : L e t y = t x a n d o b t a i n t h e p a r a m e t r i c e q u a -
t i o n s o f t h e c u r v e . T h e n u s e t h e f a c t t h a t
A r e a = 2 x d y - - y d x
x 2
d ( z )
=
i
x 2 d t
A n s . 3 a 2 / 2
y
x d y - y d x .
4 9 . V e r i f y G r e e n ' s t h e o r e m i n t h e p l a n e f o r f ( 2 x - y 3 ) d x - x y d y , w h e r e C i s t h e b o u n d a r y o f t h e r e g i o n e n -
C
c l o s e d b y t h e c i r c l e s x 2 + y 2 = 1 a n d x 2 + y 2 = 9 .
A n s . c o m m o n v a l u e = 6 0 7 T
f ( - 1 ' 0 ) - y d x + x d y
5 0 . E v a l u a t e
( i . o )
x 2 + Y 2
a l o n g t h e f o l l o w i n g p a t h s :
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D I V E R G E N C E T H E O R E M , S T O K E S ' T H E O R E M , R E L A T E D I N T E G R A L T H E O R E M S
1 3 3
( a ) s t r a i g h t l i n e s e g m e n t s f r o m ( 1 , 0 ) t o ( 1 , 1 ) , t h e n t o ( - 1 , 1 ) , t h e n t o ( - 1 , 0 ) .
( b ) s t r a i g h t l i n e s e g m e n t s f r o m ( 1 , 0 ) t o ( 1 , - 1 ) , t h e n t o ( - 1 , - i ) , t h e n t o ( - 1 , 0 ) .
S h o w t h a t a l t h o u g h a R = a N , t h e l i n e i n t e g r a l i s d e p e n d e n t o n t h e p a t h j o i n i n g ( 1 , 0 ) t o ( - 1 , 0 ) a n d e x p l a i n .
y
x
A n s . ( a ) 7 ( b ) - 7
5 1 . B y c h a n g i n g v a r i a b l e s f r o m ( x , y ) t o ( u , v ) a c c o r d i n g t o t h e t r a n s f o r m a t i o n x = x ( u , v ) , y = y ( u , v ) , s h o w t h a t
t h e a r e a A o f a r e g i o n R b o u n d e d b y a s i m p l e c l o s e d c u r v e C i s g i v e n b y
A
f f
J ( u , v )
d u d v
w h e r e
R
J ( u v )
a x
a y
a u a u
a x a y
a v
a v I
i s t h e J a c o b i a n o f x a n d y w i t h r e s p e c t t o u a n d v .
W h a t r e s t r i c t i o n s s h o u l d y o u m a k e ? I l l u s t r a t e t h e r e -
s u l t w h e r e u a n d v a r e p o l a r c o o r d i n a t e s .
H i n t : U s e t h e r e s u l t A = i f x d y - y d x ,
t r a n s f o r m t o u , v c o o r d i n a t e s a n d t h e n u s e G r e e n ' s t h e o r e m .
5 2 . E v a l u a t e f f F n d S ,
w h e r e F = 2 x y i + y z 2 j + x z k a n d S i s :
S
( a ) t h e s u r f a c e o f t h e p a r a l l e l e p i p e d b o u n d e d b y x = 0 , y = 0 , z = 0 , x = 2 , y = 1 a n d z
= 3 ,
( b ) t h e s u r f a c e o f t h e r e g i o n b o u n d e d b y x = 0 , y = 0 , y = 3 , z = 0 a n d x + 2 z =
6 .
A n s . ( a ) 3 0
( b ) 3 5 1 / 2
5 3 . V e r i f y t h e d i v e r g e n c e t h e o r e m f o r A = 2 x - 2 y i - y 2 j + 4 x z 2 k t a k e n o v e r t h e r e g i o n i n t h e f i r s t o c t a n t
b o u n d e d b y y 2 + z 2 = 9 a n d x = 2 .
A n s . 1 8 0
5 4 . E v a l u a t e f f r n d S w h e r e ( a ) S i s t h e s p h e r e o f r a d i u s 2 w i t h c e n t e r a t ( 0 , 0 , 0 ) , ( b ) S i s t h e s u r f a c e o f
S
t h e c u b e b o u n d e d b y x = - 1 , y = - 1 , z = - 1 , x = 1 , y = 1 , z = 1 ,
( c ) S i s t h e s u r f a c e b o u n d e d b y t h e p a r a b o l o i d
z = 4 - ( x 2 + y 2 ) a n d t h e x y p l a n e .
A n s . ( a ) 3 2 7 ( b ) 2 4 ( c ) 2 4 7
5 5 . I f S i s a n y c l o s e d s u r f a c e e n c l o s i n g a v o l u m e V a n d A = a x i + b y j + c z k ,
p r o v e t h a t f f A n d S
( a + b + c ) V .
S
5 6 . I f R = c u r l A , p r o v e t h a t f f H n d S = 0 f o r a n y c l o s e d s u r f a c e S .
S
5 7 . I f n i s t h e u n i t o u t w a r d d r a w n n o r m a l t o a n y c l o s e d s u r f a c e o f a r e a S , s h o w t h a t f f f d i v n d Y = S .
V
5 8 . P r o v e
f f f
2
= f f
T e n d S
.
r
7
S
5 9 . P r o v e f f r 5 n d S = f f f s r s r d V .
S
V
6 0 . P r o v e
f f n d S = 0
f o r a n y c l o s e d s u r f a c e S .
S
6 1 . S h o w t h a t G r e e n ' s s e c o n d i d e n t i t y c a n b e w r i t t e n
f f f ( c 1 5 V 2 q i
-
b V 2 c p ) d V
=
f f w / d
-
d - O )
d S
V
S
6 2 . P r o v e f f r x d S = 0
f o r a n y c l o s e d s u r f a c e S .
3
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1 3 4
D I V E R G E N C E T H E O R E M , S T O K E S ' T H E O R E M , R E L A T E D I N T E G R A L T H E O R E M S
6 3 . V e r i f y S t o k e s ' t h e o r e m f o r A = ( y - z + 2 ) i + ( y z + 4 ) i - x z k ,
w h e r e S i s t h e s u r f a c e o f t h e c u b e x = 0 ,
y = 0 , z = 0 , x = 2 , y = 2 , z = 2 a b o v e t h e x y p l a n e . A n s . c o m m o n v a l u e = - 4
6 4 . V e r i f y S t o k e s ' t h e o r e m f o r F = x z i - y j + x 2 y k , w h e r e S i s t h e s u r f a c e o f t h e r e g i o n b o u n d e d b y
x = 0 ,
y = 0 , z = 0 , 2 x + y + 2 z = 8 w h i c h i s n o t i n c l u d e d i n t h e x z p l a n e .
A n s . c o m m o n v a l u e = 3 2 / 3
6 5 . E v a l u a t e
f f ( V x A ) . n d S ,
w h e r e A = ( x 2 + y - 4 ) i + 3 x y j + ( 2 z z + z 2 ) k a n d S i s t h e s u r f a c e o f ( a ) t h e
S
h e m i s p h e r e x 2 + y 2 + z 2 = 1 6 a b o v e t h e x y p l a n e , ( b ) t h e p a r a b o l o i d z = 4 - ( x 2 + y 2 ) a b o v e t h e x y p l a n e .
A n s . ( a ) - 1 6 7 7 , ( b ) - 4 7 7
6 6 . I f A = 2 y z i - ( x + 3 y - 2 ) j + ( x 2 + z ) k , e v a l u a t e
f f ( V x A ) . n d S
o v e r t h e s u r f a c e o f i n t e r s e c t i o n o f t h e
S
2
c y l i n d e r s x 2 + y 2 = a 2 , x 2 + z 2 = a 2 w h i c h i s i n c l u d e d i n t h e f i r s t o c t a n t .
A n s . - 1 2 ( 3 7 7 + 8 a )
6 7 . A v e c t o r B i s a l w a y s n o r m a l t o a g i v e n c l o s e d s u r f a c e S . S h o w t h a t
f f f c u r l B d V
= 0 , w h e r e V i s t h e
r e g i o n b o u n d e d b y S .
V
6 8 . I f
E d r = -
c a t
f f n d S , w h e r e S i s
a n y s u r f a c e b o u n d e d b y t h e c u r v e C , s h o w t h a t V x E _
C S
1 a H
C a t
6 9 . P r o v e
f o d r = f f d S x V o .
S
7 0 . U s e t h e o p e r a t o r e q u i v a l e n c e o f S o l v e d P r o b l e m 2 5 t o a r r i v e a t ( a ) V 0 , ( b ) V . A , ( c ) V x A i n r e c t a n g u l a r
c o o r d i n a t e s .
7 1 . P r o v e
f f f v c 7 5 . A d v = f f A . n d s
- f f f V . A d v .
V S
V
7 2 . L e t r b e t h e p o s i t i o n v e c t o r o f a n y p o i n t r e l a t i v e t o a n o r i g i n 0 . S u p p o s e 0 h a s c o n t i n u o u s d e r i v a t i v e s o f
o r d e r t w o , a t l e a s t , a n d l e t S b e a c l o s e d s u r f a c e b o u n d i n g a v o l u m e V . D e n o t e 0 a t 0 b y 0 o . S h o w t h a t
f f [ 3 V q _ g 5 V ( 3 ) ] . d S
=
f f f i d v + a
S
w h e r e a = 0 o r 4 7 7 0 , a c c o r d i n g a s 0 i s o u t s i d e o r i n s i d e S .
7 3 . T h e p o t e n t i a l O ( P ) a t a p o i n t P ( x , y , z ) d u e t o a s y s t e m o f c h a r g e s ( o r m a s s e s )
v e c t o r s r 1 , r 2 , . . . , r n w i t h r e s p e c t t o P i s g i v e n b y
n
P r o v e G a u s s ' l a w
f f E . d s
= 4 7 7 Q
S
g l , g 2 , . . . , q n
h a v i n g p o s i t i o n
n
w h e r e E = - V V i s t h e e l e c t r i c f i e l d i n t e n s i t y , S i s a s u r f a c e e n c l o s i n g a l l t h e c h a r g e s a n d Q = Y q n
i s t h e t o t a l c h a r g e w i t h i n S .
' a = I
7 4 . I f a r e g i o n V b o u n d e d b y a s u r f a c e S h a s a c o n t i n u o u s c h a r g e ( o r m a s s ) d i s t r i b u t i o n o f d e n s i t y p , t h e p o -
t e n t i a l
( P ) a t a p o i n t P i s d e f i n e d b y
=
f f f _ - . .
D e d u c e t h e f o l l o w i n g u n d e r s u i t a b l e a s s u m p t i o n s :
( a ) f f E - d S = 4 7 7 f f f p d V ,
w h e r e E _ - V .
S V
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T R A N S F O R M A T I O N O F C O O R D I N A T E S . L e t t h e r e c t a n g u l a r c o o r d i n a t e s ( x , y , z ) o f a n y p o i n t b e
e x p r e s s e d a s f u n c t i o n s o f ( u 1 , u 2 , u 3 ) s o t h a t
( 1 )
x = x ( u 1 , u 2 , u 3 ) ,
y = y ( u 1 , u 2 , u 3 ) ,
z = z ( u 1 , u 2 , u 3 )
S u p p o s e t h a t ( 1 ) c a n b e s o l v e d f o r u 1 , u 2 , u 3 i n t e r m s o f x , y , z , i . e . ,
( 2 )
u 1 = u 1 ( x , y , z ) ,
u 2 = u 2 ( x , y , z ) ,
u s = u 3 ( x , y , z )
T h e f u n c t i o n s i n ( 1 ) a n d ( 2 ) a r e a s s u m e d t o b e s i n g l e - v a l u e d a n d t o h a v e c o n t i n u o u s d e r i v a t i v e s s o
t h a t t h e c o r r e s p o n d e n c e b e t w e e n ( x , y , z ) a n d ( u 1 , u 2 , u 3 ) i s u n i q u e . i n p r a c t i c e t h i s a s s u m p t i o n m a y
n o t a p p l y a t c e r t a i n p o i n t s a n d s p e c i a l c o n s i d e r a t i o n i s r e q u i r e d .
G i v e n a p o i n t P w i t h r e c t a n g u l a r c o o r d i n a t e s ( x , y , z ) w e c a n , f r o m ( 2 ) a s s o c i a t e a u n i q u e s e t
o f c o o r d i n a t e s ( u 1 , u 2 , u 3 ) c a l l e d t h e c u r v i l i n e a r c o o r d i n a t e s o f P . T h e s e t s o f e q u a t i o n s ( 1 ) o r ( 2 )
d e f i n e a t r a n s f o r m a t i o n o f c o o r d i n a t e s .
z
O R T H O G O N A L C U R V I L I N E A R C O O R D I N A T E S .
T h e s u r f a c e s u 1 = c 1 , u 2 = c 2 , u 3 = c 3 ,
w h e r e
c 1 , r 2 , c 3
a r e c o n s t a n t s , a r e c a l l e d c o o r d i n a t e s u r -
f a c e s a n d e a c h p a i r o f t h e s e s u r f a c e s i n t e r s e c t i n
c u r v e s c a l l e d c o o r d i n a t e c u r v e s o r l i n e s ( s e e F i g . 1 ) .
I f t h e c o o r d i n a t e s u r f a c e s i n t e r s e c t a t r i g h t a n g l e s
t h e c u r v i l i n e a r c o o r d i n a t e s y s t e m i s c a l l e d o r t h o g o -
n a l . T h e u 1 , u 2 a n d u 3 c o o r d i n a t e c u r v e s o f a c u r v i -
l i n e a r s y s t e m a r e a n a l o g o u s t o t h e x , y a n d z c o o r -
d i n a t e a x e s o f a r e c t a n g u l a r s y s t e m .
F i g . 1
U N I T V E C T O R S I N C U R V I L I N E A R S Y S T E M S . L e t r = x i + y 3 + z k b e t h e p o s i t i o n v e c t o r o f a p o i n t
P . T h e n ( 1 ) c a n b e w r i t t e n r = r ( u 1 , u 2 , u 3 ) , A t a n -
g e n t v e c t o r t o t h e u 1 c u r v e a t P ( f o r w h i c h u 2 a n d u 3 a r e c o n s t a n t s ) i s a u 1
. T h e n a u n i t t a n g e n t
' a r
v e c t o r i n t h i s d i r e c t i o n i s e 1 = b - - / ,
-
-
y
s o t h a t
a u 1 = h 1
e 1 w h e r e h 1 =
,
a u 1
I
.
S i m i l a r l y , i f
e 2 a n d e 3 a r e u n i t t a n g e n t v e c t o r s t o t h e u 2 a n d u 3 c u r v e s a t P r e s p e c t i v e l y , t h e n a
h 2 e 2 a n d
u 2
u s
=
h 3 e 3 w h e r e h 2 =
a
h 3
T h e
a r e
i n t h e d i r e c t i o n s o f i n c r e a s i n g u 1 , u 2 , U S . r e s p e c t i v e l y .
S i n c e V u 1 i s a v e c t o r a t P n o r m a l t o t h e s u r f a c e u 1 = c 1 ,
a u n i t v e c t o r i n t h i s d i r e c t i o n i s g i v -
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1 3 6
C U R V I L I N E A R C O O R D I N A T E S
e n b y E 1 =
V u 1 / I V u 1 I
.
S i m i l a r l y , t h e u n i t v e c t o r s E 2 = V u 2 / I D u 2 I
a n d
E 3 = V u 3 / I
V u 3
I
a t P
a r e n o r m a l t o t h e s u r f a c e s u 2 = c 2 a n d u 3 = c 3 r e s p e c t i v e l y .
T h u s a t e a c h p o i n t P o f a c u r v i l i n e a r s y s t e m t h e r e
e x i s t , i n g e n e r a l , t w o s e t s o f u n i t v e c t o r s , e 1 , e 2 , e 3 t a n -
g e n t t o t h e c o o r d i n a t e c u r v e s a n d E 1 , E 2 , E 3 n o r m a l t o
t h e c o o r d i n a t e s u r f a c e s ( s e e F i g . 2 ) . T h e s e t s b e c o m e
i d e n t i c a l i f a n d o n l y i f t h e c u r v i l i n e a r c o o r d i n a t e s y s t e m
i s o r t h o g o n a l ( s e e P r o b l e m 1 9 ) . B o t h s e t s a r e a n a l o g o u s
u
t o t h e i , j , k u n i t v e c t o r s i n r e c t a n g u l a r c o o r d i n a t e s b u t
r ' t -
- e 2
a r e u n l i k e t h e m i n t h a t t h e y m a y c h a n g e d i r e c t i o n s f r o m
p o i n t t o p o i n t .
I t c a n b e s h o w n ( s e e P r o b l e m 1 5 ) t h a t t h e
s e t s
-
a u , a u
a n d V u 1 , V u 2 , V u 3 c o n s t i t u t e r e c i p -
r o c a l s y s t e m s
o f
v e c t o r s .
f o r m
F i g . 2
A v e c t o r A c a n b e r e p r e s e n t e d i n t e r m s o f t h e u n i t b a s e v e c t o r s e 1 , e 2 , e 3 o r E 1 , E 2 , E 3 i n t h e
A =
A . e . + A 2 e 2 + A 3 e 3
=
a 1 E , + a 2 E 2 + a 3 E .
w h e r e A 1 , A 2 , A s a n d a 1 , a 2 , a s a r e t h e r e s p e c t i v e c o m p o n e n t s o f A i n e a c h s y s t e m .
- 6 r
W e c a n a l s o r e p r e s e n t A i n t e r m s o f t h e b a s e v e c t o r s
a u ,
- -
, a u
o r
V u 1 , V u 2 , V u 3 w h i c h
l . ' a
a r e c a l l e d u n i t a r y b a s e v e c t o r s b u t a r e n o t u n i t v e c t o r s i n g e n e r a l . I n 2 t h i s c a s e
A
_
C .
a u
+
C 2
a u
+
C .
a u
C 1 a 1
+
C 2 a 2 + C 3 a s
1
2
3
a n d
A
C 1 V u 1 +
C 2 V u 2 + C 3 V u 3
C 1 6 4 1 + C 2 1 r 2 + C 3 M 3
w h e r e C 1 , C 2 , C . a r e c a l l e d t h e c o n t r a v a r i a n t c o m p o n e n t s o f A a n d c 1 , c 2 , c 3 a r e c a l l e d t h e c o v a r i a n t
c o m p o n e n t s o f A ( s e e P r o b l e m s 3 3 a n d 3 4 ) . N o t e t h a t a 0 = a u ,
l 3 ¢ = V u ,
, p = 1 , 2 , 3 .
A R C L E N G T H A N D V O L U M E E L E M E N T S . F r o m r = r ( u 1 , u 2 i u 3 ) w e h a v e
d r =
a u d u l + a u d u 2 + a u d u 3
1
2 3
h 1 d u 1 e 1 + h 2 d u 2 e 2 + h 3 d u 3 e 3
T h e n t h e d i f f e r e n t i a l o f a r e l e n g t h d s i s d e t e r m i n e d f r o m
d s 2 = d r d r .
F o r o r t h o g o n a l s y s t e m s ,
e 1 e 2 = e 2 . e 3 =
e 3 e 1 = 0 a n d
d s 2 =
h 2 d u 2 + h 2 d u 2 + h 3 d u 3
F o r n o n - o r t h o g o n a l o r g e n e r a l c u r v i l i n e a r s y s t e m s s e e
P r o b l e m 1 7 .
A l o n g a u 1 c u r v e , u 2 a n d u 3 a r e c o n s t a n t s s o t h a t
d r = h 1 d u 1 e 1 .
T h e n t h e d i f f e r e n t i a l o f a r c l e n g t h d s 1
a l o n g u 1 a t P i s h 1 d u 1 .
S i m i l a r l y t h e d i f f e r e n t i a l a r c
l e n g t h s a l o n g u 2 a n d u 3 a t P a r e d s 2 = h 2 d u 2 , d s 3 = h 3 d u 3 .
u
R e f e r r i n g t o F i g . 3 t h e v o l u m e e l e m e n t f o r a n o r -
t h o g o n a l c u r v i l i n e a r c o o r d i n a t e s y s t e m i s g i v e n b y
F i g . 3
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C U R V I L I N E A R C O O R D I N A T E S
1 3 7
d V
=
I ( h 1 d u 1 e 1 ) ( h 2 d u 2
e 2 ) x ( h 3 d u 3 e 3 ) I
=
h 1 h 2 h 3 d u 1 d u 2 d u 3
s i n c e
I e 1 . e 2 x e 3 l
=
1 .
T H E G R A D I E N T , D I V E R G E N C E A N D C U R L c a n b e e x p r e s s e d i n t e r m s o f c u r v i l i n e a r c o o r d i n a t e s .
I f 4 ) i s a s c a l a r f u n c t i o n a n d A = A l e 1 + A 2 e 2 + A . e 3
a v e c t o r f u n c t i o n o f o r t h o g o n a l c u r v i l i n e a r c o o r d i n a t e s u 1 , u 2 , u 3 , t h e n t h e f o l l o w i n g r e s u l t s a r e v a l i d .
1 . © = g r a d < 1 ) _ 1 a
e 1 +
2
a u 2 - e 2 + h a T e 3
2 . . A =
d i v A =
h h h
a u ( h 2 h 3 A 1 )
+ a u ( h 3 h 1 A 2 ) + u
( A . h 2 A 3 )
1 2 3
1 2
3
h 1 e 1
3 . V x A =
c u r l A =
i
h 1 h 2 A S
h 2 e 2
h 3 e 3
a
a u 3
h 3 A 3
4 . V 2
=
L a p l a c i a n o f
_
1
a
h 2 A S a
A S h 1 - 6 ( D
)
a
h 1 h 2 a
h 1 h 2 h 3
a u 1 {
h 1 a u 1 ) + a u 2 {
h 2
a u 2
+ a u 3 {
h 3
a u 3
)
I f h 1 = h 2 = A S = 1
a n d
e 1 , e 2 , e 3 a r e r e p l a c e d b y i , j , k ,
t h e s e r e d u c e t o t h e u s u a l e x p r e s s i o n s i n
r e c t a n g u l a r c o o r d i n a t e s w h e r e ( u 1 , u 2 i u 3 ) i s r e p l a c e d b y ( x , y , z )
.
E x t e n s i o n s o f t h e a b o v e r e s u l t s a r e a c h i e v e d b y a m o r e g e n e r a l t h e o r y o f c u r v i l i n e a r s y s t e m s
u s i n g t h e m e t h o d s o f t e n s o r a n a l y s i s w h i c h i s c o n s i d e r e d i n C h a p t e r 8 .
S P E C I A L O R T H O G O N A L C O O R D I N A T E S Y S T E M S .
1 . C y l i n d r i c a l C o o r d i n a t e s ( p , 0 , z ) . S e e F i g . 4 b e l o w .
x = p c o s 4 ,
y = p s i n q 5 ,
z = z
w h e r e
p ? 0 , 0 < _ 0 < 2 7 r , - c o < z < o o
h P = i , h o = p , h z = 1
2 . S p h e r i c a l C o o r d i n a t e s ( r , 6 , 0 ) . S e e F i g . 5 b e l o w .
x = r s i n 6 c o s 0 ,
y = r s i n 6 s i n 0 ,
z
r c o s 6
w h e r e r > 0 , 0 < O < 2 7 T ,
0 < 6 < 7 r
h r = 1 ,
h e = r , h o = r s i n 6
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C U R V I L I N E A R C O O R D I N A T E S
z
F i g . 4
F i g . 5
3 .
P a r a b o l i c C y l i n d r i c a l C o o r d i n a t e s ( u , v , z ) . S e e F i g . 6 b e l o w .
x =
2
( u 2 - v 2 ) ,
y = u v ,
z = z
w h e r e - c o < u < c o , v > 0 , - o o < z < c o
h u = h v =
u 2 + v 2 ,
h z = 1
I n c y l i n d r i c a l c o o r d i n a t e s ,
u = 2 c o s
± ,
v = 2 s i n
.
,
z = z
T h e t r a c e s o f t h e c o o r d i n a t e s u r f a c e s o n t h e x y p l a n e a r e s h o w n i n F i g . 6 b e l o w . T h e y a r e
c o n f o c a l p a r a b o l a s w i t h a c o m m o n a x i s .
F i g . 6
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C U R V I L I N E A R C O O R D I N A T E S
1 3 9
4 . P a r a b o l o i d a l C o o r d i n a t e s ( u , v , 0 ) .
x = u v c o s 4 ,
y =
u v s i n ( p ,
z = z
( u 2
- v 2 )
w h e r e
u > O ,
v > O ,
0 < < f < 2 7 T
h u = b y =
u 2 + v 2 , h = u v
T w o s e t s o f c o o r d i n a t e s u r f a c e s a r e o b t a i n e d b y r e v o l v i n g t h e p a r a b o l a s o f F i g . 6 a b o v e
a b o u t t h e x a x i s w h i c h i s r e l a b e l e d t h e z a x i s . T h e t h i r d s e t o f c o o r d i n a t e s u r f a c e s a r e p l a n e s
p a s s i n g t h r o u g h t h i s a x i s .
5 .
E l l i p t i c C y l i n d r i c a l C o o r d i n a t e s ( u , v , z ) . S e e F i g . 7 b e l o w .
x = a c o s h u c o s y , y = a s i n h u s i n v ,
z = z
w h e r e
u > 0 ,
0 s v < 2 7 r ,
- o o < z < o o
h u = h v = a s i n h 2 u + s i r ? v ,
h 2 = 1
T h e t r a c e s o f t h e c o o r d i n a t e s u r f a c e s o n t h e x y p l a n e a r e s h o w n i n F i g . 7 b e l o w . T h e y a r e
c o n f o c a l e l l i p s e s a n d h y p e r b o l a s .
6 .
P r o l a t e S p h e r o i d a l C o o r d i n a t e s ( , 7 7 , 0 ) .
x = a s i n h 6 s i n 7 7 c o s 0 ,
y = a s i n h
s i n ? s i n ,
z = a c o s h 6 c o s
w h e r e ? 0 , 0 : 5 ? 7
7 r ,
4 < 2 7 T
a s i n h e s i n ? 7
T w o s e t s o f c o o r d i n a t e s u r f a c e s a r e o b t a i n e d b y r e v o l v i n g t h e c u r v e s o f F i g . 7 a b o v e a b o u t
t h e x a x i s w h i c h i s r e l a b e l e d t h e z a x i s . T h e t h i r d s e t o f c o o r d i n a t e s u r f a c e s a r e p l a n e s p a s s i n g
t h r o u g h t h i s a x i s .
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C U R V I L I N E A R C O O R D I N A T E S
7 . O b l a t e S p h e r o i d a l C o o r d i n a t e s ( 6 , 7 7 , q b ) .
x = a c o s h 6 c o s r c o s 0 ,
y
a c o s h e c o s 7 7 s i n 4 ) ,
w h e r e > _ p ,
2
2 , 0 < 9 5 < 2 7 T
z = a s i n h e s i n 7 7
h e = h 7 7 = a s i n h 2 6 + s i n 2 7 )
h j , = a c o s h C o s - q
T w o s e t s o f c o o r d i n a t e s u r f a c e s a r e o b t a i n e d b y r e v o l v i n g t h e c u r v e s o f F i g . 7 a b o v e a b o u t
t h e y a x i s w h i c h i s r e l a b e l e d t h e z a x i s . T h e t h i r d s e t o f c o o r d i n a t e s u r f a c e s a r e p l a n e s p a s s i n g
t h r o u g h t h i s a x i s .
8 . E l l i p s o i d a l C o o r d i n a t e s ( X , µ , v ) .
, , . 2
2
2
a 2 - X
b 2 -
c 2 - A
a 2
x 2
+
b 2
y 2
+
c 2
z 2
=
1 , c 2 < A < b 2 < a 2
/
2 x 2
+ 2 y 2
+ 2 z 2
=
1 ,
c 2 < b 2 < v < a 2
a - v
b - v c - v
h
_
1
( 1 a - X ) ( v - X )
h
=
1
( y - µ ) ( X - / . L )
2
2
( a 2 - 1 - p ) ( b 2 - 1 - i ) ( c 2 - / )
h
=
1
v
2
( a 2 - v ) ( b 2 - v ) ( c 2 - v )
9 . B i p o l a r C o o r d i n a t e s ( u , v , z ) .
S e e F i g . 8 b e l o w .
x 2 + ( y - a c o t u ) 2 = a 2 c s c 2 u ,
( x - a c o t h v ) 2 + y 2 = a 2 c s c h 2 v ,
z = z
+ -
Y
+
z
=
1 ,
) < c 2 < b 2 < a 2
F i g . 8
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C U R V I L I N E A R C O O R D I N A T E S
a s i n k v
a s i n u
o r
x =
c o s h v - c o s u '
y =
c o s h v - c o s u '
z = z
w h e r e O s u < 2 7 r , - o o < v < o o , - o o < z < o o
h u = h v =
,
h z = i
1 4 1
T h e t r a c e s o f t h e c o o r d i n a t e s u r f a c e s o n t h e x y p l a n e a r e s h o w n i n F i g . 8 a b o v e . B y r e -
v o l v i n g t h e c u r v e s o f F i g . 8 a b o u t t h e y a x i s a n d r e l a b e l i n g t h i s t h e z a x i s a t o r o i d a l c o o r d i n a t e
s y s t e m i s o b t a i n e d .
S O L V E D P R O B L E M S
a
c o s h v - c o s u
1 . D e s c r i b e t h e c o o r d i n a t e s u r f a c e s a n d c o o r d i n a t e c u r v e s f o r ( a ) c y l i n d r i c a l a n d ( b ) s p h e r i c a l c o -
o r d i n a t e s .
( a )
T h e c o o r d i n a t e s u r f a c e s ( o r l e v e l s u r f a c e s ) a r e :
P =
Z =
T h e c o o r d i n a t e c u r v e s a r e :
I n t e r s e c t i o n o f p = c 1 a n d
= c 2 ( z c u r v e ) i s a s t r a i g h t l i n e .
I n t e r s e c t i o n o f p = c 1 a n d z = c 3 ( r p c u r v e ) i s a c i r c l e ( o r p o i n t ) .
I n t e r s e c t i o n o f 0 = c 2 a n d z = c 3 ( p c u r v e ) i s a s t r a i g h t l i n e .
c 1
c y l i n d e r s c o a x i a l w i t h t h e z a x i s ( o r z a x i s i f c 1 = 0 ) .
c 2 p l a n e s t h r o u g h t h e z a x i s .
c 3
p l a n e s p e r p e n d i c u l a r t o t h e z a x i s .
( b ) T h e c o o r d i n a t e s u r f a c e s a r e :
r = c 1
s p h e r e s h a v i n g c e n t e r a t t h e o r i g i n ( o r o r i g i n i f c 1 = 0 ) .
B = c 2
c o n e s h a v i n g v e r t e x a t t h e o r i g i n ( l i n e s i f c 2 = 0 o r I T , a n d t h e x y p l a n e i f c 2 = 7 T / 2 ) .
= c 3 p l a n e s t h r o u g h t h e z a x i s .
T h e c o o r d i n a t e c u r v e s a r e ,
I n t e r s e c t i o n o f
r = c 2 a n d 8 = c 2
c u r v e ) i s a c i r c l e ( o r p o i n t ) .
I n t e r s e c t i o n o f r = c 1 a n d
= c 3 ( 8 c u r v e ) i s a s e m i - c i r c l e ( c 1
0 ) .
I n t e r s e c t i o n o f 8 = c 2 a n d
= c 3 ( r c u r v e ) i s a l i n e .
2 . D e t e r m i n e t h e t r a n s f o r m a t i o n f r o m c y l i n d r i c a l t o r e c t a n g u l a r c o o r d i n a t e s .
T h e e q u a t i o n s d e f i n i n g t h e t r a n s f o r m a t i o n f r o m r e c t a n g u l a r t o c y l i n d r i c a l c o o r d i n a t e s a r e
( 1 ) x = p c o s 0 ,
( 2 ) y = p s i n p ,
( 3 )
z = z
S q u a r i n g ( 1 ) a n d ( 2 ) a n d a d d i n g ,
p 2 ( c o s 2 o + s i n 2 o ) = x 2 + y 2
o r
p =
x 2 + y 2 , s i n c e c o s 2 0 + s i n 2 4 = 1 a n d p i s p o s i t i v e .
D i v i d i n g e q u a t i o n ( 2 ) b y ( 1 ) ,
y
=
p
s i n
= t a n 0 o r 0 = a r c t a n
y
.
X
p c o s q
X
T h e n t h e r e q u i r e d t r a n s f o r m a t i o n i s
( 4 ) p = V x 2 ' + y 2 ,
( 5 ) 0 = a r e t a n z ,
( 6 ) z = z .
F o r p o i n t s o n t h e z a x i s ( x = 0 , y = 0 ) , n o t e t h a t 0 i s i n d e t e r m i n a t e . S u c h p o i n t s a r e c a l l e d s i n g u l a r
p o i n t s o f t h e t r a n s f o r m a t i o n .
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C U R V I L I N E A R C O O R D I N A T E S
3 . P r o v e t h a t a c y l i n d r i c a l c o o r d i n a t e s y s t e m i s o r t h o g o n a l .
T h e p o s i t i o n v e c t o r o f a n y p o i n t i n c y l i n d r i c a l c o o r d i n a t e s i s
r
=
x i + y j + z k
=
p c o s c p i + p s i n g j
+ z k
T h e t a n g e n t v e c t o r s t o t h e p , 0 a n d z c u r v e s a r e g i v e n r e s p e c t i v e l y b y a P ,
a n d a s w h e r e
= k
a p
=
c o s
i
+ s i n
j ,
a
= - p s i n 0 i + p c o s 0 j ,
' 3 Z
T h e u n i t v e c t o r s i n t h e s e d i r e c t i o n s a r e
e 1
_ e p =
a r / a p
J
c o s 0 i
+ s i n 4 ) j
=
c o s 0 i
+ s i n 0 j
I a r / a p I
c o s t g 5 + s i n e g 5
e 2
e
_
- p s i n Q 5 i + p c o s 0 j
= - s i n 4 i + c o s 0 i
' I a r / a I y
p 2 s i n 2 c 5 + p 2 c o s 2 g 5
e 3
= e z =
a r / a z
k
j a r / a z I
T h e n
e 1 . e 2
= ( c o s g 5 i + s i n g 5 j ) . ( - s i n g 5 i + c o s 0 j )
=
0
e 1 . e 3
= ( c o s c p i + s i n g 5 j ) ( k ) = 0
e 2 e 3 =
( - s i n c b i + c o s g 5 j )
( k ) =
0
a n d s o e 1 , e 2 a n d e 3 a r e m u t u a l l y p e r p e n d i c u l a r a n d t h e c o o r d i n a t e s y s t e m i s o r t h o g o n a l .
4 . R e p r e s e n t t h e v e c t o r A = z i - 2 x j + y k i n c y l i n d r i c a l c o o r d i n a t e s . T h u s d e t e r m i n e A O , 4 a n d A z .
F r o m P r o b l e m 3 ,
( 1 ) e p
=
c o s c p i
+ s i n 0 j ( 2 ) s i n 0 i
+ c o s 4 5 j
( 3 ) e 2 = k
S o l v i n g ( 1 ) a n d ( 2 ) s i m u l t a n e o u s l y ,
i =
c o s c 5 e p - s i n 0 e o ,
j =
s i n 0 e p + c o s c p e ( k
T h e n A = z i - 2 x j + y k
=
z ( c o s 4 e p - s i n g 5 e e ) - - 2 p c o s g 5 ( s i n 0 e p + c o s 0 e ( + p s i n c a e 2
( z c o s c a - 2 p c o s ( 5 s i n g 5 ) e p - ( z s i n 6 + 2 p c o s 2 0 ) e 4 + p s i n 0 e 2
a n d A P =
z c o s o - 2 p c o s g 5 s i n o , A 0 = - z s i n c p - 2 p c o s 2 g 5 ,
A z = p s i n g 5 .
5 . P r o v e
d t
e p =
e , , a t e .
c b e p
w h e r e d o t s d e n o t e d i f f e r e n t i a t i o n w i t h r e s p e c t t o t i m e t .
F r o m P r o b l e m 3 ,
e p
=
c o s g 5 i + s i n g 5 j ,
e ,
s i n 0 i + c o s g 5 j
T h e n
d t
e p
= - ( s i n 0 ) g 5 i
+ ( c o s 0 )
s i n 0 i + c o s 0 j ) g 5
_
g 5 e ( k
d
e ( h
_
- ( c o s g 5 ) g S I - ( s i n
- ( c o s g 5 i + s i n 0 j ) c p
=
- r , e p
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C U R V I L I N E A R C O O R D I N A T E S
6 . E x p r e s s t h e v e l o c i t y v a n d a c c e l e r a t i o n a o f a p a r t i c l e i n c y l i n d r i c a l c o o r d i n a t e s .
1 4 3
I n r e c t a n g u l a r c o o r d i n a t e s t h e p o s i t i o n v e c t o r i s
r = x i + y j + z k a n d t h e v e l o c i t y a n d a c c e l e r a t i o n
v e c t o r s a r e
v
= d t
=
z i + y j + A
a n d
a
I n c y l i n d r i c a l c o o r d i n a t e s , u s i n g P r o b l e m 4 ,
r
d t r
=
z i + y j + a k
x i + y j + z k
= ( p c o s 0 ) ( c o s c b e p - s i n 0
( p s i n ( ; b ) ( s i n o e p + c o s 0 e d , )
+
p e p + z e z
z e z
d p
d e
e z = P e p + p q e o + z e Z
h e n
v
`
d t
d t e p
+ p d t p +
d z
u s i n g P r o b l e m 5 .
D i f f e r e n t i a t i n g a g a i n ,
a
=
d 2
=
d t
( p
d t
d e p
. ,
. d e b
d t + P e p + P , d t
+ p e o +
e + z e z
p e o + P .
+ p
( - e p ) + P 4 e e + P c e o +
e z
( p - p 2 ) e p + ( p
+ 2 p ) e b +
e z
u s i n g P r o b l e m 5 .
7 . F i n d t h e s q u a r e o f t h e e l e m e n t o f a r c l e n g t h i n c y l i n d r i c a l c o o r d i n a t e s a n d d e t e r m i n e t h e c o r r e -
s p o n d i n g s c a l e f a c t o r s .
F i r s t M e t h o d .
x = p c o s 0 ,
y = p s i n o ,
z = z
d x
p s i n c p d o + c o s o d p ,
d y
= p c o s 0 d o + s i n 0 d p ,
d z = d z
T h e n
d s 2 = d x 2 + d y 2 + d z 2 =
( - p s i n o
d o +
c o s 0
d p ) 2 +
( p c o s o d c /
+ s i n %
d p ) 2 +
( d z ) 2
( d p )
+ p 2 ( d c b ) 2 + ( d z ) 2 = h 1 ( d p ) 2 + h 2 ( d o ) 2 + h 2
( d z ) 2
a n d
h 1 = h = 1 , h 2 = h q = p ,
h s = h z = I
a r e t h e s c a l e f a c t o r s .
S e c o n d M e t h o d . T h e p o s i t i o n v e c t o r i s r = p c o s 0 i
+ p s i n
j
+ z k .
T h e n
d r
=
a p d p + . 0
r d o
+
a z
d z
= ( c o s
i + s i n 0 j ) d p + ( - p s i n
i + p c o s 0 j ) d o + k d z
( c o s
d p - p s i n 0 d c p ) i
+ ( s i n
d p + p c o s c p d o ) j
+ k d z
T h u s
d s 2 = d r d r =
( c o s
0 d p - p s i n 0 d c ) 2 + ( s i n 0 d p + p c o s 0 d o ) 2 + ( d z ) 2
( d p f + p 2 ( d c b ) 2 +
( d z ) 2
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1 4 4
C U R V I L I N E A R C O O R D I N A T E S
8 .
W o r k P r o b l e m 7 f o r ( a ) s p h e r i c a l a n d ( b ) p a r a b o l i c c y l i n d r i c a l c o o r d i n a t e s .
( a )
x =
r s i n 6 c o s 0 , y = r s i n 6 s i n 0 ,
z = r c o s 6
T h e n
d x
= - r s i n B s i n 0 d o + r c o s 6 c o s 0 d 6 + s i n 6 c o s 4 d r
d y = r s i n 6 c o s q d o + r c o s 6 s i n % d 6 + s i n 6 s i n 0 d r
d z
= - r s i n 6 d 6 + c o s 6 d r
( b )
a n d
( d s ) 2
=
( d x ) 2 +
( d y ) 2 + ( d z )
( d r ) 2 + r 2 ( d O ) + r 2 s i n 2 6 ( d o
T h e s c a l e f a c t o r s a r e h 1 = h r = 1 ,
h 2 = h 8 = r , h 3 = h o = r s i n 6 .
x =
2 ( u 2 - v 2 ) ,
y = u v ,
z = z
T h e n
d x =
u d u - v d v ,
d y = u d v + v d u ,
d z = d z
a n d
( d s ) 2
( d x ) 2 + ( d y ) 2 + ( d z ) 2
=
( u 2 + v 2 ) ( d u ) 2 +
( u 2 + v 2 ) ( d v ) 2
+ ( d z ) 2
T h e s c a l e f a c t o r s a r e
h 1 = h u = V u 2 + v 2 ,
h 2 = h
v =
u 2 + v 2 , h 3 = h z = 1 .
9 .
S k e t c h a v o l u m e e l e m e n t i n ( a ) c y l i n d r i c a l a n d ( b ) s p h e r i c a l c o o r d i n a t e s g i v i n g t h e m a g n i t u d e s
o f i t s e d g e s .
( a ) T h e e d g e s o f t h e v o l u m e e l e m e n t i n c y l i n d r i c a l c o o r d i n a t e s ( F i g . ( a ) b e l o w ) h a v e m a g n i t u d e s p d o , d p
a n d d z . T h i s c o u l d a l s o b e s e e n f r o m t h e f a c t t h a t t h e e d g e s a r e g i v e n b y
d s 1 = h 1 d u 1 = ( 1 ) ( d p ) = d p ,
d s 2 = h 2 d u 2 = p d o ,
d s 3 = ( 1 ) ( d 7 ) = d z
u s i n g t h e s c a l e f a c t o r s o b t a i n e d f r o m P r o b l e m 7 .
( p d o ) ( d p ) ( d z )
p d p d , d z
Y
Y
F i g . ( a ) V o l u m e e l e m e n t i n c y l i n d r i c a l c o o r d i n a t e s .
F i g . ( b ) V o l u m e e l e m e n t i n s p h e r i c a l c o o r d i n a t e s .
( b ) T h e e d g e s o f t h e v o l u m e e l e m e n t i n s p h e r i c a l c o o r d i n a t e s ( F i g . ( b ) a b o v e ) h a v e m a g n i t u d e s d r , r d 6 a n d
r s i n 6 d o . T h i s c o u l d a l s o b e s e e n f r o m t h e f a c t t h a t t h e e d g e s a r e g i v e n b y
d s 1 = h 1 d u 1 = ( 1 ) ( d r ) = d r ,
d s 2 = h 2 d u 2 = r d 6 ,
d s 3 = h 3 d u 3 = r s i n 6 d o
u s i n g t h e s c a l e f a c t o r s o b t a i n e d f r o m P r o b l e m 8 ( a ) .
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C U R V I L I N E A R C O O R D I N A T E S
1 4 5
1 0 . F i n d t h e v o l u m e e l e m e n t d V i n ( a ) c y l i n d r i c a l , ( b ) s p h e r i c a l a n d ( c ) p a r a b o l i c c y l i n d r i c a l c o o r -
d i n a t e s .
T h e v o l u m e e l e m e n t i n o r t h o g o n a l c u r v i l i n e a r c o o r d i n a t e s u 1 , u 2 , a 3 i s
d V
=
h 1 h 2 h 3 d u 1 d u 2 d u 3
( a ) I n c y l i n d r i c a l c o o r d i n a t e s u 1 p , u 2 = 4 ) , u 3 = z , h 1 = 1 , h 2 = p , h 3 = 1 ( s e e P r o b l e m 7 ) .
T h e n
d V =
( 1 ) ( p ) ( 1 ) d p d o d z =
p d p d o d z
T h i s c a n a l s o b e o b s e r v e d d i r e c t l y f r o m F i g . ( a ) o f P r o b l e m 9 .
( b ) I n s p h e r i c a l c o o r d i n a t e s
u 1 = r , u 2 = 6 , u 3 = 0 , h 1 = 1 , h 2 = r , h 3 = r s i n 6 ( s e e P r o b l e m 8 ( a ) ) .
T h e n
d V
=
( 1 ) ( r ) ( r s i n 6 ) d r d 8 d o
= r 2 s i n 6 d r d 6 d o
T h i s c a n a l s o b e o b s e r v e d d i r e c t l y f r o m F i g . ( b ) o f P r o b l e m 9 .
< c ) I n p a r a b o l i c c y l i n d r i c a l c o o r d i n a t e s u 1 = u , u 2 = v , u 3 = z , h 1 =
h 2 =
u 2 + v 2 , h 3 = 1 ( s e e P r o b -
l e m 8 ( b ) ) . T h e n
d V
=
( u 2 + v 2 ) ( u 2 + v 2 ) ( 1 ) d u d v d z
=
( u 2 + v 2 ) d u d v d z
1 1 . F i n d ( a ) t h e s c a l e f a c t o r s a n d ( b ) t h e v o l u m e e l e m e n t d V i n o b l a t e s p h e r o i d a l c o o r d i n a t e s .
( a )
x = a c o s h 6 c o s 7 ) c o s 0 ,
y = a c o s h 6 c o s 7 ) s i n o ,
z = a s i n h
s i n ' r )
d x = - a c o s h 6 c o s 7 ) s i n q d o - a c o s h 5 s i n 7 ) c o s 4 d 7 ) + a s i n h
c o s 7 ) c o s c a d
d y = a c o s h 6 c o s 7 ) c o s c p d o - a c o s h 6 s i n 7 ) s i n 0 d 7 7 + a s i n h
c o s ? ) s i n 0 d e
d z = a s i n h
c o s 7 j d 7 7 + a c o s h e s i n 7 7 d e
T h e n
( d s ) 2 =
( d x ) 2 + ( d y ) 2 +
( d z ) 2 = a 2 ( s i n h 2 + s i n 2 7 ) ) ( d e ) 2
+ a 2 ( s i n h 2 e + s i n 2 7 ) ) ( d 7 7 ) 2
+ a 2 c o s h 2 6 c o s 2 7 ) ( d o ) 2
a n d
h 1 = h e = a s i n h 2 e + s i n 2 7 ) ,
h 2 = h , = a s i n h 2 e + s i n 2 7 ) ,
h 3 = h o = a c o s h
c o s 7 ) .
( b )
d V =
( a c o s h
c o s 7 7 ) d 6 d 7 ) d o
= a 3 ( s i n h 2
+ s i n e 7 ) ) c o s h
c o s 7 ) d e d 7 ) d o
1 2 . F i n d e x p r e s s i o n s f o r t h e e l e m e n t s o f a r e a i n o r t h o g o n a l c u r v i l i n e a r c o o r d i n a t e s .
R e f e r r i n g t o F i g u r e 3 , p . 1 3 6 , t h e a r e a e l e m e n t s a r e g i v e n b y
d A 1
=
I
( h 2 d u 2 e 2 ) x ( h 3 d u 3 e 3 ) I
=
h 2 h 3 e 2 x e 3 I
d u e d u 3
=
h 2 h 3 d u e d u 3
s i n c e
I e 2 x e 3
=
I
e 1 I
=
1 .
S i m i l a r l y
d A 2
=
I ( h 1 d u 1 e 1 ) x ( h 3 d u 3 e 3 )
I
=
h 1 h 3 d u 1 d u 3
d A 3
=
I
( h 1 d u 1 e 1 ) x ( h 2 d u 2 e 2 )
I
=
h 1 h 2 d u 1 d u 2
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1 4 6
C U R V I L I N E A R C O O R D I N A T E S
1 3 .
I f u 1 , u 2 , u 3 a r e o r t h o g o n a l c u r v i l i n e a r c o o r d i n a t e s , s h o w t h a t t h e J a c o b i a n o f x , y , z w i t h r e s p e c t
t o u 1 , u 2 , u 3 i s
R
x , y , z
( x , y , Z )
U 1 , U 2 , u 3
( u 1 + u 2 + u 3 )
h 1 h 2 4
a x
a y
a Z
a u 1
a u ,
a u 1
a x
a y
a z
a u 2
a u 2
a u 2
a x
a y
a z
a u 3 a u 3
a u 3
B y P r o b l e m 3 8 o f C h a p t e r 2 , t h e g i v e n d e t e r m i n a n t e q u a l s
a x
- a y - a Z - a x
- a z
- a x
- a - a Z
i + a u i + a u
i +
a y i +
a
k ) x ( . a 3 i + a u i + a 3 k )
1 1
2 2
2
3
a r
a r
x
a r
= h 1 e 1 , h 2 e 2 x h 3 e 3
a U l a u 2
a u 3
h 1 h 2 h 3 e 1 e 2 x e 3
= h 1 h 2 h 3
I f t h e J a c o b i a n e q u a l s z e r o i d e n t i c a l l y t h e n
a r
,
a r
a r
a r e c o p l a n a r v e c t o r s a n d t h e c u r v i -
a U 1
a u 2 a u 3
l i n e a r c o o r d i n a t e t r a n s f o r m a t i o n b r e a k s d o w n , i . e . t h e r e i s a r e l a t i o n b e t w e e n x , y , z
h a v i n g t h e f o r m
F ( x , y , z ) = 0 . W e s h a l l t h e r e f o r e r e q u i r e t h e J a c o b i a n t o b e d i f f e r e n t f r o m z e r o .
1 4 .
E v a l u a t e f f f ( x 2 + y 2 + z 2 ) d x d y d z w h e r e V i s a s p h e r e h a v i n g c e n t e r a t t h e o r i g i n a n d r a -
v
d i u s e q u a l t o a .
z
F i g . ( a )
F i g . ( b )
T h e r e q u i r e d i n t e g r a l i s e q u a l t o e i g h t t i m e s t h e i n t e g r a l e v a l u a t e d o v e r t h a t p a r t o f t h e s p h e r e c o n -
t a i n e d i n t h e f i r s t o c t a n t ( s e e F i g . ( a ) a b o v e ) .
T h e n i n r e c t a n g u l a r c o o r d i n a t e s t h e i n t e g r a l e q u a l s
( ' a
V a t - x 2 a 2 - x 2 _ y 2
8
J
J
f
( x 2 + y 2 + z 2 ) d z d y d x
x = o y = 0 z = 0
b u t t h e e v a l u a t i o n , a l t h o u g h p o s s i b l e , i s t e d i o u s .
I t i s e a s i e r t o u s e s p h e r i c a l c o o r d i n a t e s f o r t h e e v a l -
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C U R V I L I N E A R C O O R D I N A T E S
1 4 7
u a t i o n .
I n c h a n g i n g t o s p h e r i c a l c o o r d i n a t e s , t h e i n t e g r a n d x 2 + y 2 + z 2 i s r e p l a c e d b y i t s e q u i v a l e n t r 2
w h i l e t h e v o l u m e e l e m e n t d x d y d z
i s r e p l a c e d b y t h e v o l u m e e l e m e n t r 2 s i n 8 d r d e d o ( s e e P r o b l e m
1 0 ( b ) ) . T o c o v e r t h e r e q u i r e d r e g i o n i n t h e f i r s t o c t a n t , f i x 8 a n d 0 ( s e e F i g . ( b ) a b o v e ) a n d i n t e g r a t e f r o m
r = 0 t o r = a ; t h e n k e e p 0 c o n s t a n t a n d i n t e g r a t e f r o m 6 = 0 t o 7 T / 2 ; f i n a l l y i n t e g r a t e w i t h r e s p e c t t o 0
f r o m 9 5 = 0 t o 0 _ 7 T / 2 .
H e r e w e h a v e p e r f o r m e d t h e i n t e g r a t i o n i n t h e o r d e r r , 8 , o a l t h o u g h a n y o r d e r c a n
b e u s e d . T h e r e s u l t i s
7 1 / 2
7 1 / 2 a 7 1 / 2
f X 7 1 / 2 ( '
a
8 f
J
J
( r 2 ) ( r 2 s i n 6 d r d 8 d o )
=
8
f J
J
r ' s i n 6 d r d 6 d o
( k = o
0 = o r = o
= o
0 = 0
r = 0
7 1 / 2
7 1 / 2
a
7 1 / 2
7 1 / 2
8
f
f
5 s i n 8 I r = o d 8 d o
=
8 6 f f
s i n 6 d 8 d ( p
< h = o
a = o
7 1 / 2
7 1 / 2
8 a 5 f
- c o s 6
I B = o d o
5
= O
f
a s
d o
4 7 T a s
-
5
- o
P h y s i c a l l y t h e i n t e g r a l r e p r e s e n t s t h e m o m e n t o f i n e r t i a o f t h e s p h e r e w i t h r e s p e c t t o t h e o r i g i n , i . e . t h e
p o l a r m o m e n t o f i n e r t i a , i f t h e s p h e r e h a s u n i t d e n s i t y .
I n g e n e r a l , w h e n t r a n s f o r m i n g m u l t i p l e i n t e g r a l s f r o m r e c t a n g u l a r t o o r t h o g o n a l c u r v i l i n e a r c o o r d i -
n a t e s t h e v o l u m e e l e m e n t d x d y d z i s r e p l a c e d b y h 1 h 2 h 3 d u l d u . 2 d u 3 o r t h e e q u i v a l e n t J ( u i 1 ' u 2 z u 3 ) d u l d u 2 d u 3
w h e r e J i s t h e J a c o b i a n o f t h e t r a n s f o r m a t i o n f r o m x , y , z t o u 1 , u 2 , u 3 ( s e e P r o b l e m 1 3 ) .
' 3 r
' a r
- a r
1 5 .
I f
u 1 , u 2 , u 3 a r e g e n e r a l c o o r d i n a t e s , s h o w t h a t
a u '
a u a n d V u , , V u 2 , V u 3 a r e r e c i p r o -a u
1
2 3
c a l s y s t e m s o f v e c t o r s .
W e m u s t s h o w t h a t
a r
. V u -
W e h a v e
d u p
` +
0 i f p A q
c = o
0 = 0
l t / 2
w h e r e p a n d q c a n h a v e a n y o f t h e v a l u e s 1 , 2 , 3 .
d r
=
a u d u 1
+
a u d u e
+
a u d u 3
1 2
3
M u l t i p l y b y V u 1
.
T h e n
o r
=
d u 1
=
( V u 1
- D - r ) d u 1
+
( V u 1
_ r _ )
d u e
+
( V , ,
a r )
d u 3
-
1
i f p = q
a u 1
a u 2
a u 3
a
1 = 1 ,
v u 1 - = 0
,
V U 1
u 3 = 0
S i m i l a r l y , u p o n m u l t i p l y i n g b y V u 2 '
a n d V u 3
t h e r e m a i n i n g r e l a t i o n s a r e p r o v e d .
- a r
1 6 . P r o v e
a u
. a u X a r
V u 1
V u 2 X V u 3
= 1 .
1 2
3
F r o m P r o b l e m 1 5 ,
- a
' a u ' a u
a n d V u 1 r V u 2 , V u 3 a r e r e c i p r o c a l s y s t e m s o f v e c t o r s . T h e n t h e
- a U l 2
3
r e q u i r e d r e s u l t f o l l o w s f r o m P r o b l e m 5 3 ( c ) o f C h a p t e r 2 .
T h e r e s u l t i s e q u i v a l e n t t o a t h e o r e m o n J a c o b i a n s f o r
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1 4 8
C U R V I L I N E A R C O O R D I N A T E S
a n d s o
J ( x , Y , z
) J ( u 1 , u 2 . u 3 )
_
3 u 1
a u 1
a u 1
a x
a y
a z
a u 2
a u 2
a u 2
a x
a y
a z
a u 3 a u 2 a u 3
a x
a y
a z
1
u s i n g P r o b l e m 1 3 .
=
J ( u 1 , U 2 ' u 3 )
x , y , z
1 7 . S h o w t h a t t h e s q u a r e o f t h e e l e m e n t o f a r e l e n g t h i n g e n e r a l c u r v i l i n e a r c o o r d i n a t e s c a n b e e x -
p r e s s e d b y
P = 1 q = 1
T h i s i s c a l l e d t h e f u n d a m e n t a l q u a d r a t i c f o r m o r m e t r i c f o r m . T h e q u a n t i t i e s g 0 a r e c a l l e d m e t r i c
c o e f f i c i e n t s a n d a r e s y m m e t r i c , 2 i . e . g , , : 3
2
g q p .
I f
9 0 q = 0 , p / q , t h e n t h e c o o r d i n a t e s y s t e m i s o r t h o g o n a l .
I n t h i s c a s e g 1 1 = h i . g 2 2 = h 2
, 9 3 3 =
. T h e m e t r i c f o r m e x t e n d e d t o h i g h e r d i m e n s i o n a l s p a c e i s o f
f u n d a m e n t a l i m p o r t a n c e i n t h e t h e o r y o f r e l a t i v i t y ( s e e C h a p t e r 8 ) .
G R A D I E N T , D I V E R G E N C E A N D C U R L I N O R T H O G O N A L C O O R D I N A T E S .
u i , u 2 , u 3
x , y , z
W e h a v e
=
a 1 d u i + d 2 d u e + d d u 3
r
=
- a u l .
+
a u
d u 2 +
a u s
1
2 3
T h e n
d s 2
3
3
d s 2
g p q d u , d u q
1 , = 1
q = 1
d r
d r
=
C E ,
4 2 , d u i + a , - 4 2 d u i d u 2 + a i a 3 d u i d u 3
+ C 6 2 a 1 d u 2 d u i + d 2 I t 2 d u 2 + a 2 Q 3 d u 2 d u 3
+ a 3 Q i d u 3 d u 1 + a c 3 Q 2 d u 3 d u 2 + U 3 a 3 d u 3
3 3
g 0 q d u o d u q
w h e r e g p q
i t i p 0 1 q
1 8 . D e r i v e a n e x p r e s s s i o n f o r v 4 ) i n o r t h o g o n a l c u r v i l i n e a r c o o r d i n a t e s .
L e t 7 1 ) =
f 1 e i + f 2 e 2 + f 3 e 3 w h e r e f 1 , f 2 , f 3 a r e t o b e d e t e r m i n e d .
S i n c e
d r
a u d u i + a u d u 2 + a u
d u 3
1
2
3
h 1 e 1 d u i + h 2 e 2 d u 2 + h 3 e 3 d u 3
w e h a v e
( 1 )
d < P
=
V
d r
h 1 f i d u i + h 2 f 2 d u 2 + h 3 f 3 d u 3
B u t
( 2 )
d 4 )
=
a
d u i +
a -
d u g +
a c p
d u 3
i
u 2
3
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C U R V I L I N E A R C O O R D I N A T E S
E q u a t i n g ( 1 ) a n d ( 2 ) ,
T h e n
f =
f = 1 a
1 - h 1 a u l '
2
h 2 a u 2
3
h 3 a u 3
T h i s i n d i c a t e s t h e o p e r a t o r e q u i v a l e n c e
e l a
e 2 a 4 e 3 a
h 1 a u 1 + h 2 a u - 2 + h 3 a u 3
e 1
e 2
e 3
h 1 a u 1 + h 2 a u 2 + h 3 a u 3
w h i c h r e d u c e s t o t h e u s u a l e x p r e s s i o n f o r t h e o p e r a t o r V i n r e c t a n g u l a r c o o r d i n a t e s .
1 9 . L e t u 1 , u 2 , u 3 b e o r t h o g o n a l c o o r d i n a t e s . ( a ) P r o v e t h a t
I V u p l = h p
p = 1 , 2 , 3 .
( b ) S h o w t h a t e p = E p .
1 4 9
( a ) L e t
= u 1 i n P r o b l e m 1 8 . T h e n V u t = h 1 a n d s o I V u 1 i
=
I e 1 1 1 h , = h 1 1 ,
s i n c e
I e 1 I = 1 .
S i m i -
1
- 1
1
l a r l y b y l e t t i n g C I ) = u 2 a n d u s ,
I v u 2 I
= h 2
a n d
I v u 3
h 3 -
.
V u p
( b ) B y d e f i n i t i o n E p =
I
v u p I .
F r o m p a r t ( a ) , t h i s c a n b e w r i t t e n E p = h p v u p = e p a n d t h e r e s u l t i s p r o v e d .
2 0 . P r o v e e 1 = h 2 h 3 v u 2 x V u 3 w i t h s i m i l a r e q u a t i o n s f o r e 2 a n d e 3 , w h e r e u l , u 2 , u 3 a r e o r t h o g o n a l
c o o r d i n a t e s .
F r o m P r o b l e m 1 9 , v u 1 =
h l
= e 2
3 =
e 2
- ,
V U
2
h
u
h
T h e n
v u 2 x v u 3 =
e 2 x e 3
-
e 1
a n d
e 1 = h 2 h 3 v u 2 x V 3 3 .
h 2 . h 3
- h 2 h 3
S i m i l a r l y
e 2 = h 3 h 1 v u 3 x V u 1 - a n d e 3 = h 1 h 2 V u 1 x v u 2 .
2 1 . S h o w t h a t i n o r t h o g o n a l c o o r d i n a t e s
( a )
V
.
( A l e 1 )
h h h
a u
( A l , h 2 h 3 )
1 2 3 1
( b ) V x ( A 1 e 1 )
h 3 2
- 6 u 3 ( A 1 h 1 ) - h e h 2 a u 2 ( A 1 h 1 )
w i t h s i m i l a r r e s u l t s f o r v e c t o r s A 2 e 2 a n d A 3 e 3 .
( a ) F r o m P r o b l e m 2 0 ,
V ' ( A l e , )
=
V
( A 1 h 2 h 3 V u 2 x V u 3 )
=
V ( A 1 h 2 h 3 ) , V u 2 x V u 3 + A 1 h 2 h 3 V ( V u 2 X v u 3 )
h 2
h 3
V ( A 1 h 2 h 3 )
X
e 1
a
h 1
u 1
( A 1 h 2 h 3 ) +
a
1
+
0
= V ( A h h )
3
e 1
1
h 2 h 3
2
3
e
h
a u
( A 1 h 2 h 3 ) +
h
a u ( A 1 h 2 h 3 )
I
h h
2 3
3
2 3
h 1 h 2 h 3
a u 1 ( A 1 h 2 h 3 )
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1 5 0
C U R V I L I N E A R C O O R D I N A T E S
( b ) O x ( A i e i )
=
V x ( A i h i V u i )
= V ( A i h i ) x V u i + A i h i V x V u i
Q ( A i h i ) x h i +
1
0
h l
- a u ,
( A 1 h i ) + h 2
a
2 2
( A i h i ) + h
u 3
( A i h i )
x
h i
e 2
a
A h
h 3 h i
a u 3 (
1
1 ) e 3 a ( A i h i )
h i h 2
a u 2
2 2 . E x p r e s s d i v A = V - A i n o r t h o g o n a l c o o r d i n a t e s .
V I A = V . ( A 1 e i + A 2 e 2 + A 3 e 3 )
= V ( A 1 e 1 ) +
V ( A 2 e 2 ) + V ' ( A 3 e 3 )
_
1
a
( A 1 h 2 h 3 ) +
2
( A 2 h 3 h i ) +
a
( A 3 h 1 h 2 )
T
h 1 h 2 h 3 a u i a u 2
a u 3
u s i n g P r o b l e m 2 1 ( a ) .
2 3 . E x p r e s s c u r l A = V x A i n o r t h o g o n a l c o o r d i n a t e s .
V x A
e 2
( A 1 h 1 ) -
e 3
( A i h i )
h 3 h i
u 3
h i h 2 a u 2
+
h h a u i
( ' 4 2 h 2 ) -
h e
h 3 a u 3 ( A 2 h 2 )
= V x ( A i e i + A 2 e 2 + A 3 e 3 )
=
V x ( A i e i ) + V x ( A 2 e 2 ) + V x ( A 3 e 3 )
e i
a
+
h 2 h 3 a u 2
( A 3 h 3 )
e i
( A 3 h 3 ) - ( A 2 h 2 ) +
h 2 h 3
a u 2
u 3
e 2
a ( A 3
h 3 )
h 3 h i a u i
e 2
a
( A 1 h 1 ) -
a
( A 3 h 3 )
h 3 h i
a u 3
a u i
e 3
a
( A 2 h 2 ) -
a
( A i h i )
h i h 2
a u i
a u 2
u s i n g P r o b l e m 2 1 ( b ) . T h i s c a n b e w r i t t e n
V X A
= 1
h 1 h 2 h 3
h i e 1
a
a u 1
A i h 1
2 4 . E x p r e s s V 2 q i n o r t h o g o n a l c u r v i l i n e a r c o o r d i n a t e s .
e i a l ' J
e 2
V b
r o m P r o b l e m 1 8 ,
h i a u i
h 2 a u 2
A 2 h 2 A 3 h 3
e 3 a
h 3 a u 3
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C U R V I L I N E A R C O O R D I N A T E S
I f A = v q , t h e n
_
1
a
_ 1
a /
_ a
A i
h a u
A 2
h 2 '
A 3
h a u
1
2
2 3
3
a n d b y P r o b l e m 2 2 ,
v - A =
v 2 J
1
a ( h 2 1 1 3
+
( h 3 h 1 a t e )
+
( h 1 h 2 a q
h i h 2 h 3
a u i
h 1
- a u ,
a u 2
h 2
a u 2
a u 3
h 3
a u 3
2 5 . U s e t h e i n t e g r a l d e f i n i t i o n
f f A
n d S
l i m
A S
A V
( s e e P r o b l e m 1 9 , C h a p t e r 6 ) t o e x p r e s s V A
i n o r t h o g o n a l c u r v i l i n e a r c o o r d i n a t e s .
C o n s i d e r t h e v o l u m e e l e m e n t A V ( s e e a d j a -
c e n t f i g u r e ) h a v i n g e d g e s h j A u 1 , h 2 A u 2 , h 3 A u 3 .
L e t A = A i e 1 + A 2 e 2 + A 3 e 3 a n d l e t n b e
t h e o u t w a r d d r a w n u n i t n o r m a l t o t h e s u r f a c e A S o f
A V . O n f a c e J K L P , n = - e 1 . T h e n w e h a v e a p -
p r o x i m a t e l y ,
e 1
f f A n d S
=
( A n a t p o i n t P ) ( A r e a o f J K L P )
J K L P
[ ( A 1 e 1 + A 2 e 2 + A 3 e 3 )
( - e l ) I ( h 2 h 3 A u 2 A u 3 )
- A l h 2 h 3 A u 2 A u 3
O n f a c e E F G H , t h e s u r f a c e i n t e g r a l i s
1 5 1
e 2
A l h 2 h 3 D u 2 A u 3
+
a u ( A 1 h 2 h 3 A u 2 A u 3 ) A U 1
1
a p a r t f r o m i n f i n i t e s i m a l s o f o r d e r h i g h e r t h a n A 1 A 2 A 3 .
T h e n t h e n e t c o n t r i b u t i o n t o t h e s u r f a c e
i n t e g r a l f r o m t h e s e t w o f a c e s i s
a u
( A , . h 2 h 2 A u 2 A u 3 ) A
1
1
T h e c o n t r i b u t i o n f r o m a l l s i x f a c e s o f A V i s
a
( A 1 h 2 h 3 ) A u 1 A u 2 A u 3
1
a u i
( A 1 h 2 h 3 ) +
a - ( A 2 h 1 h 3 ) + a u 3 ( A 3 h l h 2 )
A u 1 A u 2 A u 3
U 2
I
D i v i d i n g t h i s b y t h e v o l u m e h 1 h 2 h 3 A 1 A 2 A u 3 a n d t a k i n g t h e l i m i t a s A u 1 , A u 2 , A 3 3 a p p r o a c h z e r o ,
w e f i n d
d i v A
=
v A
1
[ -
( A 1 h 2 h 3 ) +
( A 2 h 1 h 3 ) + a ( A 3 h 1 h 2 )
h i h 2 h 3
a u 1
o u 2
- 3 u 3
N o t e t h a t t h e s a m e r e s u l t w o u l d b e o b t a i n e d h a d w e c h o s e n t h e v o l u m e e l e m e n t A V s u c h t h a t P i s
a t i t s c e n t e r .
I n t h i s c a s e t h e c a l c u l a t i o n w o u l d p r o c e e d i n a m a n n e r a n a l o g o u s t o t h a t o f P r o b l e m 2 1 ,
C h a p t e r 4 .
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1 5 2
2 6 . U s e t h e i n t e g r a l d e f i n i t i o n
J
A
d r
( c u r l A )
n = ( O x A ) n =
l i m
A S - 0
L A S
( s e e P r o b l e m 3 5 , C h a p t e r 6 ) t o e x p r e s s V x A
i n o r t h o g o n a l c u r v i l i n e a r c o o r d i n a t e s .
L e t u s f i r s t c a l c u l a t e ( c u r l A )
e 1 .
T o d o
t h i s c o n s i d e r t h e s u r f a c e S 1 n o r m a l t o e 1 a t P , a s
s h o w n i n t h e a d j o i n i n g f i g u r e . D e n o t e t h e b o u n d a r y
o f S 1 b y C 1 . L e t A = A l e 1 + A 2 e 2 + A 3 e 3 .
W e
h a v e
P Q
Q b
T h e f o l l o w i n g a p p r o x i m a t i o n s h o l d
( 1 )
f A
d r
=
( A a t P )
( h 2 A u 2 e 2 )
P Q
T h e n
o r
C U R V I L I N E A R C O O R D I N A T E S
+
f A . d r
+
f A . d r
L M
( A 1 e 1 + A 2 e 2 + A 3 e 3 )
h 2 u 2 e 2 )
f
A d r
= A 2 h 2 A U - 2
M L
( 2 )
f A . d r
L M
S i m i l a r l y ,
o r
f A .
d r
P M
- A 2 h 2 A - 2
- a
( A 2 h 2 A u 2 ) L \ u 3
- a u s
a u 3 ( A 2
h 2 A u 2 ) A 3
( A a t P ) ( h 3 D u 3 e 3 )
=
A 3 h 3 A u 3
( 3 )
f
A - d r
=
- A 3 h 3 D u 3
M P
a n d
( 4 )
f A . d r
=
A s h 3 D u 3 + -
u
( A 3 h 3 A u 3 ) A u 2
Q L
A d d i n g ( 1 ) , ( 2 ) , ( 3 ) , ( 4 ) w e h a v e
J
A d r =
c 1
2
M P
A 2 h 2 A u 2
a ( A 3 h 3
A 3 ) u 2
-
a ( A 2 h 2 A u 2 ) A u 3
2 3
=
a u 2
( A 3 h 3 ) -
a 3 3
( A 2 h 2 )
A U 2 A 3
a p a r t f r o m i n f i n i t e s i m a l s o f o r d e r h i g h e r t h a n A 2 A u 3 .
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C U R V I L I N E A R C O O R D I N A T E S
z e r o ,
1
( c u r l A )
- e 1
=
h h 3
a 2 2
( A 3 h 3 ) - " a u 3 ( A 2 h 2 )
S i m i l a r l y , b y c h o o s i n g a r e a s S 2 a n d S 3 p e r p e n d i c u l a r t o e 2 a n d e 3 a t P r e s p e c t i v e l y , w e f i n d ( c u r l A ) - e 2
a n d ( c u r l A ) - e 3 . T h i s l e a d s t o t h e r e q u i r e d r e s u l t
c u r l A =
1 5 3
D i v i d i n g b y t h e a r e a o f S 1 e q u a l t o h 2 h 3 D u 2 t a u 3 a n d t a k i n g t h e l i m i t a s A U 2 a n d A 3 a p p r o a c h
e i
' 6
a u ( A 3 h 3 ) - a ( A 2 h 2 )
h 2 h 3
2
3
+
e h
a u ( A 1 h 1 )
- a u
( A 3 h 3 )
h 3 1
3
1
+ e
I - a u ,
( A 2 h 2 '
a u ( A 1 h 1 )
h 1 h 2
2
h i e 1
h 2 e 2
h 3 e 3
a a
a
- 3 u 1
a u 2
a u 3
h 1 A 1
h 2 A 2 h 3 A 3
T h e r e s u l t c o u l d a l s o h a v e b e e n d e r i v e d b y c h o o s i n g P a s t h e c e n t e r o f a r e a S i ; t h e c a l c u l a t i o n
w o u l d t h e n p r o c e e d a s i n P r o b l e m 3 6 , C h a p t e r 6 .
2 7 . E x p r e s s i n c y l i n d r i c a l c o o r d i n a t e s t h e q u a n t i t i e s ( a ) V < P , ( b ) V - A , ( c ) V x A , ( d ) V 2 < P .
a n d
1 a w e 1 +
a y e 2
+
a y e 3
h 1 a u ,
h 2 a u 2
h 3 a u 3
1 a
e
+
1
e +
1 a T
1 a p
P
p a 0
1
a z
a
1 a
a
a p
e p +
T
p
e o
+
a Z
e z
e z
( b )
V -
A
1 - 3 u , ( h 2 h 3 A 1 ) + a u ( h 3 h 1 A 2 ) +
1 h 2 h 3
2
a
a u 3
1
a
( ( p ) ( 1 ) A p }
+
( i c )
( 1 ) ( p ) a
a p
a A
p a p
( p A p )
+
+
a z ( p A z )
F o r c y l i n d r i c a l c o o r d i n a t e s ( p , q , z ) ,
u 1 = p u 2 = 0 , u 3 = z ;
e 1 e p ,
e 3 = e z
h 1 = h p = 1 ,
h 2 = h o = p ,
h 3 = h z = 1
1
h 1 h 2 h 3
( h 1 h 2 A 3 )
+
a z
( ( 1 ) ( p ) A Z ) ]
w h e r e A = A P e 1 + A o e 2 + A Z e 3 ,
i . e . A I = A , , A 2 = A k , A 3 = A Z .
( c )
V X A
= 1
h 1 h 2 h 3
h 1 e 1
h 2 e 2
h 3 e 3
a
a
a
a u 1
a u 2
a u 3
1
=
P
e p p e g
e z
a a
a
a p 4 a z
h 1 A 1
h 2 A 2 h 3 A 3
I A p p A 0 A Z
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1 5 4
1
P
C U R V I L I N E A R C O O R D I N A T E S
a A z
_
a
( P A k )
e p
+
a
L A P
a p
( P A ) _ )
o
e z
( d )
V 2
x
1
a
( ? t h
a
h 3 h I
+
h 1 h 2
V 2 4 )
h 1 h 2 h 3 L a u 1
h 1
a u ,
a u 2
h 2
a u 2
a u 3
h 3
1
a ( ( p ) ( 1 )
- a ( D )
+
a ( ( 1 ) ( 1 ) a
+
a
( 1 ) ( P ) a q )
a z
( 1 )
a z
r
( 1 ) ( P ) ( 1 )
a P
( 1 )
a P
P
a ( a i l
P a p ` p a p l
2 8 . E x p r e s s ( a ) V x A a n d ( b ) V 2
H e r e
u 1 = r , u 2 = e , U 3 = 0 ;
O 1 A =
a )
h 1 h 2 h 3
P
a A p
P
a A z
a z
a P
( 1 ) ( r ) ( r s i n e )
h 1 e 1
a
a u ,
1 a 2 a 2 < p
+
P 2
4 2
+ a z 2
i n s p h e r i c a l c o o r d i n a t e s .
e 2 =
h , 2 =
h 2 e 2
h 3 e 3
a
a
- 3 U 2
a u 3
h 1 A 1 h 2 A 2 h 3 A 3
r 2 s i n e
- 6 0
( r s i n e A o ) - - a ( r A e )
a A r
1 ( r s i n 0 A O )
( b ) V 2
_
1
a ( h 2 h s P \ +
a
h 1 h 2 h 3
- 3 u ,
h l
- 3 u ,
a u 2
r e 6
+
e r
e , ; +
e r
r e e r s i n 0 e ( k
a
a
a
a r
a &
a 0
A r r A e
r s i n e
a ( r A e ) _ a A r
r s i n e e ( h
a r
a e
h 3 h 1 a
+
a
( h - j h - 2 a q
J
h 2
a u 2 a u 3
h 3
a u 3
1
[ - 3 ( ( r ) ( r s i n e ) a
( 1 ) ( r ) ( r s i n e )
a r
( 1 )
a r
+
a
( r s i n e ) 0 ) ( 1 ) ' a q j
' 3 0
r a e
+
1 s i n e
a ( r : 2 a
+
s i n e
r 2 s i n e
a r
a r a e
a e
1
a
r 2
+
1
a ( s i n O ' \
r 2 - 6 r
a r
r 2 s i n e ' 3 0
a e
2 9 . W r i t e L a p l a c e ' s e q u a t i o n i n p a r a b o l i c c y l i n d r i c a l c o o r d i n a t e s .
2 q
r 2 s i n 2 9
F r o m P r o b l e m 8 ( b ) ,
u 1 = u , u 2 = v , u 3 = z ;
h 1 =
u 2 + v 2 , h 2 = u 2 + v 2 , h 3
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C U R V I L I N E A R C O O R D I N A T E S
T h e n
V 2
` f '
=
2 2
I
u
a +
+ v
a a v
a 2 ` Y +
a - 2
q
a 2
u 2 + v 2
a u 2
a v 2
a z 2
a n d L a p l a c e ' s e q u a t i o n i s V 2 ' = 0 o r
+ j
( u + v
2
z
a 2
+
a 2
+
( u 2 + v 2 )
a 2
= 0
a u
a v
a t p
a v
a z 2
3 0 . E x p r e s s t h e h e a t c o n d u c t i o n e q u a t i o n
a U
= K
V 2 U i n e l l i p t i c c y l i n d r i c a l c o o r d i n a t e s .
H e r e
u 1 = u , u 2 - , = v , u 3 = z ;
h l = h , 2 = a s i n h 2 u + s i n e v ,
h 3 = 1 .
T h e n
0 2 U
=
1 a
( i u ) +
a
( L U ) +
a ( a 2 ( s i n h 2 u
a 2 ( s i n h 2 u + s i n 2 v )
a u
a u
a v
a v
a z
1
r a 2 U
+
a 2 U
+
a 2 U
a 2 ( s i n h 2 u + s i n 2 v ) L a u 2
- a v 2
a z 2
a n d t h e h e a t c o n d u c t i o n e q u a t i o n i s
a U
a t
1
a 2 U + a 2 U
+
a 2 U
a 2 ( s i n h 2 u + s i n 2 v )
[ a U 2
a v 2
- a Z 2
S U R F A C E C U R V I L I N E A R C O O R D I N A T E S
+ s i n 2 v )
3 1 . S h o w t h a t t h e s q u a r e o f t h e e l e m e n t o f a r c l e n g t h o n t h e s u r f a c e r = r ( u , v ) c a n b e w r i t t e n
d s 2
=
E d u 2
+
2 F d u d v
+
G d v 2
W e h a v e
T h e n
d s 2 =
d r d r
d r
=
a d .
+ a a
- v d v
_
a r
a r
2
a r
a r
a r
a r
a u
a u
d u
+
2
a u
a v
d u d v
+
a v a v
d v 2
E d u 2
+
2 F d u d v
+
G d v 2
3 2 . S h o w t h a t t h e e l e m e n t o f s u r f a c e a r e a o f t h e s u r f a c e r = r ( u , v ) i s g i v e n b y
d S
=
E
d u d v
T h e e l e m e n t o f a r e a i s g i v e n b y
d S
f ( d u ) x ( i - d v )
+
-
a r
x
a r
d u d v
=
a u
a v
J
1 5 5
) d u d v
x
a v )
a u
x
a v
u
T h e q u a n t i t y u n d e r t h e s q u a r e r o o t s i g n i s e q u a l t o ( s e e P r o b l e m 4 8 , C h a p t e r 2 )
( a r
a r ) ( a r
a r )
-
( a r
a r ) ( a r
a r )
=
E G - F 2
a n d t h e r e s u l t f o l l o w s .
a u a u
a v a v
a u
a v
a v
a u
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1 5 6
C U R V I L I N E A R C O O R D I N A T E S
M I S C E L L A N E O U S P R O B L E M S O N G E N E R A L C O O R D I N A T E S .
3 3 . L e t A b e a g i v e n v e c t o r d e f i n e d w i t h r e s p e c t t o t w o g e n e r a l c u r v i l i n e a r c o o r d i n a t e s y s t e m s
( u i , u 2 , u 3 ) a n d ( u i , u 2 , u 3 ) .
F i n d t h e r e l a t i o n b e t w e e n t h e c o n t r a v a r i a n t c o m p o n e n t s o f t h e v e c t o r
i n t h e t w o c o o r d i n a t e s y s t e m s .
S u p p o s e t h e t r a n s f o r m a t i o n e q u a t i o n s f r o m a r e c t a n g u l a r
( x , y , z ) s y s t e m t o t h e ( u i , u 2 i u 3 ) a n d
( u i , u 2 , i . L 3 ) s y s t e m s a r e g i v e n b y
( 1 )
x = X 1 ( u i , u 2 . U 3 ) ,
Y = Y 1 ( u 1 , u 2 , u 3 ) ,
x = x . , 0 7 1 , 4 2 , u 3 ) ,
Y = Y - 2 0 7 1 , u 2 , u 3 ) ,
z = z i ( u 1 , u 2 , u 3 )
z = z 2 0 7 1 , u 2 , u 3 )
T h e n t h e r e e x i s t s a t r a n s f o r m a t i o n d i r e c t l y f r o m t h e ( u 1 , u 2 , u 3 ) s y s t e m t o t h e ( u i , u 2 , u 3 ) s y s t e m d e f i n e d b y
( 2 )
u 1 = u 1 6 1 , u 2 , u 3 ) ,
u 2 = u 2 ( u 1 , u 2 , u 3 ) ,
a n d c o n v e r s e l y . F r o m ( 1 ) ,
u 3 =
u 3 0 i 1 , u 2 , u 3 )
d r
=
a d u 1
+
a u d u 2
+
a u d u 3
a 1 d u 1
+
a 2 d u 2
+
a 3 d u 3
1
2
3
d r
=
a r
d u i
+
a r
d u 2
+
a r
d u 3
=
a i d u i +
a 2 d u 2
+ a n d u 3
- a u ,
a u ` 2 a u 3
T h e n
( 3 )
a 1 d u 1
+
a t 2 d u 2
+
d o d u 3
=
a i d u i
+
a 2 d i 1 2
+
a 3 d u 3
F r o m ( 2 ) ,
d u i
=
a u 1 d u i
+
a u 1 d 7 2
+
a u 1 d u 3
1 2
d u 2
=
a u 2
d u i
+
a u 2
d u 2
+ a u 2
t h i s
a u i
a u 2
a u 3
d u 3
=
a u 3
d u i
+ a u
d u 2
+
a u 3
d u 3
a u i
a u 2
a u 3
S u b s t i t u t i n g i n t o ( 3 ) a n d e q u a t i n g c o e f f i c i e n t s o f d u i , d u 2 , d u ` 3 o n b o t h s i d e s , w e f i n d
a 1
a u i
a u 2
' U . 1
a l
- 3 7 ; , L
+
a 2
a i
+
a 3
a u i
( 4 )
° ` 2
= a 1
1
+
a 2 u
2
+
a 3
u 3
u
a
a u 2
2
a 3
a u i
a
+
2
3
a u 3
a u 3
a u 3
N o w A c a n b e e x p r e s s e d i n t h e t w o c o o r d i n a t e s y s t e m s a s
( 5 )
A
C , C 1 1 + C 2 a 2 + C 3 & a
a n d
A
c 1 a 1 + c 2 a 2 + c 3 a 3
w h e r e C 1 , C 2 , C 3 a n d C 1 , C 2 , C 3 a r e t h e c o n t r a v a r i a n t c o m p o n e n t s o f A i n t h e t w o s y s t e m s . S u b s t i t u t i n g
( 4 ) i n t o ( 5 ) ,
c 1 a 1
+
C 2 a c 2
+
C 3 a 3 =
c i a i
+
c 2 a 2 +
c 3 a 3
a u 2
a u 3
-
+
W
a u i
- a u , - a u , a u 2
-
a u 2
a u 2 -
a u 3
- a u 3
-
a U 3
( C 1
a u i
+ c 2
a u 2 +
a u 3 ' a l
+
( C l a u i
+
a u 2
+
- a - a s )
a 2 +
( C 1
a u i
+ C 2
a 2
+ c 3 u n ) a 3
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C U R V I L I N E A R C O O R D I N A T E S
T h e n
6 )
C l
=
C 2
=
a u 1
- C
a u l
C 1
a u 1 +
2
a U 2
+
- a u 2
- a u 2
+C 2
C 1
a u 1
C 3
a u 3
a u 2
C 3
-
3
=
a - u 1
a 2
a u 3
- a u 3
C 1 a a ,
+
C 2
a u 2
+
a - u 3
a u 3
C 3
a u 3
o r i n s h o r t e r n o t a t i o n
( 7 )
C ,
=
a u
C 1
u P
+
C 2
P
+
a u
C 3
P
p = 1 , 2 , 3
a u ' 1
a u 2 a u 3
a n d i n e v e n s h o r t e r n o t a t i o n
( 8 ) C
3
-
a u
C
k
=
1
2
3p
,
,
a u
9
S i m i l a r l y , b y i n t e r c h a n g i n g t h e c o o r d i n a t e s w e s e e t h a t
9
C
3
a u p
C
=
1
2 3
)
p
E
q
a
p ,
,
u
9
1 5 7
T h e a b o v e r e s u l t s l e a d u s t o a d o p t t h e f o l l o w i n g d e f i n i t i o n . I f t h r e e q u a n t i t i e s C 1 , C 2 , C 3 o f a c o -
o r d i n a t e s y s t e m ( u 1 , u 2 , u 3 ) a r e r e l a t e d t o t h r e e o t h e r q u a n t i t i e s C 1 , C 2 , C 3 o f a n o t h e r c o o r d i n a t e s y s t e m
( Z 1 , 2 , u 3 ) b y t h e t r a n s f o r m a t i o n e q u a t i o n s ( 6 ) , ( 7 ) , ( 8 ) o r ( 9 ) , t h e n t h e q u a n t i t i e s a r e c a l l e d c o m p o n e n t s o f
a c o n t r a v a r i a n t v e c t o r o r a c o n t r a v a r i a n t t e n s o r o f t h e f i r s t r a n k .
3 4 . W o r k P r o b l e m 3 3 f o r t h e c o v a r i a n t c o m p o n e n t s o f A .
W r i t e t h e c o v a r i a n t c o m p o n e n t s o f A i n t h e s y s t e m s
( u 1 , u 2 , u 3 ) a n d ( u 1 , u 2 , u 3 ) a s
c 1 , c 2 , c 3 a n d
c 1 , c 2 , c 3 r e s p e c t i v e l y . T h e n
( 1 ) A
=
C 1 D u 1 +
C 2 V U 2 + C 3 V U 3
=
c 1 D u 1 +
C 2 V U 2 +
c 3 V U - 3
N o w s i n c e
u p = i p ( u s , u 2 , u 3 ) w i t h p = 1 , 2 , 3 ,
( 2 )
z
a ; j , a u 1
a u 1 a z
a u p a u 1
a u 1 a x
a ' u p a u 1
a u 1 a y
A l s o ,
( 3 )
C 1 V u 1 + C 2 V u 2 + C 3 V u 3
a u p a u 2 a u k - 6 U 3
a u 2 a x
a u 3 a x
a a p a u 2
a u p a u 3
a u 2 a y a u 3 a y
- a u o a u 2
a u i - 3 u 3
a u 2 a z
a u 3 a Z
a u 1
+
a u 2 a u 3 )
( c l a x
c 2
a x
+ c 3 a x i
p = 1 , 2 , 3
a u 1 a u 2
a u 3
a u 1
a u 2
a u 3
+ ( C 1
a y
+ c 2 - j -
+ c 3 a y )
+ ( C 1
a Z + C 2 a Z + C 3 a z
) k
a n d
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1 5 8
( 4 )
c 1 V U , + c 2 V u 2 + C 3 V u 3
a u 1
a u 2 a u 3
a u l
_
a u 2
F S
u 3
+ { C 1
y
+ C 2
a y
y
+ C 3 a )
+ { C 1
a Z + C 2 a Z +
a Z
k
E q u a t i n g c o e f f i c i e n t s o f
a u 1
C l
a x
( 5 )
a u l
C l a
a u 1
C l a Z
C U R V I L I N E A R C O O R D I N A T E S
i , 3 , k i n ( 3 ) a n d ( 4 ) ,
a u 2
C 2
+x
a u 3
_ a u 1
_ - a - U 2
_ a u g
c g
a x
c l
a x +
c 2
a x +
c 3
a x
a u 2
a u 3
a u l
a u 2
a u 3
+ c 2 a y + c 3 a y
=
c l a y + c 2
a y
+ c 3 a y
+ C
a u 2
+
C
U 3
c
a u 1
+ C
- a - U 2
+ c
- a - U 3
2
a z
3
a z
l
a z
2 a Z
3
a z
S u b s t i t u t i n g e q u a t i o n s ( 2 ) w i t h p = 1 , 2 , 3 i n a n y o f t h e e q u a t i o n s ( 5 ) a n d
a u 2 a u 3
a u 1
a u 2 a u 3
a u l
a u 2
a u 3
o n e a c h s i d e , w e f i n d
'
' '
z
x
a x
a y
a y a y
d
_
a 1 4 1
_ a u 2
_
a u g )
( C l a x + c 2 a x + c g a x
1
a z
a Z
a u , .
a u 2
C l .
=
+
a u ,
a u l
c 2 a u 1
+
_ a u 2
( 6 )
C 2
C l
a u 2
+
C 2
a u 2
+
, . ,
a u 1
a u 2
C S
a u 3
+
C 2
- a a U 3
+
w h i c h c a n b e w r i t t e n
a u l
a u 2
c
7 )
c
+
1
2
+
o r
a u k a u p
( 8 )
S i m i l a r l y , w e c a n s h o w t h a t
( 9 )
3
_ a u q
c l ,
=
q = 1
c q a u 0
C l ,
E 3
q = l
p = 1 , 2 , 3
p = 1 , 2 , 3
o f
T h e a b o v e r e s u l t s l e a d u s t o a d o p t t h e f o l l o w i n g d e f i n i t i o n .
I f t h r e e q u a n t i t i e s c 1 , c 2 , c 3 o f a c o -
o r d i n a t e s y s t e m ( u 1 , u 2 , u 3 ) a r e r e l a t e d t o t h r e e o t h e r q u a n t i t i e s
c 1 , c 2 , c 3 o f a n o t h e r c o o r d i n a t e s y s t e m
( u 1 , u 2 , u 3 ) b y t h e t r a n s f o r m a t i o n e q u a t i o n s ( 6 ) , ( 7 ) , ( 8 ) o r ( 9 ) , t h e n t h e q u a n t i t i e s a r e c a l l e d c o m p o n e n t s o f
a c o v a r i a n t v e c t o r o r a c o v a r i a n t t e n s o r o f t h e f i r s t r a n k .
I n g e n e r a l i z i n g t h e c o n c e p t s i n t h i s P r o b l e m a n d i n P r o b l e m 3 3 t o h i g h e r d i m e n s i o n a l s p a c e s , a n d
i n g e n e r a l i z i n g t h e c o n c e p t o f v e c t o r , w e a r e l e d t o t e n s o r a n a l y s i s w h i c h w e t r e a t i n C h a p t e r 8 . I n t h e
p r o c e s s o f g e n e r a l i z a t i o n i t i s c o n v e n i e n t t o u s e a c o n c i s e n o t a t i o n i n o r d e r t o e x p r e s s f u n d a m e n t a l i d e a s
i n c o m p a c t f o r m . I t s h o u l d b e r e m e m b e r e d , h o w e v e r , t h a t d e s p i t e t h e n o t a t i o n u s e d , t h e b a s i c i d e a s t r e a t -
e d i n C h a p t e r 8 a r e i n t i m a t e l y c o n n e c t e d w i t h t h o s e t r e a t e d i n t h i s c h a p t e r .
a u 3
c 3 a u 1
e q u a t i n g c o e f f i c i e n t s
c 3
a u 3
p = 1 , 2 , 3
a u o
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1 6 0
C U R V I L I N E A R C O O R D I N A T E S
S U P P L E M E N T A R Y P R O B L E M S
A n s w e r s t o t h e S u p p l e m e n t a r y P r o b l e m s a r e g i v e n a t t h e e n d o f t h i s C h a p t e r .
3 6 . D e s c r i b e a n d s k e t c h t h e c o o r d i n a t e s u r f a c e s a n d c o o r d i n a t e c u r v e s f o r ( a ) e l l i p t i c c y l i n d r i c a l , ( b ) b i p o l a r ,
a n d ( c ) p a r a b o l i c c y l i n d r i c a l c o o r d i n a t e s .
3 7 . D e t e r m i n e t h e t r a n s f o r m a t i o n f r o m ( a ) s p h e r i c a l t o r e c t a n g u l a r c o o r d i n a t e s , ( b ) s p h e r i c a l t o c y l i n d r i c a l
c o o r d i n a t e s .
3 8 . E x p r e s s e a c h o f t h e f o l l o w i n g l o c i i n s p h e r i c a l c o o r d i n a t e s :
( a ) t h e s p h e r e
x 2 + y 2 + z 2 = 9
( c ) t h e p a r a b o l o i d
z = x 2 + y 2
( b ) t h e c o n e z 2 = 3 ( x 2 + y 2 )
( d ) t h e p l a n e
z = 0
( e ) t h e p l a n e
y = x .
3 9 .
I f p , 0 , z a r e c y l i n d r i c a l c o o r d i n a t e s , d e s c r i b e e a c h o f t h e f o l l o w i n g l o c i a n d w r i t e t h e e q u a t i o n o f e a c h
l o c u s i n r e c t a n g u l a r c o o r d i n a t e s : ( a ) p = 4 , z = 0 ; ( b ) p = 4 ; ( c ) 0 = 7 T / 2 ;
( d ) 0 = 7 T / 3 , z = 1 .
4 0 . I f u , v , z a r e e l l i p t i c c y l i n d r i c a l c o o r d i n a t e s w h e r e a = 4 , d e s c r i b e e a c h o f t h e f o l l o w i n g l o c i a n d w r i t e t h e
e q u a t i o n o f e a c h l o c u s i n r e c t a n g u l a r c o o r d i n a t e s :
( a ) v = 7 T / 4 ;
( b ) u = 0 , z = 0 ; ( c ) u = 1 n 2 , z = 2 ; ( d ) v = 0 , z = 0 .
4 1 .
I f u , v , z a r e p a r a b o l i c c y l i n d r i c a l c o o r d i n a t e s , g r a p h t h e c u r v e s o r r e g i o n s d e s c r i b e d b y e a c h o f t h e f o l -
l o w i n g : ( a ) u = 2 , z = 0 ; ( b ) v = 1 , z = 2 ; ( c )
3 , z = 0 ; ( d ) 1 < u < 2 , 2 < v < 3 , z = 0 .
4 2 .
( a ) F i n d t h e u n i t v e c t o r s e r , e e a n d
o f a s p h e r i c a l c o o r d i n a t e s y s t e m i n t e r m s o f i , j a n d k .
( b ) S o l v e f o r i , j a n d k i n t e r m s o f e r , e e a n d e o .
4 3 . R e p r e s e n t t h e v e c t o r A = 2 y i - z j + 3 x k i n s p h e r i c a l c o o r d i n a t e s a n d d e t e r m i n e A r , A e a n d
4 4 . P r o v e t h a t a s p h e r i c a l c o o r d i n a t e s y s t e m i s o r t h o g o n a l .
4 5 . P r o v e t h a t ( a ) p a r a b o l i c c y l i n d r i c a l , ( b ) e l l i p t i c c y l i n d r i c a l , a n d ( c ) o b l a t e s p h e r o i d a l c o o r d i n a t e s y s t e m s
a r e o r t h o g o n a l .
4 6 . P r o v e e r = B e e + s i n 6
e " ,
e e =
e . ,
e 0 = - s i n 6
e r - c o s 6
e e .
4 7 . E x p r e s s t h e v e l o c i t y v a n d a c c e l e r a t i o n a o f a p a r t i c l e i n s p h e r i c a l c o o r d i n a t e s .
4 8 . F i n d t h e s q u a r e o f t h e e l e m e n t o f a r e l e n g t h a n d t h e c o r r e s p o n d i n g s c a l e f a c t o r s i n ( a ) p a r a b o l o i d a l ,
( b ) e l l i p t i c c y l i n d r i c a l , a n d ( c ) o b l a t e s p h e r o i d a l c o o r d i n a t e s .
4 9 . F i n d t h e v o l u m e e l e m e n t d V i n ( a ) p a r a b o l o i d a l , ( b ) e l l i p t i c c y l i n d r i c a l , a n d ( c ) b i p o l a r c o o r d i n a t e s .
5 0 . F i n d ( a ) t h e s c a l e f a c t o r s a n d ( b ) t h e v o l u m e e l e m e n t d V f o r p r o l a t e s p h e r o i d a l c o o r d i n a t e s .
5 1 . D e r i v e e x p r e s s i o n s f o r t h e s c a l e f a c t o r s i n ( a ) e l l i p s o i d a l a n d ( b ) b i p o l a r c o o r d i n a t e s .
5 2 . F i n d t h e e l e m e n t s o f a r e a o f a v o l u m e e l e m e n t i n ( a ) c y l i n d r i c a l , ( b ) s p h e r i c a l , a n d ( c ) p a r a b o l o i d a l c o -
o r d i n a t e s .
5 3 . P r o v e t h a t a n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n t h a t a c u r v i l i n e a r c o o r d i n a t e s y s t e m b e o r t h o g o n a l i s t h a t
g P q = 0 f o r p I q .
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C U R V I L I N E A R C O O R D I N A T E S
1 6 1
5 4 . F i n d t h e J a c o b i a n J ( x ' y ' z )
f o r
( a ) c y l i n d r i c a l , ( b ) s p h e r i c a l , ( c ) p a r a b o l i c c y l i n d r i c a l , ( d ) e l l i p t i c
u 1 , u 2 . u 3
c y l i n d r i c a l , a n d ( e ) p r o l a t e s p h e r o i d a l c o o r d i n a t e s .
5 5 . E v a l u a t e
1 f f
x 2 + y 2 d x d y d z , w h e r e V i s t h e r e g i o n b o u n d e d b y z = x 2 + y 2 a n d
V
H i n t : U s e c y l i n d r i c a l c o o r d i n a t e s .
z = 8 - ( X
2 + y 2 )
.
5 6 . F i n d t h e v o l u m e o f t h e s m a l l e r o f t h e t w o r e g i o n s b o u n d e d b y t h e s p h e r e x 2 + y 2 + z 2 = 1 6 a n d t h e c o n e
z 2 = x 2 + y 2 .
5 7 . U s e s p h e r i c a l c o o r d i n a t e s t o f i n d t h e v o l u m e o f t h e s m a l l e r o f t h e t w o r e g i o n s b o u n d e d b y a s p h e r e o f
r a d i u s a a n d a p l a n e i n t e r s e c t i n g t h e s p h e r e a t a d i s t a n c e h f r o m i t s c e n t e r .
5 8 .
( a ) D e s c r i b e t h e c o o r d i n a t e s u r f a c e s a n d c o o r d i n a t e c u r v e s f o r t h e s y s t e m
x 2 - y 2 = 2 u 1 c o s u 2 ,
x y = u 1 s i n U 2 ,
x
z
z = u 3
( b ) S h o w t h a t t h e s y s t e m i s o r t h o g o n a l . ( c ) D e t e r m i n e J (
' y '
) f o r t h e s y s t e m .
( d ) S h o w t h a t u 1 a n d
u 1 , u 2 , u 3
u 2 a r e r e l a t e d t o t h e c y l i n d r i c a l c o o r d i n a t e s p a n d 0 a n d d e t e r m i n e t h e r e l a t i o n s h i p .
5 9 . F i n d t h e m o m e n t o f i n e r t i a o f t h e r e g i o n b o u n d e d b y x 2 - y 2 = 2 ,
x 2 - y 2 = 4 , x y = 1 , x y = 2 , z = 1 a n d
z = 3 w i t h r e s p e c t t o t h e z a x i s i f t h e d e n s i t y i s c o n s t a n t a n d e q u a l t o K . H i n t : L e t x 2 - y 2 = 2 u ,
x y = v .
6 0 . F i n d
a u ,
r
, o u r ,
D u 1 , O u t , Q u a i n ( a ) c y l i n d r i c a l , ( b ) s p h e r i c a l , a n d ( c ) p a r a b o l i c c y l i n d r i c a l c o -
t
a u 2
3
o r d i n a t e s . S h o w t h a t e 1 = E 1 , e 2 = E 2 , e 3 = E 3 f o r t h e s e s y s t e m s .
6 1 . G i v e n t h e c o o r d i n a t e t r a n s f o r m a t i o n u 1 = x y , 2 u 2 = x 2 + y 2 , u 3 = z .
( a ) S h o w t h a t t h e c o o r d i n a t e s y s t e m i s
n o t o r t h o g o n a l .
( b ) F i n d J (
x ,
y '
z
) .
( c ) F i n d d s 2 .
u 1 i u 2 , u 3
6 2 . F i n d T D , d i v A a n d c u r l A i n p a r a b o l i c c y l i n d r i c a l c o o r d i n a t e s .
6 3 . E x p r e s s ( a ) V J i a n d ( b ) V A i n s p h e r i c a l c o o r d i n a t e s .
6 4 . F i n d V q i n o b l a t e s p h e r o i d a l c o o r d i n a t e s .
a 2 . : p
a 2 ( 1 )
6 5 . W r i t e t h e e q u a t i o n
a x e
+
a 2
=
i n e l l i p t i c c y l i n d r i c a l c o o r d i n a t e s .
Y
6 6 . E x p r e s s M a x w e l l ' s e q u a t i o n V x E
- a n
i n p r o l a t e s p h e r o i d a l c o o r d i n a t e s .
2
6 7 . E x p r e s s S c h r o e d i n g e r ' s e q u a t i o n o f q u a n t u m m e c h a n i c s V q + $
2
m ( E
- V ( x , y , z ) ) i i = 0 i n p a r a b o l i c
c y l i n d r i c a l c o o r d i n a t e s w h e r e m , h a n d E a r e c o n s t a n t s .
6 8 . W r i t e L a p l a c e ' s e q u a t i o n i n p a r a b o l o i d a l c o o r d i n a t e s .
6 9 . E x p r e s s t h e h e a t e q u a t i o n a U = K V 2 U i n s p h e r i c a l c o o r d i n a t e s i f U i s i n d e p e n d e n t o f ( a ) 0 , ( b ) Q S a n d
e , ( c ) r a n d t , ( d ) a n d t .
7 0 . F i n d t h e e l e m e n t o f a r e l e n g t h o n a s p h e r e o f r a d i u s a .
7 1 . P r o v e t h a t i n a n y o r t h o g o n a l c u r v i l i n e a r c o o r d i n a t e s y s t e m , d i v c u r l A = 0 a n d c u r l g r a d
= 0 .
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1 6 2
C U R V I L I N E A R C O O R D I N A T E S
7 2 . P r o v e t h a t t h e s u r f a c e a r e a o f a g i v e n r e g i o n R o f t h e s u r f a c e r = r ( u , v ) i s
f f / E G _ F 2
d u d v .
U s e
R
t h i s t o d e t e r m i n e t h e s u r f a c e a r e a o f a s p h e r e .
7 3 . P r o v e t h a t a v e c t o r o f l e n g t h p w h i c h i s e v e r y w h e r e n o r m a l t o t h e s u r f a c e
r = r ( u , v ) i s g i v e n b y
A =
± p ( a r x
a r )
E E G - F 2
a u
a v
7 4 .
( a ) D e s c r i b e t h e p l a n e t r a n s f o r m a t i o n x = x ( u , v ) , y = y ( u , v ) .
( b ) U n d e r w h a t c o n d i t i o n s w i l l t h e i t , v c o o r d i n a t e l i n e s b e o r t h o g o n a l ?
7 5 . L e t ( x , y ) b e c o o r d i n a t e s o f a p o i n t P i n a r e c t a n g u l a r x y p l a n e a n d ( u , v ) t h e c o o r d i n a t e s o f a p o i n t Q i n
a r e c t a n g u l a r u v p l a n e .
I f x = x ( u , v ) a n d y = y ( u , v ) w e s a y t h a t t h e r e i s a c o r r e s p o n d e n c e o r m a p p i n g
b e t w e e n p o i n t s P a n d Q .
( a ) I f x = 2 u + v a n d y = u - 2 v , s h o w t h a t t h e l i n e s i n t h e x y p l a n e c o r r e s p o n d t o l i n e s i n t h e
u v p l a n e .
( b ) W h a t d o e s t h e s q u a r e b o u n d e d b y x = 0 , x = 5 , y = 0 a n d y = 5 c o r r e s p o n d t o i n t h e u v
p l a n e ?
( c ) C o m p u t e t h e J a c o b i a n J ( X ,
v )
a n d s h o w t h a t t h i s i s r e l a t e d t o t h e r a t i o s o f t h e a r e a s o f t h e s q u a r e
a n d i t s i m a g e i n t h e u v p l a n e .
7 6 .
I f x = 2 ( u 2 - v 2 ) , y = u v v
d e t e r m i n e t h e i m a g e ( o r i m a g e s ) i n t h e u v p l a n e o f a s q u a r e b o u n d e d b y x = 0 ,
x = l , y = 0 . y = 1 i n t h e x y p l a n e .
7 7 . S h o w t h a t u n d e r s u i t a b l e c o n d i t i o n s o n F a n d G ,
f
0 0 0 "
0 0
f e - s t
0
0
t
F ( u ) G ( t - u ) d u
d t
- s ( x + y ) F ( x ) G ( y ) d x d y
H i n t : U s e t h e t r a n s f o r m a t i o n x + y = t , x = v f r o m t h e x y p l a n e t o t h e v t p l a n e . T h e r e s u l t i s i m p o r t a n t i n
t h e t h e o r y o f L a p l a c e t r a n s f o r m s .
7 8 .
( a ) I f x = 3 u 1 + u 2 - u 3 , y = u , + 2 u 2 + 2 u 3 , z = 2 u 1 - u 2 - u 3 ,
f i n d t h e v o l u m e s o f t h e c u b e b o u n d e d b y
x = 0 , x = 1 5 , y = 0 , y = 1 0 ,
= 0 a n d z = 5 , a n d t h e i m a g e o f t h i s c u b e i n t h e u 1 u 2 u 3 r e c t a n g u l a r c o o r -
d i n a t e s y s t e m .
( b ) R e l a t e t h e r a t i o o f t h e s e v o l u m e s t o t h e J a c o b i a n o f t h e t r a n s f o r m a t i o n .
7 9 . L e t ( x , y , z ) a n d ( u 1 , u 2 i u 3 ) b e r e s p e c t i v e l y t h e r e c t a n g u l a r a n d c u r v i l i n e a r c o o r d i n a t e s o f a p o i n t .
( a ) I f x = 3 u 1 + u 2 - - U 3 , y = u 1 + 2 u 2 + 2 u 3 ; z = 2 u 1 - u 2 - U 3 , i s t h e s y s t e m u 1 u 2 u 3 o r t h o g o n a l ?
( b ) F i n d d s 2 a n d g f o r t h e s y s t e m .
( c ) W h a t i s t h e r e l a t i o n b e t w e e n t h i s a n d t h e p r e c e d i n g p r o b l e m ?
2
2
+ +
`
b
b i
0
d t h
a ( x , y , z )
V i f
t h t 2 =
, y = u 1
( a
g a n
)
e J a c o
a n2 , z = u 3 - - - i t ,
f i n
I f x = u 1
A N S W E R S T O S U P P L E M E N T A R Y P R O B L E M S .
- 3 ( U - 1 ' U 2 , u s )
e r
y
a ] g .
3 6 .
( a ) u = c 1 a n d v = c 2 a r e e l l i p t i c a n d h y p e r b o l i c c y l i n d e r s r e s p e c t i v e l y , h a v i n g z a x i s a s c o m m o n a x i s .
z = c 3 a r e p l a n e s . S e e F i g . 7 , p a g e 1 3 9 .
( b ) i s = c 1 a n d v = c 2 a r e c i r c u l a r c y l i n d e r s w h o s e i n t e r s e c t i o n s w i t h t h e x y p l a n e a r e c i r c l e s w i t h c e n t e r s
o n t h e y a n d x a x e s r e s p e c t i v e l y a n d i n t e r s e c t i n g a t r i g h t a n g l e s . T h e c y l i n d e r s u = c 1 a l l p a s s
t h r o u g h t h e p o i n t s ( - a , 0 , 0 ) a n d ( a , 0 , 0 ) .
z = c 3 a r e p l a n e s .
S e e F i g . 8 , p a g e 1 4 0 .
( c ) i s = c 1 a n d v = c 2 a r e p a r a b o l i c c y l i n d e r s w h o s e t r a c e s o n t h e x y p l a n e a r e i n t e r s e c t i n g m u t u a l l y p e r -
p e n d i c u l a r c o a x i a l p a r a b o l a s w i t h v e r t i c e s o n t h e x a x i s b u t o n o p p o s i t e s i d e s o f t h e o r i g i n .
z = c 3
a r e p l a n e s . S e e F i g . 6 , p a g e 1 3 8 .
T h e c o o r d i n a t e c u r v e s a r e t h e i n t e r s e c t i o n s o f t h e c o o r d i n a t e s u r f a c e s .
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1 6 4
5 2 .
5 6 .
6 4 7 T ( 2 - V )
5 4 . ( a ) p , ( b ) r 2 s i n 0 ,
( c ) u 2 + v 2 ,
( d ) a 2 ( s i n h 2 u + s i n 2 v ) ,
( e ) a 3 ( s i n h 2 6 + s i n e ? ) ) s i n h
s i n 7 )
5 5 .
( a ) p d p d o ,
p d c / d z , d p d z
( b ) r s i n 0 d r d o , r 2 s i n 0 d 9 d o , r d r d O
( c )
( u 2 + v 2 ) d u d v , u v u 2 + v 2 d u d % ,
u v u 2 + v 2 d v d o
2 5 6 7 T
1 5
5 9 .
2 K
r c o s 9 c o s 0 i
+ r c o s 9 s i n 0 j - r s i n 9 k
6 0 . ( a )
r r
=
c o s c ) i + s i n 0 j .
P
a _ r
=
_ p s i n 0 i
+ p c o s c a j ,
a r
=
k .
v z = k
a z
5 8 .
( c )
z ; ( d ) u z = 2 P 2 , u 2 = 2 0
x i + y j
O p =
- c o s 0 i + s i n
j
x 2 + y 2
= - S i n e i + c o s 4 j
p
( b ) a r
=
s i n e c o s
i + s i n e s i n
j
+ c o s 9 k
a r
a 9
a r
a q
- r s i n 9 s i n ¢ i + r s i n e c o s 0 j
V r
=
x i + y j + z k
=
s i n e c o s 0 i
+ s i n a s i n j + c o s 9 k
x 2 + y 2 + z 2
V 9
=
x z i + y z j - ( x 2 + y 2 ) k
c o s e c o s
i + c o s 9 s i n 0 j - s i n 9 k
( x 2 + y 2 + z 2 ) x 2 + y 2
r
v
y i + x j
- s i n
i
+ c o s t j
x 2 + y 2
r s i n e
( c ) a u = u i + v i ,
a v = - v i + u j ,
a a r
= k
v u
u i + v j
C U R V I L I N E A R C O O R D I N A T E S
3
v v =
V z = k
5 7 .
3
( 2 a 3 - 3 a 2 h + h 3 )
- v i + u j
u 2 + v 2
, u 2 + v 2
,
6 1 . ( b )
y 2
1 x 2
,
( c ) d s 2
u
a u u 2 + v 2 a v
+
1
a
e
+
2 .
v 4 )
=
1
a
e
d i v A
c u r l A
u 2
( x 2 + y 2 ) ( d u i + d u e ) - 4 x y d u 1 d u 2
+ d u 2
=
( x 2 - y 2 ) 2
s
v
u 2 + v 2
A u )
1
a A z
-
u 2 + v 2
a v
a -
a e
z
z
A v l
+ a A z
/ J a Z
( 1 / 2 A ) }
u V
` + v 2 e u
a z
u 2 ( d u i + d u e ) - 2 u 1 d u 1 d u e
+ d u _
2
2 ( u 2 - u 1 )
+
a
(
u 2 + v 2 A
- a A z
u ,
+ v 2
v
- )
z
u
a u
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C h a p t e r 8
P H Y S I C A L L A W S m u s t b e i n d e p e n d e n t o f a n y p a r t i c u l a r c o o r d i n a t e s y s t e m s u s e d i n d e s c r i b i n g t h e m
m a t h e m a t i c a l l y , i f t h e y a r e t o b e v a l i d . A s t u d y o f t h e c o n s e q u e n c e s o f t h i s r e - L
q u i r e m e n t l e a d s t o t e n s o r a n a l y s i s , o f g r e a t u s e i n g e n e r a l r e l a t i v i t y t h e o r y , d i f f e r e n t i a l g e o m e t r y ,
m e c h a n i c s , e l a s t i c i t y , h y d r o d y n a m i c s , e l e c t r o m a g n e t i c t h e o r y a n d n u m e r o u s o t h e r f i e l d s o f s c i e n c e
a n d e n g i n e e r i n g .
S P A C E S O F N D I M E N S I O N S . I n t h r e e d i m e n s i o n a l s p a c e a p o i n t i s a s e t o f t h r e e n u m b e r s , c a l l e d
c o o r d i n a t e s , d e t e r m i n e d b y s p e c i f y i n g a p a r t i c u l a r c o o r d i n a t e s y s t e m
o r f r a m e o f r e f e r e n c e .
F o r e x a m p l e ( x , y , z ) , ( p , c , z ) , ( r , 8 , 5 5 )
a r e c o o r d i n a t e s o f a p o i n t i n r e c t a n -
g u l a r , c y l i n d r i c a l a n d s p h e r i c a l c o o r d i n a t e s y s t e m s r e s p e c t i v e l y . A p o i n t i n N d i m e n s i o n a l s p a c e i s ,
b y a n a l o g y , a s e t o f N n u m b e r s d e n o t e d b y ( x 1 , x 2 , . . . , x N ) w h e r e
1 , 2 , . . . , N a r e t a k e n n o t a s e x p o -
n e n t s b u t a s s u p e r s c r i p t s , a p o l i c y w h i c h w i l l p r o v e u s e f u l .
T h e f a c t t h a t w e c a n n o t v i s u a l i z e p o i n t s i n s p a c e s o f d i m e n s i o n h i g h e r t h a n t h r e e h a s o f c o u r s e
n o t h i n g w h a t s o e v e r t o d o w i t h t h e i r e x i s t e n c e .
C O O R D I N A T E T R A N S F O R M A T I O N S . L e t ( x 1 , x 2 , . . . , x N ) a n d ( x 1 , x 2 , . . . , R N ) b e c o o r d i n a t e s o f a p o i n t
i n t w o d i f f e r e n t f r a m e s o f r e f e r e n c e .
S u p p o s e t h e r e e x i s t s N
i n d e p e n d e n t r e l a t i o n s b e t w e e n t h e c o o r d i n a t e s o f t h e t w o s y s t e m s h a v i n g t h e f o r m
1 _
- X ' 1
2
2 2 1 2
( 1 )
w h i c h w e c a n i n d i c a t e b r i e f l y b y
( 2 )
x N
=
z N ( x 1 , x 2 ,
. . . ,
x N )
x k
=
x k ( x 1 , x 2 ,
. . . ,
x N ) k = 1 , 2 ,
. . . , N
w h e r e i t i s s u p p o s e d t h a t t h e f u n c t i o n s i n v o l v e d a r e s i n g l e - v a l u e d , c o n t i n u o u s , a n d h a v e c o n t i n u o u s
d e r i v a t i v e s .
T h e n c o n v e r s e l y t o e a c h s e t o f c o o r d i n a t e s ( x 1 , x 2 , . . . , x N ) t h e r e w i l l c o r r e s p o n d a
u n i q u e s e t ( x 1 , x 2 ,
. . . ,
x N ) g i v e n b y
k
= 1
2
N
( 3 )
X
k
x , x , . . . , x ) k = 1 , 2 , . . . , N
T h e r e l a t i o n s ( 2 ) o r ( 3 ) d e f i n e 4 t r a n s f o r m a t i o n o f c o o r d i n a t e s f r o m o n e f r a m e o f r e f e r e n c e t o a n o t h e r .
1 6 6
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T E N S O R A N A L Y S I S
1 6 7
T H E S U M M A T I O N C O N V E N T I O N . I n w r i t i n g a n e x p r e s s i o n s u c h a s a 1 x 1 + a 2 x 2 + . . . + a 1 y x 1 w e c a n
X
u s e t h e s h o r t n o t a t i o n j Z 1
x i . A n e v e n s h o r t e r n o t a t i o n i s s i m -
p l y t o w r i t e i t a s a j x i , w h e r e w e a d o p t t h e c o n v e n t i o n t h a t w h e n e v e r a n i n d e x ( s u b s c r i p t o r s u p e r -
s c r i p t ) i s r e p e a t e d i n a g i v e n t e r m w e a r e t o s u m o v e r t h a t i n d e x f r o m 1 t o N u n l e s s o t h e r w i s e s p e c -
i f i e d . T h i s i s c a l l e d t h e s u m m a t i o n c o n v e n t i o n . C l e a r l y , i n s t e a d o f u s i n g t h e i n d e x j w e c o u l d h a v e
u s e d a n o t h e r l e t t e r , s a y p , a n d t h e s u m c o u l d b e w r i t t e n a o x O . A n y i n d e x w h i c h i s r e p e a t e d i n a g i v -
e n t e r m , s o t h a t t h e s u m m a t i o n c o n v e n t i o n a p p l i e s , i s c a l l e d a d u m m y i n d e x o r u m b r a l i n d e x .
A n i n d e x o c c u r r i n g o n l y o n c e i n a g i v e n t e r m i s c a l l e d a f r e e i n d e x a n d c a n s t a n d f o r a n y o f t h e
n u m b e r s 1 , 2 , . . . , N s u c h a s k i n e q u a t i o n ( 2 ) o r ( 3 ) , e a c h o f w h i c h r e p r e s e n t s N e q u a t i o n s .
C O N T R A V A R I A N T A N D C O V A R I A N T V E C T O R S . I f N q u a n t i t i e s A 1 , A 2 , . . . ,
A N i n
a c o o r d i n a t e s y s -
t e m ( x 1 , x 2 , . . . , x 1 ) a r e r e l a t e d t o N o t h e r q u a n t i t i e s
A 1 , A 2 ,
. . . ,
f f
i n a n o t h e r c o o r d i n a t e s y s t e m ( x 1 , x 2 , . . . , x N ) b y t h e t r a n s f o r m a t i o n e q u a t i o n s
A
_
a x 9 A q
p =
1 , 2 , . . . , N
q = 1
w h i c h b y t h e c o n v e n t i o n s a d o p t e d c a n s i m p l y b e w r i t t e n a s
A =
a x P A q
a x q
t h e y a r e c a l l e d c o m p o n e n t s o f a c o n t r a v a r i a n t v e c t o r o r c o n t r a v a r i a n t t e n s o r o f t h e f i r s t r a n k o r f i r s t
o r d e r . T o p r o v i d e m o t i v a t i o n f o r t h i s a n d l a t e r t r a n s f o r m a t i o n s , s e e P r o b l e m s 3 3 a n d 3 4 o f C h a p t e r 7 .
I f N q u a n t i t i e s
A 1 i A 2 ,
. . . ,
A N
i n a c o o r d i n a t e s y s t e m ( x 1 , 1 2 , . . . , x 1 ) a r e r e l a t e d t o N o t h e r
q u a n t i t i e s A t , A 2 , . . . , A f f i n a n o t h e r c o o r d i n a t e s y s t e m ( x 1 , x 2 , . . . , x N ) b y t h e t r a n s f o r m a t i o n e q u a t i o n s
A p
=
a x p A q
p =
1 , 2 , . . . , N
q = 1
o r
A P
a x q A
a z p
q
t h e y a r e c a l l e d c o m p o n e n t s o f a c o v a r i a n t v e c t o r o r c o v a r i a n t t e n s o r o f t h e f i r s t r a n k o r f i r s t o r d e r .
N o t e t h a t a s u p e r s c r i p t i s u s e d t o i n d i c a t e c o n t r a v a r i a n t c o m p o n e n t s w h e r e a s a s u b s c r i p t i s
u s e d t o i n d i c a t e c o v a r i a n t c o m p o n e n t s ; a n e x c e p t i o n o c c u r s i n t h e n o t a t i o n f o r c o o r d i n a t e s .
I n s t e a d o f s p e a k i n g o f a t e n s o r w h o s e c o m p o n e n t s a r e A p o r A P w e s h a l l o f t e n r e f e r s i m p l y t o
t h e t e n s o r A P o r A P A . N o c o n f u s i o n s h o u l d a r i s e f r o m t h i s .
C O N T R A V A R I A N T , C O V A R I A N T A N D M I X E D T E N S O R S . I f N 2 q u a n t i t i e s A q s i n a c o o r d i n a t e s y s t e m
_ ( x 1 , x 2 , . . . , x 1 ) a r e r e l a t e d t o N 2 o t h e r q u a n -
t i t i e s A
i n a n o t h e r c o o r d i n a t e s y s t e m ( x 1 , x 2 , . . . , x " ) b y t h e t r a n s f o r m a t i o n e q u a t i o n s
A i r
a x 9 a x s A q s
p , r = 1 , 2 , . . . , N
S = 1 q = 1
x
x
o r
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T E N S O R A N A L Y S I S
a x q
a x r A g s
a x q a x s
b y t h e a d o p t e d c o n v e n t i o n s , t h e y a r e c a l l e d c o n t r a v a r i a n t c o m p o n e n t s o f a t e n s o r o f t h e s e c o n d r a n k
o r r a n k t w o .
T h e N 2 q u a n t i t i e s A q s a r e c a l l e d c o v a r i a n t c o m p o n e n t s o f a t e n s o r o f t h e s e c o n d r a n k i f
A P r
a x q a x s
A
a x p - a y r
q s
S i m i l a r l y t h e N 2 q u a n t i t i e s A S a r e c a l l e d c o m p o n e n t s o f a m i x e d t e n s o r o f t h e s e c o n d r a n k i f
A P
=
a x P a x s
A
q
a x q o x r
s
T H E K R O N E C K E R D E L T A , w r i t t e n 8 k , i s d e f i n e d b y
S k
J 0
i f j A k
1
i f j = k
A s i t s n o t a t i o n i n d i c a t e s , i t i s a m i x e d t e n s o r o f t h e s e c o n d r a n k .
T E N S O R S O F R A N K G R E A T E R T H A N T W O a r e e a s i l y d e f i n e d .
F o r e x a m p l e , A k i t
a r e t h e c o m p o -
n e n t s o f a m i x e d t e n s o r o f r a n k 5 , c o n t r a v a r i a n t o f o r d e r
3 a n d c o v a r i a n t o f o r d e r 2 , i f t h e y t r a n s f o r m a c c o r d i n g t o t h e r e l a t i o n s
A f i r m
=
a x p a x r a z m a x k a x 1 .
A g s t
t i
a x q a x s a x t a x i a x 9
k i
S C A L A R S O R I N V A R I A N T S . S u p p o s e 0 i s a f u n c t i o n o f t h e c o o r d i n a t e s x k , a n d l e t
d e n o t e t h e
f u n c t i o n a l v a l u e u n d e r a t r a n s f o r m a t i o n t o a n e w s e t o f c o o r d i n a t e s x k
T h e n c b i s c a l l e d a s c a l a r o r i n v a r i a n t w i t h r e s p e c t t o t h e c o o r d i n a t e t r a n s f o r m a t i o n i f
_ . A
s c a l a r o r i n v a r i a n t i s a l s o c a l l e d a t e n s o r o f r a n k z e r o .
T E N S O R F I E L D S . I f t o e a c h p o i n t o f a r e g i o n i n N d i m e n s i o n a l s p a c e t h e r e c o r r e s p o n d s a d e f i n i t e
t e n s o r , w e s a y t h a t a t e n s o r f i e l d h a s b e e n d e f i n e d . T h i s i s a v e c t o r f i e l d o r
a s c a l a r f i e l d a c c o r d i n g a s t h e t e n s o r i s o f r a n k o n e o r z e r o .
I t s h o u l d b e n o t e d t h a t a t e n s o r o r
t e n s o r f i e l d i s n o t j u s t t h e s e t o f i t s c o m p o n e n t s i n o n e s p e c i a l c o o r d i n a t e s y s t e m b u t a l l t h e p o s s i -
b l e s e t s u n d e r a n y t r a n s f o r m a t i o n o f c o o r d i n a t e s .
S Y M M E T R I C A N D S K E W - S Y M M E T R I C T E N S O R S . A t e n s o r i s c a l l e d s y m m e t r i c w i t h r e s p e c t t o t w o
c o n t r a v a r i a n t o r t w o c o v a r i a n t i n d i c e s i f i t s c o m -
p o n e n t s r e m a i n u n a l t e r e d u p o n i n t e r c h a n g e o f t h e i n d i c e s . T h u s i f A q s r = A Q S r t h e t e n s o r i s s y m -
m e t r i c i n m a n d p . I f a t e n s o r i s s y m m e t r i c w i t h r e s p e c t t o a n y t w o c o n t r a v a r i a n t a n d a n y t w o c o -
v a r i a n t i n d i c e s , i t i s c a l l e d s y m m e t r i c .
A t e n s o r i s c a l l e d s k e w - s y m m e t r i c w i t h r e s p e c t t o t w o c o n t r a v a r i a n t o r t w o c o v a r i a n t i n d i c e s
i f i t s c o m p o n e n t s c h a n g e s i g n u p o n i n t e r c h a n g e o f t h e i n d i c e s . T h u s i f A q s r = - A q s r t h e t e n s o r i s
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1 6 9
s k e w - s y m m e t r i c i n m a n d p .
I f a t e n s o r i s s k e w - s y m m e t r i c w i t h r e s p e c t t o a n y t w o c o n t r a v a r i a n t a n d
a n y t w o c o v a r i a n t i n d i c e s i t i s c a l l e d s k e w - s y m m e t r i c .
F U N D A M E N T A L O P E R A T I O N S W I T H T E N S O R S .
1 . A d d i t i o n . T h e s u m o f t w o o r m o r e t e n s o r s o f t h e s a m e r a n k a n d t y p e ( i . e . s a m e n u m b e r o f c o n t r a -
v a r i a n t i n d i c e s a n d s a m e n u m b e r o f c o v a r i a n t i n d i c e s ) i s a l s o a t e n s o r o f t h e s a m e r a n k a n d t y p e .
T h u s i f A Q 0 a n d B q 0 a r e t e n s o r s , t h e n C O = A q 0 +
B r i s a l s o a t e n s o r .
A d d i t i o n o f t e n s o r s
i s c o m m u t a t i v e a n d a s s o c i a t i v e .
2 . S u b t r a c t i o n . T h e d i f f e r e n c e o f t w o t e n s o r s o f t h e s a m e r a n k a n d t y p e i s a l s o a t e n s o r o f t h e s a m e
r a n k a n d t y p e . T h u s i f A q 0 a n d B r a r e t e n s o r s , t h e n
D r = A q
O - B q 0 i s a l s o a t e n s o r .
3 . O u t e r M u l t i p l i c a t i o n . T h e p r o d u c t o f t w o t e n s o r s i s a t e n s o r w h o s e r a n k i s t h e s u m o f t h e r a n k s
o f t h e g i v e n t e n s o r s . T h i s p r o d u c t w h i c h i n v o l v e s o r d i n a r y m u l t i p l i c a t i o n o f t h e c o m p o n e n t s o f
t h e t e n s o r i s c a l l e d t h e o u t e r p r o d u c t . F o r e x a m p l e ,
A q r B S
= C q s ' i s t h e o u t e r p r o d u c t o f
A l i r
a n d B S .
H o w e v e r , n o t e t h a t n o t e v e r y t e n s o r c a n b e w r i t t e n a s a p r o d u c t o f t w o t e n s o r s o f l o w e r
r a n k . F o r t h i s r e a s o n d i v i s i o n o f t e n s o r s i s n o t a l w a y s p o s s i b l e .
4 . C o n t r a c t i o n . I f o n e c o n t r a v a r i a n t a n d o n e c o v a r i a n t i n d e x o f a t e n s o r a r e s e t e q u a l , t h e r e s u l t i n -
d i c a t e s t h a t a s u m m a t i o n o v e r t h e e q u a l i n d i c e s i s t o b e t a k e n a c c o r d i n g t o t h e s u m m a t i o n c o n -
v e n t i o n .
T h i s r e s u l t i n g s u m i s a t e n s o r o f r a n k t w o l e s s t h a n t h a t o f t h e o r i g i n a l t e n s o r . T h e
p r o c e s s i s c a l l e d c o n t r a c t i o n .
F o r e x a m p l e , i n t h e t e n s o r o f r a n k 5 , A g P r , s e t r = s t o o b t a i n
A g r r
= B q
"
a t e n s o r o f r a n k 3 .
F u r t h e r , b y s e t t i n g p = q w e o b t a i n 8 0
= C 2 a t e n s o r o f r a n k 1 .
5 . I n n e r M u l t i p l i c a t i o n . B y t h e p r o c e s s o f o u t e r m u l t i p l i c a t i o n o f t w o t e n s o r s f o l l o w e d b y a c o n t r a c -
t i o n , w e o b t a i n a n e w t e n s o r c a l l e d a n i n n e r p r o d u c t o f t h e g i v e n t e n s o r s . T h e p r o c e s s i s c a l l e d
i n n e r m u l t i p l i c a t i o n . F o r e x a m p l e , g i v e n t h e t e n s o r s A ' O a n d B s t , t h e o u t e r p r o d u c t i s A q 1 B r
s t *
L e t t i n g q = r , w e o b t a i n t h e i n n e r p r o d u c t A r k B
.
L e t t i n g q = r a n d p = s , a n o t h e r i n n e r p r o d u c t
A r 1 ' B r i s o b t a i n e d . I n n e r a n d o u t e r m u l t i p l i c a t i o n o f t e n s o r s i s c o m m u t a t i v e a n d
a s s o c i a t i v e .
6 . Q u o t i e n t L a w . S u p p o s e i t i s n o t k n o w n w h e t h e r a q u a n t i t y X i s a t e n s o r o r n o t .
I f a n i n n e r p r o d -
u c t o f X w i t h a n a r b i t r a r y t e n s o r i s i t s e l f a t e n s o r , t h e n X i s a l s o a t e n s o r . T h i s i s c a l l e d t h e
q u o t i e n t l a w .
M A T R I C E S . A m a t r i x o f o r d e r m b y n i s a n a r r a y o f q u a n t i t i e s a p q , c a l l e d e l e m e n t s , a r r a n g e d i n m
r o w s a n d n c o l u m n s a n d g e n e r a l l y d e n o t e d b y
a l l a 1 2
. . .
a l n
a l l a 1 2
. . .
a 2 1 a 2 2 . . .
a 2 n a n a 2 2
o r
a . 4 1 a n n
a , n s a n i 2 . . .
a i n n
a s n
a 2 n
o r i n a b b r e v i a t e d f o r m b y ( a 1 , q ) o r [ a p q ]
p = 1 , . . . , m ; q = 1 , . . . , n .
I f m = n t h e m a t r i x i s a s q u a r e
m a t r i x o f o r d e r m b y m o r s i m p l y m ; i f m = 1
i t i s a r o w m a t r i x o r r o w v e c t o r ; i f n = 1 i t i s a c o l u m n
m a t r i x o r c o l u m n v e c t o r .
T h e d i a g o n a l o f a s q u a r e m a t r i x c o n t a i n i n g t h e e l e m e n t s a s s , a t e , . . . , a n n i s c a l l e d t h e p r i n c i -
p a l o r m a i n d i a
o g T 1 . A s q u a r e m a t r i x w h o s e e l e m e n t s a r e e q u a l t o o n e i n t h e p r i n c i p a l d i a g o n a l a n d
z e r o e l s e h i s c a l l e d a u n i t m a t r i x a n d i s d e n o t e d b y 1 . A n u l l m a t r i x , d e n o t e d b y 0 , i s a m a t r i x
a l l o f w h o s e e l e m e n t s a r e z e r o .
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T E N S O R A N A L Y S I S
M A T R I X A L G E B R A . I f A = ( a p q ) a n d B = ( b p q ) a r e m a t r i c e s h a v i n g t h e s a m e o r d e r ( m b y n ) t h e n
1 . A = B i f a n d o n l y i f a p q = b 1 , q .
2 . T h e s u m S a n d d i f f e r e n c e D a r e t h e m a t r i c e s d e f i n e d b y
S = A + B = ( a j , q + b p q ) ,
D = A - B = ( a p q - b p q )
3 . T h e p r o d u c t P = A B i s d e f i n e d o n l y w h e n t h e n u m b e r n o f c o l u m n s i n A e q u a l s t h e n u m b e r o f r o w s
i n B a n d i s t h e n g i v e n b y
P = A B = ( a p q ) ( b p q ) = ( a p r b r q )
n
w h e r e
a l , r b r q =
a p r b r q b y t h e s u m m a t i o n c o n v e n t i o n .
M a t r i c e s w h o s e p r o d u c t i s d e f i n e d
r . 1
a r e c a l l e d c o n f o r m a b l e .
I n g e n e r a l , m u l t i p l i c a t i o n o f m a t r i c e s i s n o t c o m m u t a t i v e , i . e . A B A B A . H o w e v e r t h e a s s o -
c i a t i v e l a w f o r m u l t i p l i c a t i o n o f m a t r i c e s h o l d s , i . e . -
C ) _ ( A B ) C p r o v i d e d t h e m a t r i c e s a r e
c o n f o r m a b l e .
A l s o t h e d i s t r i b u t i v e l a w s h o l d , i . e . A ( B + C ) = A B + A C , ( A + B ) C = A C + B C .
4 . T h e d e t e r m i n a n t o f a s q u a r e m a t r i x A = ( a , q ) i s d e n o t e d b y
I A I , d e t A ,
I
I
o r d e t ( a j q ) .
I f P = A B t h e n
I P I = I A I B .
5 . T h e i n v e r s e o f a s q u a r e m a t r i x A i s a m a t r i x
A - 1 s u c h t h a t A A - 1
= 1 , w h e r e I i s t h e u n i t m a t r i x .
A n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n t h a t A - 1 e x i s t i s t h a t d e t A
0 .
I f d e t A = 0 , A i s c a l l e d
s i n g u l a r .
6 . T h e p r o d u c t o f a s c a l a r ? . b y a m a t r i x A
d e n o t e d b y X A , i s t h e m a t r i x ( X a p q ) w h e r e e a c h
e l e m e n t o f A i s m u l t i p l i e d b y X .
7 . T h e t r a n s p o s e o f a m a t r i x A i s a m a t r i x A T w h i c h i s f o r m e d f r o m A b y i n t e r c h a n g i n g i t s r o w s a n d
c o l u m n s . T h u s i f A = ( a p q ) , t h e n A T = ( a q p ) . T h e t r a n s p o s e o f A i s a l s o d e n o t e d b y A .
T H E L I N E E L E M E N T A N D M E T R I C T E N S O R . I n r e c t a n g u l a r c o o r d i n a t e s ( x , y , z ) t h e d i f f e r e n t i a l
a r e l e n g t h d s i s o b t a i n e d f r o m
B y t r a n s f o r m i n g t o g e n e r a l c u r v i l i n e a r c o o r d i n a t e s ( s e e P r o b l e m 1 7 , C h a p t e r 7 ) t h i s b e c o m e s d s
3
3
E I g o q d u p d u q .
S u c h s p a c e s a r e c a l l e d t h r e e d i m e n s i o n a l E u c l i d e a n s p a c e s .
P = 1 q = 1
A g e n e r a l i z a t i o n t o N d i m e n s i o n a l s p a c e w i t h c o o r d i n a t e s ( x 1 , x 2 , . . . , x N ) i s i m m e d i a t e . W e d e -
f i n e t h e l i n e e l e m e n t d s i n t h i s s p a c e t o b e g i v e n b y t h e q u a d r a t i c f o r m , c a l l e d t h e m e t r i c f o r m o r
m e t r i c ,
I n t h e s p e c i a l c a s e w h e r e t h e r e e x i s t s a t r a n s f o r m a t i o n o f c o o r d i n a t e s f r o m x I t o
x k
s u c h t h a t
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T E N S O R A N A L Y S I S
1 7 1
t h e m e t r i c f o r m i s t r a n s f o r m e d i n t o ( d z 1 ) 2 + ( d x 2 ) 2 + . . . + ( d x N ) 2 o r d x k d x k ,
t h e n t h e s p a c e i s c a l l -
e d N d i m e n s i o n a l E u c l i d e a n s p a c e .
I n t h e g e n e r a l c a s e , h o w e v e r , t h e s p a c e i s c a l l e d R i e m a n n i a n .
T h e q u a n t i t i e s g p q a r e t h e c o m p o n e n t s o f a c o v a r i a n t t e n s o r o f r a n k t w o c a l l e d t h e m e t r i c
t e n s o r o r f u n d a m e n t a l t e n s o r .
W e c a n a n d a l w a y s w i l l c h o o s e t h i s t e n s o r t o b e s y m m e t r i c ( s e e P r o b -
l e m 2 9 ) .
C O N J U G A T E O R R E C I P R O C A L T E N S O R S . L e t g =
g q
d e n o t e t h e d e t e r m i n a n t w i t h e l e m e n t s
p qa n d s u p p
e g A 0 . D e f i n e g b y -
g p q
p q
c o f a c t o r o f g p q
g
T h e n g p q i s a s y m m e t r i c c o n t r a v a r i a n t t e n s o r o f r a n k t w o c a l l e d t h e c o n j u g a t e o r r e c i p r o c a l t e n s o r
o f g p q ( s e e P r o b l e m 3 4 ) .
I t c a n b e s h o w n ( P r o b l e m 3 3 ) t h a t
g p q g r q
s p
r
A S S O C I A T E D T E N S O R S . G i v e n a t e n s o r , w e c a n d e r i v e o t h e r t e n s o r s b y r a i s i n g o r l o w e r i n g i n d i c e s .
F o r e x a m p l e , g i v e n t h e t e n s o r A p q w e o b t a i n b y r a i s i n g t h e i n d e x p , t h e ,
t e n s o r A
. q ,
t h e d o t i n d i c a t i n g t h e o r i g i n a l p o s i t i o n o f t h e m o v e d i n d e x . B y r a i s i n g t h e i n d e x q a l s o
w e o b t a i n . 4 ' ?
. W h e r e n o c o n f u s i o n c a n a r i s e w e s h a l l o f t e n o m i t t h e d o t s ; t h u s A p q c a n b e w r i t t e n
A p q .
T h e s e d e r i v e d t e n s o r s c a n b e o b t a i n e d b y f o r m i n g i n n e r p r o d u c t s o f t h e g i v e n t e n s o r w i t h t h e
m e t r i c t e n s o r g p q o r i t s c o n j u g a t e g p q
.
T h u s , f o r e x a m p l e
p
r p
A . q = g
A r q ,
A p q
=
g r p g s q A r s
A r s = g r q A - p s
A
q % n t k
_
g p k g
g r m
A
T h e s e b e c o m e c l e a r i f w e i n t e r p r e t m u l t i p l i c a t i o n b y g r p a s m e a n i n g : l e t r = p ( o r p = r ) i n w h a t e v e r
f o l l o w s a n d r a i s e t h i s i n d e x .
S i m i l a r l y w e i n t e r p r e t m u l t i p l i c a t i o n b y g r q a s m e a n i n g : l e t r = q ( o r
q = r ) i n w h a t e v e r f o l l o w s a n d l o w e r t h i s i n d e x .
A l l t e n s o r s o b t a i n e d f r o m a g i v e n t e n s o r b y f o r m i n g i n n e r p r o d u c t s w i t h t h e m e t r i c t e n s o r a n d
i t s c o n j u g a t e a r e c a l l e d a s s o c i a t e d t e n s o r s o f t h e g i v e n t e n s o r . F o r e x a m p l e A ' 4 a n d A . a r e a s s o -
c i a t e d t e n s o r s , t h e f i r s t a r e c o n t r a v a r i a n t a n d t h e s e c o n d c o v a r i a n t c o m p o n e n t s . T h e r e l a t i o n b e -
t w e e n t h e m i s g i v e n b y
A P
= g p q A q
o r A P = g p q A q
F o r r e c t a n g u l a r c o o r d i n a t e s g p q = 1
i f p = q ,
a n d 0 i f p A q ,
s o t h a t A p = A p , w h i c h e x p l a i n s w h y
n o d i s t i n c t i o n w a s m a d e b e t w e e n c o n t r a v a r i a n t a n d c o v a r i a n t c o m p o n e n t s o f a v e c t o r i n e a r l i e r c h a p -
t e r s .
L E N G T H O F A V E C T O R , A N G L E B E T W E E N V E C T O R S . T h e q u a n t i t y A P B P ,
w h i c h i s t h e i n n e r
p r o d u c t o f A P a n d B q
,
i s a s c a l a r a n a l -
o g o u s t o t h e s c a l a r p r o d u c t i n r e c t a n g u l a r c o o r d i n a t e s . W e d e f i n e t h e l e n g t h L o f t h e v e c t o r A O o r
A P a s g i v e n b y
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1 7 2
T E N S O R A N A L Y S I S
L 2
=
A P A P
= g 1 g A 1 ' A q
=
g 1 g A P A q
W e c a n d e f i n e t h e a n g l e 6 b e t w e e n
A P a n d B 1 '
a s g i v e n b y
A 1 B 1
c o s 6
=
( A 1 A 1 ' ) ( B 1 B 1 ' )
T H E P H Y S I C A L C O M P O N E N T S o f a v e c t o r
A 1 ' o r A 1 '
, d e n o t e d b y A u , A V , a n d A . a r e t h e p r o j e c -
t i o n s o f t h e v e c t o r o n t h e t a n g e n t s t o t h e c o o r d i n a t e c u r v e s a n d a r e
g i v e n i n t h e c a s e o f o r t h o g o n a l c o o r d i n a t e s b y
A u = v
A l
=
A l
V _ 9 _ 1
1
A v = 2 2 A 2 =
A w =
, s A s =
A
9 2 2
9 3 3
S i m i l a r l y t h e p h y s i c a l c o m p o n e n t s o f a t e n s o r
A
A
g
A
1 2
C H R I S T O F F E L ' S S Y M B O L S . T h e s y m b o l s
A 1 2
V ' 9 1 1 9 2 2
1 3
A 1 3
A u w = g 1 g A =
e t c .
- 1 1 - 9 3 31 9
a r e c a l l e d t h e C h r i s t o f f e l s y m b o l s o f t h e f i r s t a n d s e c o n d k i n d r e s p e c t i v e l y . O t h e r s y m b o l s u s e d i n -
s t e a d o
i s n o t t r u t - t f e n e r a l .
a n d 1 q .
T h e l a t t e r s y m b o l s u g g e s t s h o w e v e r a t e n s o r c h a r a c t e r , w h i c h
T R A N S F O R M A T I O N L A W S O F C H R I S T O F F E L ' S S Y M B O L S . I f w e d e n o t e b y a b a r a s y m b o l i n a c o -
o r d i n a t e s y s t e m x k , t h e n
[ j k m
[ p q , r ]
a 0 a x q a x r
+
g
a x 1 '
a 2 x q
a x k a x k a : x '
p q
a x ' s a x j a x k
n s
a x n a x 1 a x q
a z n
a 2 x q
1 k p q
a x s a x q a x k
a x q a x 3 a z k
a r e t h e l a w s o f t r a n s f o r m a t i o n o f t h e C h r i s t o f f e l s y m b o l s s h o w i n g t h a t t h e y a r e n o t t e n s o r s u n l e s s
t h e s e c o n d t e r m s o n t h e r i g h t a r e z e r o .
G E O D E S I C S . T h e d i s t a n c e s b e t w e e n t w o p o i n t s t 1 a n d t 2 o n a c u r v e x r = x ' ^ ( t )
i n a R i e m a n n i a n
s p a c e i s g i v e n b y
s =
J ; 1 t 2 / ; p q
2 d x
d t
t a t
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T E N S O R A N A L Y S I S
1 7 3
T h a t c u r v e i n t h e s p a c e w h i c h m a k e s t h e d i s t a n c e a m i n i m u m i s c a l l e d a g e o d e s i c o f t h e s p a c e . B y
u s e o f t h e c a l c u l u s o f v a r i a t i o n s ( s e e P r o b l e m s 5 0 a n d 5 1 ) t h e g e o d e s i c s a r e f o u n d f r o m t h e d i f f e r e n -
t i a l e q u a t i o n
d 2 x r
+
r
d x p d x q
=
d s 2
p q
d s
d s
0
w h e r e s i s t h e a r e l e n g t h p a r a m e t e r . A s e x a m p l e s , t h e g e o d e s i c s o n a p l a n e a r e s t r a i g h t l i n e s w h e r e -
a s t h e g e o d e s i c s o n a s p h e r e a r e a r c s o f g r e a t c i r c l e s .
T H E C O V A R I A N T D E R I V A T I V E o f a t e n s o r A p w i t h r e p e c t t o x q i s d e n o t e d b y A p , q a n d i s d e -
f i n e d b y
_
a A p
s
A p , q
a x q
_
p q
A s
a c o v a r i a n t t e n s o r o f r a n k t w o .
T h e c o v a r i a n t d e r i v a t i v e o f a t e n s o r A p w i t h r e s p e c t t o x 9 i s d e n o t e d b y A p q a n d i s d e f i n e d b y
A p
-
a A p
q
a x q
a m i x e d t e n s o r o f r a n k t w o .
1 P 1 ) , A S
q s
F o r r e c t a n g u l a r s y s t e m s , t h e C h r i s t o f f e l s y m b o l s a r e z e r o a n d t h e c o v a r i a n t d e r i v a t i v e s a r e t h e
u s u a l p a r t i a l d e r i v a t i v e s . C o v a r i a n t d e r i v a t i v e s o f t e n s o r s a r e a l s o t e n s o r s ( s e e P r o b l e m 5 2 ) .
T h e a b o v e r e s u l t s c a n b e e x t e n d e d t o c o v a r i a n t d e r i v a t i v e s o f h i g h e r r a n k t e n s o r s . T h u s
A P i . . . p n
_
a A P l
n C
i -
r
i . . . n , q
a x q
s
1 , 4 S r 2 . . . r n
-
r
S
A p .
s
r a p
r n
2
i q
q
s p .
. P l n
+ { P i A r 2 r
n
+ P 2
A p l s p 3 . . . P i n
q s
i . . . r n
p i . . . p i n
q
i s t h e c o v a r i a n t d e r i v a t i v e o f A r i
r n
w i t h r e s p e c t t o x
A p i . . . p X
r i . . . r n _ i s
+ . . .
+
P i n
A p i . .
p i n -
i
s
q s
1 . . . r n
T h e r u l e s o f c o v a r i a n t d i f f e r e n t i a t i o n f o r s u m s a n d p r o d u c t s o f t e n s o r s a r e t h e s a m e a s t h o s e
f o r o r d i n a r y d i f f e r e n t i a t i o n .
I n p e r f o r m i n g t h e d i f f e r e n t i a t i o n s , t h e t e n s o r s g p q , g p q a n d 8 0 m a y b e
t r e a t e d a s c o n s t a n t s s i n c e t h e i r c o v a r i a n t d e r i v a t i v e s a r e z e r o ( s e e P r o b l e m 5 4 ) .
S i n c e c o v a r i a n t
d e r i v a t i v e s e x p r e s s r a t e s o f c h a n g e o f p h y s i c a l q u a n t i t i e s i n d e p e n d e n t o f a n y f r a m e s o f r e f e r e n c e ,
t h e y a r e o f g r e a t i m p o r t a n c e i n e x p r e s s i n g p h y s i c a l l a w s .
P E R M U T A T I O N S Y M B O L S A N D T E N S O R S . D e f i n e a p q r b y t h e r e l a t i o n s
e 1 2 3 = e m 1 = e 3 1 2 = + 1 ,
e 2 1 3 = e 1 3 2 - = e 3 2 1 = - 1 ,
e p q r = 0
i f t w o o r m o r e i n d i c e s a r e e q u a l
a n d d e f i n e
e p g r
i n t h e s a m e m a n n e r . T h e s y m b o l s e p g r a n d
e p g r
a r e c a l l e d p e r m u t a t i o n s y m b o l s i n
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1 7 4
T E N S O R A N A L Y S I S
t h r e e d i m e n s i o n a l s p a c e .
F u r t h e r , l e t u s d e f i n e
I t c a n b e s h o w n t h a t E p q r a n d E p g r a r e c o v a r i a n t a n d c o n t r a v a r i a n t t e n s o r s r e s p e c t i v e l y , c a l l e d
p e r m u t a t i o n t e n s o r s i n t h r e e d i m e n s i o n a l s p a c e . G e n e r a l i z a t i o n s t o h i g h e r d i m e n s i o n s a r e p o s s i b l e .
T E N S O R F O R M O F G R A D I E N T , D I V E R G E N C E A N D C U R L .
1 . G r a d i e n t . I f < J ) i s a s c a l a r o r i n v a r i a n t t h e g r a d i e n t o f c i s d e f i n e d b y
g r a d c D
=
a C D
'
-
a x p
w h e r e < D ,
p
i s t h e c o v a r i a n t d e r i v a t i v e o f
w i t h r e s p e c t t o x p .
2 .
D i v e r g e n c e . T h e d i v e r g e n c e o f A P i s t h e c o n t r a c t i o n o f i t s c o v a r i a n t d e r i v a t i v e w i t h r e s p e c t t o
x g , i . e . t h e c o n t r a c t i o n o f A 1 , q .
T h e n
d i v A p =
A p , p
=
a k ( g A k )
g
1
h
1 f A A A -
C T
a A p a A g
u r .
e c u r o
p
i s
p , q -
q , p
a x q
a x p
d e f i n e d a s - - E p g r A p , q .
r
p q
4 . L a p l a c i a n . T h e L a p l a c i a n o f
i s t h e d i v e r g e n c e o f g r a d c P o r
v 2
d i v 4 > , p
=
1
_
( V k
a x
a x
I n c a s e g < 0 ,
m u s t b e r e p l a c e d b y = g .
B o t h c a s e s g > 0 a n d g < 0 c a n b e i n c l u d e d b y
w r i t i n g g i n p l a c e o f V V .
8 A p
T H E I N T R I N S I C O R A B S O L U T E D E R I V A T I V E o f A p a l o n g a c u r v e x q = x q ( t ) , d e n o t e d b y
S t
, i s
d e f i n e d a s t h e i n n e r p r o d u c t o f t h e c o v a r i a n t d e r i v a -
q q
t i v e o f A P a n d d t
,
i . e .
A P , q d t
a n d i s g i v e n b y
S A P d A p
s t d t
S i m i l a r l y , w e d e f i n e
8 A p
_
d A p
b t
d t
a t e n s o r o f r a n k t w o . T h e c u r l i s a l s o
d x g
A r
a t
p A r d x q
q r
d t
T h e v e c t o r s A P o r A p a r e s a i d t o m o v e p a r a l l e l l y a l o n g a c u r v e
a l o n g t h e c u r v e a r e z e r o , r e s p e c t i v e l y .
i f t h e i r i n t r i n s i c d e r i v a t i v e s
I n t r i n s i c d e r i v a t i v e s o f h i g h e r r a n k t e n s o r s a r e s i m i l a r l y d e f i n e d .
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T E N S O R A N A L Y S I S
1 7 5
R E L A T I V E A N D A B S O L U T E T E N S O R S . A t e n s o r
A p 1 . . . p ' n
i s c a l l e d a r e l a t i v e t e n s o r o f w e i g h t w
r 1 . . . r n
i f i t s c o m p o n e n t s t r a n s f o r m a c c o r d i n g t o t h e e q u a t i o n
A g 1 . . . g l
_
a x w
A p 1 . . , p m
a z g 1
a z q i
i
a x r n
. . .
p i x
a x s 1
. . .
a x
s n
1 . . . s n
a x
r 1 . . . r n
a x p 1
a x
w h e r e J =
2 z I
i s t h e J a c o b i a n o f t h e t r a n s f o r m a t i o n . I f w = 0 t h e t e n s o r i s c a l l e d a b s o l u t e a n d i s
t h e t y p e o f t e n s o r w i t h w h i c h w e h a v e b e e n d e a l i n g a b o v e .
I f w = 1 t h e r e l a t i v e t e n s o r i s c a l l e d a
t e n s o r d e n s i t y . T h e o p e r a t i o n s o f a d d i t i o n , m u l t i p l i c a t i o n , e t c . , o f r e l a t i v e t e n s o r s a r e s i m i l a r t o
t h o s e o f a b s o l u t e t e n s o r s .
S e e f o r e x a m p l e P r o b l e m 6 4 .
S O L V E D P R O B L E M S
S U M M A T I O N C O N V E N T I O N .
1 . W r i t e e a c h o f t h e f o l l o w i n g u s i n g t h e s u m m a t i o n c o n v e n t i o n .
4
1
' 3 0
( a ) d W =
a x
+
a x e
d x 2 +
+
a 0
d x N .
a x N
( b )
d z k
=
a 3 F k d x 1 + a x k d x 2
+
. . .
+
a x k d x N
d t
- a x 1
d t a x 2 d t
a x ' d t
( c ) ( x 1 ) 2 + ( x 2 ) 2 + ( x 3 ) 2 +
. . . + ( x N ) 2 .
( d ) d s 2 =
g 1 1 ( d x 1 ) 2 + g . , ( d x 2 ) 2 + g a s ( d x 3 ) 2 .
3
3
( e )
} r
g
d x p d x q
p = 1 q = 1
p 9
2 . W r i t e t h e t e r m s i n e a c h o f t h e f o l l o w i n g i n d i c a t e d s u m s .
N
( a ) a ,
x k .
} ;
a . x k =
a
x 1 + a
x 2 + . . .
+
a j N
x N
j k
b = ,
j k
=
( b ) A p q
A q r .
A p q A q r
q = 1
A p 1 A 1 r + A p 2 A 2 r
+
. . .
d % = a 0 d x q
d z k
_
a x k d x y '
d t
a x r n d t
x k x k
d s 2 =
g k k
d x k d x k
, N = 3
g p q d x p d x q ,
N = 3
+
A p N A N r
a
( c ) g
x k
N = 3 .
r s
d k
a x a z s
,
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1 7 6
T E N S O R A N A L Y S I S
3
3
a x j a x k
_
g k
j = 1 k = 1
a x r a x s
r s
a x j a x 1
j = 1
j 1 a x a z s
a x , a x 2
a x , a x 3
+ g j 2 a x r a x s + g j 3 a x r a x s
- a x ' - a x ' +
' 6 X 2 a x 1
+
_ 6 X 3 a x 1
_ g 1 1
a ` r a x s g 2 1 a x r a z s
g 3 1 a x r a z s
a x 1 a x 2
a x 2 a x 2 a x 3 a x 2
+ g 1 2 a x r a z S
+ g 2 2
a z r a x S +
g 3 2
a x r a z
a x 1 a x 3
+
_ 6 X 2 a x 3
a x 3 a x 3
+
9 1 3 - a x '
a x S
g 2 3 a z r a z s
+
g 3 3 a ' x r a z s
3 . I f
x k , k = 1 , 2 ,
. . . , N a r e r e c t a n g u l a r c o o r d i n a t e s , w h a t l o c u s i f a n y , i s r e p r e s e n t e d b y e a c h o f t h e
f o l l o w i n g e q u a t i o n s f o r N = 2 , 3 a n d
A s s u m e t h a t t h e f u n c t i o n s a r e s i n g l e - v a l u e d , h a v e c o n -
t i n u o u s d e r i v a t i v e s a n d a r e i n d e p e n d e n t , w h e n n e c e s s a r y .
( a ) a x k = 1 , w h e r e a k a r e c o n s t a n t s .
F o r N = 2 , a 1 x 1 + a 2 x 2 = 1 , a l i n e i n t w o d i m e n s i o n s , i . ' e . a l i n e i n a p l a n e .
F o r N = 3 , a 1 x 1 + a 2 x 2 + a 3 x 3 = 1 ,
a p l a n e i n 3 d i m e n s i o n s .
F o r N > 4 , a 1 x 1 + a
2
X 2 +
. . . +
a N x N = 1
i s a h y p e r p l a n e .
( b ) x k x k = 1 .
F o r N = 2 , ( x 1 ) 2 + ( x 2 ) 2 = 1 ,
a c i r c l e o f u n i t r a d i u s i n t h e p l a n e .
F o r N = 3 , ( x 1 ) 2 + ( x 2 ) 2 + ( x 3 ) 2 = 1 , a s p h e r e o f u n i t r a d i u s .
F o r N > 4 , ( x 1 ) 2 + ( x 2 ) 2 + . . . + ( x N ) 2 = 1 ,
a h y p e r s p h e r e o f u n i t r a d i u s .
( c ) x k = x k ( u ) .
F o r N = 2 , x 1 = x 1 ( u ) , x 2 = x 2 ( u ) , a p l a n e c u r v e w i t h p a r a m e t e r u .
F o r N = 3 , x 1 = x 1 ( u ) , x 2 = x 2 ( u ) , x 3 = x 3 ( u ) , a t h r e e d i m e n s i o n a l s p a c e c u r v e .
F o r N > 4 , a n N d i m e n s i o n a l s p a c e c u r v e .
( d ) x k = x k ( u , v ) .
F o r N = 2 , x 1 = x 1 ( u , v ) , x 2 = x 2 ( u , v )
i s a t r a n s f o r m a t i o n o f c o o r d i n a t e s f r o m ( u , v ) t o ( x 1 , x 2 ) .
F o r N = 3 , x 1 = x 1 ( u , v ) , x 2 = x 2 ( u , v ) , x 3 = x 3 ( u , v ) i s a 3 d i m e n s i o n a l s u r f a c e w i t h p a r a m e t e r s u a n d v .
F o r N > 4 , a h y p e r s u r f a c e .
C O N T R A V A R I A N T A N D C O V A R I A N T V E C T O R S A N D T E N S O R S .
4 . W r i t e t h e l a w o f t r a n s f o r m a t i o n f o r t h e t e n s o r s ( a ) A k , ( b ) B k ,
( c ) C m
( a )
A
=
a x s a x j a x k
A i
q r
a x i a z q a x r
j k
A s a n a i d f o r r e m e m b e r i n g t h e t r a n s f o r m a t i o n , n o t e t h a t t h e r e l a t i v e p o s i t i o n s o f i n d i c e s p , q , r o n
t h e l e f t s i d e o f t h e t r a n s f o r m a t i o n a r e t h e s a m e a s t h o s e o n t h e r i g h t s i d e . S i n c e t h e s e i n d i c e s a r e a s -
s o c i a t e d w i t h t h e z c o o r d i n a t e s a n d s i n c e i n d i c e s i , j , k a r e a s s o c i a t e d r e s p e c t i v e l y w i t h i n d i c e s p , q , r
t h e r e q u i r e d t r a n s f o r m a t i o n i s e a s i l y w r i t t e n .
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T E N S O R A N A L Y S I S
p q
_
a x P a x Q a x i a x j a x k i n n
( b ) B
s t
a x ' n - n a z r a z s a x t
i j k
( c ) C p
C m
a x " n
1 7 7
5 . A q u a n t i t y A ( j , k , 1 , m ) w h i c h i s a f u n c t i o n o f c o o r d i n a t e s x x t r a n s f o r m s t o a n o t h e r c o o r d i n a t e s y s -
t e m z 2 a c c o r d i n g t o t h e r u l e
A ( p , q , r , s )
a x j a z k a x i a . s
A ( , k 1 m )
a x p a x
a x
a x
( a ) I s t h e q u a n t i t y a t e n s o r ? ( b ) I f s o , w r i t e t h e t e n s o r i n s u i t a b l e n o t a t i o n a n d ( c ) g i v e t h e c o n -
t r a v a r i a n t a n d c o v a r i a n t o r d e r a n d r a n k .
( a ) Y e s .
( b ) A j
1 ' .
( c ) C o n t r a v a r i a n t o f o r d e r 3 , c o v a r i a n t o f o r d e r 1 a n d r a n k 3 + 1 = 4 .
6 . D e t e r m i n e w h e t h e r e a c h o f t h e f o l l o w i n g q u a n t i t i e s i s a t e n s o r .
I f s o , s t a t e w h e t h e r i t i s c o n t r a -
s
N
v a r i a n t o r c o v a r i a n t a n d g i v e i t s r a n k :
( a ) d x k ,
( b )
a 0 ( a x k
' x
( a ) A s s u m e t h e t r a n s f o r m a t i o n o f c o o r d i n a t e s
z ' 1
=
x 1 ( x 1 ,
. . . ,
x N ) . T h e n d x 1 =
a x
d x k a n d s o d x k i s a
a x k
c o n t r a v a r i a n t t e n s o r o f r a n k o n e o r a c o n t r a v a r i a n t v e c t o r . N o t e t h a t t h e l o c a t i o n o f t h e i n d e x k i s
a p p r o p r i a t e .
( b ) U n d e r t h e t r a n s f o r m a t i o n x k = x k ( x 1 , . . . , x T ) , 0 i s a f u n c t i o n o f x k a n d h e n c e 0 s u c h t h a t q ( x 1 , . . . , J ) _
i . e . c P i s a s c a l a r o r i n v a r i a n t ( t e n s o r o f r a n k z e r o ) . B y t h e c h a i n r u l e f o r p a r t i a l d i f f e r -
k
a k
k
e n t i a t i o n ,
a 4
=
= a
a x =
- a x k
t r a n s f o r m s l i k e A =
a x
A . T h e n
i s
a x i
a z j
a x k a x 7
a z k a x k
a x k
J
a x i a x k
a c o v a r i a n t t e n s o r o f r a n k o n e o r a c o v a r i a n t v e c t o r .
N o t e t h a t i n
a O
t h e i n d e x a p p e a r s i n t h e d e n o m i n a t o r a n d t h u s a c t s l i k e a s u b s c r i p t w h i c h i n d i -
a x k
c a t e s i t s c o v a r i a n t c h a r a c t e r . W e r e f e r t o t h e t e n s o r
o r e q u i v a l e n t l y , t h e t e n s o r w i t h c o m p o n e n t s
a -
a x k
a k
,
a s t h e g r a d i e n t o f
,
w r i t t e n g r a d 0 o r V O .
7 . A c o v a r i a n t t e n s o r h a s c o m p o n e n t s x y , 2 y - z 2 , x z i n r e c t a n g u l a r c o o r d i n a t e s . F i n d i t s c o v a r i a n t
c o m p o n e n t s i n s p h e r i c a l c o o r d i n a t e s .
L e t
d e n o t e t h e c o v a r i a n t c o m p o n e n t s i n r e c t a n g u l a r c o o r d i n a t e s x 1 = x , x Z = y , x 3 = z .
T h e n
A l = x y = x 1 x 2 ,
A 2 = 2 y - - z 2 = 2 x 2 - ( x 3 ) 2 ,
A 3 = x 1 x 3
w h e r e c a r e m u s t b e t a k e n t o d i s t i n g u i s h b e t w e e n s u p e r s c r i p t s a n d e x p o n e n t s .
L e t A k d e n o t e t h e c o v a r i a n t c o m p o n e n t s i n s p h e r i c a l c o o r d i n a t e s
( 1 ) A k
x 1 = r , x 2 = 6 , x = 0 .
T h e n
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1 7 8
T E N S O R A N A L Y S I S
T h e t r a n s f o r m a t i o n e q u a t i o n s b e t w e e n c o o r d i n a t e s y s t e m s a r e
x 1 = 7 1 s i n x 2 C o s x 3 ,
x 2 = x 1 s i n x s i n V ,
x 3 = V - C o s - X - 2
T h e n e q u a t i o n s ( 1 ) y i e l d t h e r e q u i r e d c o v a r i a n t c o m p o n e n t s
A l
=
A 2
=
=
( s i n 1 2 c o s z 3 ) ( x 1 x 2 )
+
( s i n x 2 s i n x 3 ) ( 2 x 2 - ( x 3 ) 2 )
+
( c o s z 2 ) ( x l x 3 )
a x 1 A l +
a x 2 A 2 + a x 3 A 3
a x 1
a x 1 ' a x '
( s i n 6 c o s ( 1 b ) ( r 2 s i n 2 6 s i n 0 c o s 0 )
+ ( s i n 6 s i n m ) ( 2 r s i n 6 s i n ( P - r 2 c o s 2 6 )
+
( c o s 6 ) ( r 2 s i n 6 c o s 6 c o s ( )
a x 1
A l +
x 2 A 2
+ x 3
A s
a x
a x 2
a z 2
( r c o s 8 c o s 0 ) ( r 2 s i n e 6 s i n
c o s ( p )
+
( r c o s 6 s i n ( p ) ( 2 r s i n 6 s i n m - r 2 c o s 2 B )
+
( - r s i n 6 ) ( r 2 s i n 6 c o s 6 c o s ( P )
A
=
a x 1
A
+
a a x . 2 A
+
a x 3 A
3
a x 3
1 V x 3
2
a x 3
3
( - r s i n 6 s i n d ) ( r 2 s i n 2 0 s i n 0 c o s g y p )
+
( r s i n 6 c o s q 5 ) ( 2 r s i n 6 s i n
- r 2 c o s 2 6 )
+
( 0 ) ( r 2 s i n 6 c o s 6 c o s ( )
8 . S h o w t h a t
a A p
a x q
i s n o t a t e n s o r e v e n t h o u g h A p i s a c o v a r i a n t t e n s o r o f r a n k o n e .
B y h y p o t h e s i s , A ,
- a x ,
A p .
D i f f e r e n t i a t i n g w i t h r e s p e c t t o - k .
a x q
a A j
d x p a A p
C x p
+
k a x q
A
a x k
a x q a x k
a x
a x p a A p a x q
a 2 x p
+
A
a x q a x q a x k
a x k D V '
a x p a x q a A p
a z j a x k a x q
a 2
X P
+ a x k a x k
A
S i n c e t h e s e c o n d t e r m o n t h e r i g h t i s p r e s e n t ,
a A p
d o e s n o t t r a n s f o r m a s a t e n s o r s h o u l d .
L a t e r w e
a x q
a A p
s h a l l s h o w h o w t h e a d d i t i o n o f a s u i t a b l e q u a n t i t y t o
q c a u s e s t h e r e s u l t t o b e a t e n s o r ( P r o b l e m 5 2 ) .
a x
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T E N S O R A N A L Y S I S
1 7 9
9 . S h o w t h a t t h e v e l o c i t y o f a f l u i d a t a n y p o i n t i s a c o n t r a v a r i a n t t e n s o r o f r a n k o n e .
k
T h e v e l o c i t y o f a f l u i d a t a n y p o i n t h a s c o m p o n e n t s
d x
i n t h e c o o r d i n a t e s y s t e m x b . I n t h e c o o r -
d i n a t e s y s t e m x ) t h e v e l o c i t y i s
d i J
.
B u t
d t
d ' l
a x l d x k
d t
a x k d t
b y t h e c h a i n r u l e , a n d i t f o l l o w s t h a t t h e v e l o c i t y i s a c o n t r a v a r i a n t t e n s o r o f r a n k o n e o r a c o n t r a v a r i a n t
v e c t o r .
T H E K R O N E C K E R D E L T A .
1 0 . E v a l u a t e ( a ) 8 q A S S ,
( b )
b q 8 q .
S i n c e
8 q
= 1
i f p = q a n d 0 i f p X q ,
( a ) 8 q
A s r
=
w e h a v e
r
A S . ( b ) 8 g 8 - =
8 r
1 1 . S h o w
t h a t a a x q =
8 q
x
I f p = q ,
a x p
=
1
s i n c e x p = x q .
q
x
a x e
P
qf p
q ,
=
0
s i n c e x a n d x
a r e i n d e p e n d e n t .
a
q
x
T h e . .
a x 4
-
8 q
a x
1 2 . P r o v e t h a t
a x f i
a x
=
8
.
a z q a x r
r
C o o r d i n a t e s x P a r e f u n c t i o n s o f c o o r d i n a t e s
b y t h e c h a i n r u l e a n d P r o b l e m 1 1 ,
x q w h i c h a r e i n t u r n f u n c t i o n s o f c o o r d i n a t e s x r . T h e n
a x P
a x p a z 9
a x r
a x q a z
a
3 . I f A
= x q A q p r o v e t h a t A q =
a x
-
A t .
a x
a x
p
M u l t i p l y e q u a t i o n
A
=
a x A q
b y
a x r
a x q
a z p
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1 8 0
T E N S O R A N A L Y S I S
- a x -
a x r a x k ' q r q r
=
A = 8 q A
= A
b y P r o b . 1 2 . P l a c i n g r = q t h e r e s u l t f o l l o w s . T h i s
h e n
a z j '
A
a
z a x q
i n d i c a t e s t h a t i n t h e t r a n s f o r m a t i o n e q u a t i o n s f o r t h e t e n s o r c o m p o n e n t s t h e q u a n t i t i e s w i t h b a r s a n d q u a n -
t i t i e s w i t h o u t b a r s c a n b e i n t e r c h a n g e d , a r e s u l t w h i c h c a n b e p r o v e d i n g e n e r a l .
1 4 . P r o v e t h a t 8 Q i s a m i x e d t e n s o r o f t h e s e c o n d r a n k .
I f
8 q i s a m i x e d t e n s o r o f t h e s e c o n d r a n k i t m u s t t r a n s f o r m a c c o r a i n g t o t h e r u l e
6 k
a x q a x q 6 p
a X P a z k
q
T h e r i g h t s i d e e q u a l s
a x ' a x k
= k b y P r o b l e m 1 2 . S i n c e 8 k
j
= 8
k
= 1
i f j = k , a n d 0 i f j
k , i t f o l -
b
a x a x
l o w s t h a t 8 q i s a m i x e d t e n s o r o f r a n k t w o , j u s t i f y i n g t h e n o t a t i o n u s e d .
N o t e t h a t w e s o m e t i m e s u s e
8 p q = 1
i f p = q
a n d 0 i f p q , a s t h e K r o n e c k e r d e l t a . T h i s i s h o w -
e v e r n o t a c o v a r i a n t t e n s o r o f t h e s e c o n d r a n k a s t h e n o t a t i o n w o u l d s e e m t o i n d i c a t e .
F U N D A M E N T A L O P E R A T I O N S W I T H T E N S O R S .
1 5 . I f
A p q
a n d
B r q
a r e t e n s o r s , p r o v e t h a t t h e i r s u m a n d d i f f e r e n c e a r e t e n s o r s .
B y h y p o t h e s i s
A r q
a n d B r q
a r e t e n s o r s , s o t h a t
- 3 - i a x k
j r
P q
A Z a x p a x q a T l
A r
a x ' s a z k a x r
f i q
B l
a x p a x q a z l
B r
A d d i n g ,
( A
j k
+
i l k )
=
a x k a x k a x r ( A P q
+ B r
Z
l
a x p a x q a l l
r
r
j k
J k
a z k a z k a x r
p q 5 q
S u b t r a c t i n g ,
( A Z - B l ) -
Z
( A r - B r )
a x p a x q a x "
4 Y +
P q p q
P q
a r e
P q
T h e n
B r
a n d A r - B r a r e t e n s o r s o f t h e s a m e r a n k a n d t y p e a s A r a n d B r .
P s
1 6 . I f A r q a n d B t a r e t e n s o r s , p r o v e t h a t C r t = A r q B t i s a l s o a t e n s o r .
W e m u s t p r o v e t h a t
r t s
i s a t e n s o r w h o s e c o m p o n e n t s a r e f o r m e d b y t a k i n g t h e p r o d u c t s o f c o m p o -
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T E N S O R A N A L Y S I S
n e n t s o f t e n s o r s
A P g
a n d B t .
S i n c e A I r
a n d
a r e
r e t e n s o r s ,
j k
-
a x J a r k a x r A P q
l
a x p a x q a r l
a r m a x t B S
a x s a m
t
M u l t i p l y i n g ,
- 1 k
A l
B m
n
a r k a z k a x r a z m a x t A P q
B s
a x p a x q a z l a x S a x n
r
t
1 8 1
w h i c h s h o w s t h a t A P g B t i s a t e n s o r o f r a n k 5 , w i t h c o n t r a v a r i a n t i n d i c e s p , q , s a n d c o v a r i a n t i n d i c e s
r , t
, t h u s w a r r a n t i n g t h e n o t a t i o n
C a t s . W e c a l l C r r =
A , , B t t h e o u t e r p r o d u c t o f
A r g a n d B t
.
1 7 . L e t
A r s
t
b e a t e n s o r . ( a ) C h o o s e p = t a n d s h o w t h a t A r q p , w h e r e t h e s u m m a t i o n c o n v e n t i o n i s
e m p l o y e d , i s a t e n s o r . W h a t i s i t s r a n k ? ( b ) C h o o s e p = t a n d q = s a n d s h o w s i m i l a r l y t h a t A r g p
i s a t e n s o r . W h a t i s i t s r a n k ?
( a ) S i n c e
A r s t i s a t e n s o r ,
( 1 )
A j k
a x k a x k a x r a x s a x t A P q
l m n
a x k ' a x q a x l a r m a x n
r s t
W e m u s t s h o w t h a t A
P q p
i s a t e n s o r . P l a c e t h e c o r r e s p o n d i n g i n d i c e s j a n d n e q u a l t o e a c h o t h e r
a n d s u m o v e r t h i s i n d e x . T h e n
A '
l T n j
A
a x J a r k a x r a x s a t
p q
a x p a x g a z l ' a x - - M a x g
A r s t
a x t a r e a x k a x r a x s
v q
r s t
1
' a x g a x l a x m
x g a x
8 t a x k a x r a x s A b q
0 a x g a r t
a r m
r s t
a r k a x r a x s A p q
a x g
a x l
a x m
r s p
a n d s o
A r s e i s a t e n s o r o f r a n k 3 a n d c a n b e d e n o t e d b y
B q s
.
T h e p r o c e s s o f p l a c i n g a c o n t r a v a r i a n t
i n d e x e q u a l t o a c o v a r i a n t i n d e x i n a t e n s o r a n d s u m m i n g i s c a l l e d c o n t r a c t i o n . B y s u c h a p r o c e s s a
t e n s o r i s f o r m e d w h o s e r a n k i s t w o l e s s t h a n t h e r a n k o f t h e o r i g i n a l t e n s o r .
P q
( b ) W e m u s t s h o w t h a t A r g p i s a t e n s o r . P l a c i n g j = n a n d k = i n i n e q u a t i o n ( 1 ) o f p a r t ( a ) a n d s u m m i n g
o v e r j a n d k ,
w e h a v e
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1 8 2
T E N S O R A N A L Y S I S
- j k
_ a x q a z k a x r a x S a x t
1 ' q
A l k j
a x p a x q a z l a z k a z k
A r s t
- a x t a x 3 a x S a z k a x r
a z k a x p a z k a x q a z l
t
s a r P q
8 8
x A
1
q a x l
r s t
a x r
p q
a x l
A r q p
A p q
r s t
w h i c h s h o w s t h a t
A r q
p i s a t e n s o r o f r a n k o n e a n d c a n b e d e n o t e d b y C r . N o t e t h a t b y c o n t r a c t i n g
t w i c e , t h e r a n k w a s r e d u c e d b y 4 .
1 8 . P r o v e t h a t t h e c o n t r a c t i o n o f t h e t e n s o r
A q
i s a s c a l a r o r i n v a r i a n t .
W e h a v e
A
=
a x q a x q A p
k a x p a z k
q
A J
=
a x q a x q A P
=
S q A p
=
A p
u t t i n g j = k a n d s u m m i n g ,
a x p a x q q
p
q p
T h e n A =
A p
a n d i t f o l l o w s t h a t A l m u s t b e a n i n v a r i a n t .
S i n c e
A P
q
i s a t e n s o r o f r a n k t w o a n d
c o n t r a c t i o n w i t h r e s p e c t t o a s i n g l e i n d e x l o w e r s t h e r a n k b y t w o , w e a r e l e d t o d e f i n e a n i n v a r i a n t a s a
t e n s o r o f r a n k z e r o .
1 9 . S h o w t h a t t h e c o n t r a c t i o n o f t h e o u t e r p r o d u c t o f t h e t e n s o r s A 0 a n d B q i s a n i n v a r i a n t .
S i n c e
A p
a n d B q a r e t e n s o r s ,
T j =
a x q A p ,
B k =
a x q
B q .
T h e n
a x
a x
- a x q a x q
k
A B
a x p a x k
q
B y c o n t r a c t i o n ( p u t t i n g j = k a n d s u m m i n g )
A B .
=
a z j a x q
A p
B =
8 p q
A p
B q
=
A p
B
I
a x p a x q
q
p
a n d s o A 1 B 1 i s a n i n v a r i a n t . T h e p r o c e s s o f m u l t i p l y i n g t e n s o r s ( o u t e r m u l t i p l i c a t i o n ) a n d t h e n c o n t r a c t -
i n g i s c a l l e d i n n e r m u l t i p l i c a t i o n a n d t h e r e s u l t i s c a l l e d a n i n n e r p r o d u c t . S i n c e A p B p i s a s c a l a r , i t i s
o f t e n c a l l e d t h e s c a l a r p r o d u c t o f t h e v e c t o r s A P a n d B q .
p
q s
2 0 . S h o w t h a t a n y i n n e r p r o d u c t o f t h e t e n s o r s A r a n d B t
i s a t e n s o r o f r a n k t h r e e .
p
Bu t e r p r o d u c t o f A
a n d B t s = A p g s
t
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T E N S O R A N A L Y S I S
1 8 3
L e t u s c o n t r a c t w i t h r e s p e c t t o i n d i c e s p a n d t , i . e . l e t p = t a n d s u m . W e m u s t s h o w t h a t t h e r e s u l t -
i n g i n n e r p r o d u c t , r e p r e s e n t e d b y
0 B g s ,
i s a t e n s o r o f r a n k t h r e e .
s
B y h y p o t h e s i s , A r a n d B t
a r e t e n s o r s ; t h e n
A
=
a ` x j a x r
A B
I n
=
a x I a x n a x t B q s
k - a
a x k
n
a x q a x s a x n
t
M u l t i p l y i n g , l e t t i n g j = n a n d s u m m i n g , w e h a v e
a x k a x r a z l a x n a x t
q s
a x P a x k a x q a x s a x k
A r B t
8 t a x r a 3 1 a x n A P
B t
P a x k a x q a x s
r
t
a x r a i l a x n A P B q s
a z k a x q a x s
r p
s h o w i n g t h a t A B q s i s a t e n s o r o f r a n k t h r e e . B y c o n t r a c t i n g w i t h r e s p e c t t o q a n d r o r s a n d r i n t h e
p r o d u c t A B t s , w e c a n s i m i l a r l y s h o w t h a t a n y i n n e r p r o d u c t i s a t e n s o r o f r a n k t h r e e .
A n o t h e r M e t h o d . T h e o u t e r p r o d u c t o f t w o t e n s o r s i s a t e n s o r w h o s e r a n k i s t h e s u m o f t h e r a n k s o f
t h e g i v e n t e n s o r s .
T h e n A P B
q s
i s a t e n s o r o f r a n k 3 + 2 = 5 .
S i n c e a c o n t r a c t i o n r e s u l t s i n a t e n s o r
w h o s e r a n k i s t w o l e s s t h a n t h a t o f t h e g i v e n t e n s o r , i t f o l l o w s t h a t a n y c o n t r a c t i o n o f A P B q s i s a t e n s o r
o f r a n k 5 - 2 = 3 .
2 1 . I f X ( p , q , r ) i s a q u a n t i t y s u c h t h a t
X ( p , q , r ) B q n = 0
f o r a n a r b i t r a r y t e n s o r B q n ,
p r o v e t h a t
X ( p , q , r ) = 0 i d e n t i c a l l y .
n
S i n c e B r
i s a n a r b i t r a r y t e n s o r , c h o o s e o n e p a r t i c u l a r c o m p o n e n t ( s a y t h e o n e w i t h q = : 2 , r = 3 ) n o t
e q u a l t o z e r o , w h i l e a l l o t h e r c o m p o n e n t s a r e z e r o . T h e n X ( p , 2 , 3 ) B a n = 0 ,
s o t h a t X ( p , 2 , 3 ) = 0 s i n c e
B a n
0 . B y s i m i l a r r e a s o n i n g w i t h a l l p o s s i b l e c o m b i n a t i o n s o f q a n d r ,
w e h a v e X ( p , q , r ) = 0 a n d t h e
r e s u l t f o l l o w s .
2 2 . A q u a n t i t y A ( p , q , r ) i s s u c h t h a t i n t h e c o o r d i n a t e s y s t e m x 2 A ( p , q , r ) B r s = C 0 w h e r e
B r s
i s a n
a r b i t r a r y t e n s o r a n d C i s a t e n s o r . P r o v e t h a t A ( p , q , r ) i s a t e n s o r .
I n t h e t r a n s f o r m e d c o o r d i n a t e s x 2 , A ( j , k , 1 ) B
k i n
= C n .
j
s
T h e n
A ( j , k , l )
a x k a z n a x r B q s
=
- 6 7 V , a x p
C l ,
a x q a x s a x l
r
a x s a x k
o r
a x s
a x k a x
Z A ( j , k , 1 )
-
a x k
A ( p , q , r )
B q s
- 0
- a x s
a x a z k
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1 8 4
T E N S O R A N A L Y S I S
I n n e r m u l t i p l i c a t i o n b y
a x ,
( i . e . m u l t i p l y i n g b y
a x n
a n d t h e n c o n t r a c t i n g w i t h t = m ) y i e l d s
a x a x t
o r
n
a z k a x r
- a x '
q s
b S
a x q a z l
. 4 ( j , k , l )
A ( p , q , r )
B r
=
0
a x k a x r - a x k
q n
a x q a x l
A ( j , k , l ) -
a x A ( p , q , r )
B r
= 0 .
n
S i n c e B r
i s a n a r b i t r a r y t e n s o r , w e h a v e b y P r o b l e m 2 1 ,
a z k a x r
k , 1 )
-
a x
A ( p , q , r )
=
0
a x q a x l
a x i
a 9 a z n
I n n e r m u l t i p l i c a t i o n b y
x
y i e l d s
a x i n a x r
k n -
a x p a x q a x ` n
b 7 n 6 1 A ( j , k , l ) -
1
A ( p , q , r )
a z a x ' s a x r
o r 4 ( j , m , n )
a x p a x q a z n
a x j a z m a x r
A ( p , q , r )
=
0
w h i c h s h o w s t h a t A ( p , q , r ) i s a t e n s o r a n d j u s t i f i e s u s e o f t h e n o t a t i o n A q .
I n t h i s p r o b l e m w e h a v e e s t a b l i s h e d a s p e c i a l c a s e o f t h e q u o t i e n t l a w w h i c h s t a t e s t h a t i f a n i n n e r
p r o d u c t o f a q u a n t i t y X w i t h a n a r b i t r a r y t e n s o r B i s a t e n s o r C , t h e n X i s a t e n s o r .
S Y M M E T R I C A N D S K E W - S Y M M E T R I C T E N S O R S .
r
2 3 . I f a t e n s o r
. 4 S q
i s s y m m e t r i c ( s k e w - s y m m e t r i c ) w i t h r e s p e c t t o i n d i c e s p a n d q i n o n e c o o r d i n a t e
s y s t e m , s h o w t h a t i t r e m a i n s s y m m e t r i c ( s k e w - s y m m e t r i c ) w i t h r e s p e c t t o p a n d q i n a n y c o o r d i -
n a t e s y s t e m .
P q
S i n c e o n l y i n d i c e s p a n d q a r e i n v o l v e d w e s h a l l p r o v e t h e r e s u l t s f o r B
I f B
P q
P q
q
s s y m m e t r i c , B
= B
.
T h e n
B j k
=
- a l l a z k B p q = a z k a z k B q p
=
k j
a x P a x q
a x q a x p
a n d B P q
r e m a i n s s y m m e t r i c i n t h e z 2 c o o r d i n a t e s y s t e m .
I f B q i s s k e w - s y m m e t r i c ,
B
j k
=
a x k a x k B ? ' q
=
-
a z k a a c l B q p
_
a x p a x q
a x q a x k
a n d B P q
r e m a i n s s k e w - s y m m e t r i c i n t h e T i c o o r d i n a t e s y s t e m .
T h e a b o v e r e s u l t s a r e , o f c o u r s e , v a l i d f o r o t h e r s y m m e t r i c ( s k e w - s y m m e t r i c ) t e n s o r s .
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T E N S O R A N A L Y S I S
1 8 5
2 4 . S h o w t h a t e v e r y t e n s o r c a n b e e x p r e s s e d a s t h e s u m o f t w o t e n s o r s , o n e o f w h i c h i s s y m m e t r i c
a n d t h e o t h e r s k e w - s y m m e t r i c i n a p a i r o f c o v a r i a n t o r c o n t r a v a r i a n t i n d i c e s .
C o n s i d e r , f o r e x a m p l e , t h e t e n s o r B . W e h a v e
B P 9
( B P 9 +
B 9 P )
+
1 ( B P 4 -
B 4 P
B u t
R p q =
( B p q + B q ' )
=
R q ' i s s y m m e t r i c , a n d S P q
=
( B P q - B q ' ) = - S q ' i s s k e w - s y m m e t r i c .
B y s i m i l a r r e a s o n i n g t h e r e s u l t i s s e e n t o b e t r u e f o r a n y t e n s o r .
2 5 . I f
= a j k
A i A k s h o w t h a t
w e c a n a l w a y s w r i t e 4 ) = b k
A j A k w h e r e b k i s s y m m e t r i c .
( D
=
a j k
A A k
=
a k J
A k
A
=
a k , .
A A k
T h e n
2 C =
a j k
A j A k +
a k . 1
A i A k
=
( a j k + a k j )
A j A k
a n d
=
z ( a j k + a k i - )
A j A k
=
b j k
A ' A k
w h e r e
b . k =
2 ( a j k + a k ' ) = b k j i s s y m m e t r i c .
M A T R I C E S .
2 6 . W r i t e t h e s u m S = A + B , d i f f e r e n c e D = A - B , a n d p r o d u c t s P = A B , Q = B A o f t h e m a t r i c e s
3
1
- 2
2
0 - 1
A
=
4 - 2
3
,
B =
- 4 1
2
- 2
1
- 1
1 - 1
0
3 + 2
1 + 0
- 2 - 1
5
1
- 3
S = A + B =
4 - 4
- 2 + 1 3 + 2
=
0 - 1 5
- 2 + 1
1 - 1
- 1 + 0
- 1
0
- 1
3 - 2 1 - 0 - 2 + 1
1
1
- 1
D = A - B =
4 + 4 - 2 - 1
3 - 2 =
8 - 3
1
- 2 - 1
1 + 1
- 1 - 0
- 3
2 - 1
( 3 ) ( 2 ) + ( 1 ) ( - 4 ) + ( - 2 ) ( 1 )
( 3 ) ( 0 ) + ( 1 ) ( 1 ) + ( - 2 ) ( - 1 )
( 3 ) ( - 1 ) + ( 1 ) ( 2 ) + ( - 2 ) ( 0 )
P = A B =
( 4 ) ( 2 ) + ( - 2 ) ( - 4 ) +
( 3 ) ( 1 )
( 4 ) ( 0 ) + ( - 2 ) ( 1 ) + ( 3 ) ( - 1 )
( 4 ) ( - 1 ) + ( - 2 ) ( 2 ) + ( 3 ) ( 0 )
( - 2 ) ( 2 ) + ( 1 ) ( - 4 ) + ( - 1 ) ( 1 )
( - - 2 ) ( 0 ) + ( 1 ) ( 1 ) + ( - 1 ) ( - 1 )
( - 2 ) ( - 1 ) +
( 1 ) ( 2 ) + ( - 1 ) ( 0 )
0 3 - 1
1 9
- 5
- 8
- - 9 2
4
Q = B A =
8
1
- 3
- 1 2
- 4
9
1
3
- 5
T h i s s h o w s t h a t A B B A , i . e . m u l t i p l i c a t i o n o f m a t r i c e s i s n o t c o m m u t a t i v e i n g e n e r a l .
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1 8 6
T E N S O R A N A L Y S I S
2 7 . I f A =
2
1
a n d B =
( - I
2
,
s h o w t h a t ( A + B ) ( A - B ) A A 2 - B 2 .
- 1 3
3 - 2
A + B = 1 3 , A - B =
3 - 1
.
T h e n ( A + B ) ( A - B ) =
1
3 - 2
1
4 5
( _ :
5
A 2 -
2
1
2
1
- 1 3
- 1 3
T h e n A 2 - B 2 =
1 - 4
1 1
4 - 2
3 5
B 2
+ - 1
2
- 1 2 = 7 - 6
y - - 5 8
' 3 - 2 3 - 2 - 9 1 0
T h e r e f o r e , ( A + B ) ( A - B )
A ` - B 2 .
H o w e v e r , ( A + B ) ( A - B ) = A 2 - A B + B A - B 2 .
2 8 . E x p r e s s i n m a t r i x n o t a t i o n t h e t r a n s f o r m a t i o n e q u a t i o n s f o r ( a ) a c o v a r i a n t v e c t o r , ( b ) a c o n t r a -
v a r i a n t t e n s o r o f r a n k t w o , a s s u m i n g N = 3
.
a 9
( a ) T h e t r a n s f o r m a t i o n e q u a t i o n s A P =
x -
A q c a n b e w r i t t e n
a x
a x I a x 2
a x 3
a x 1 o x
o x
a x 1
a x 2
a x 3
a x 2 a x 2 a x 2
a x 1 a x 2
a x 3
a x 3
a x 3
a x 3
i n t e r m s o f c o l u m n v e c t o r s , o r e q u i v a l e n t l y i n t e r m s o f r o w v e c t o r s
a x 1
- a x '
a x 1
a x 1 a x 2
a x 3
( A 1 A 2 A 3 )
_
( A 1 A 2 A 3 )
a x 2
a x 2
a x 2
a y 1
a z 2
- 3 x 3
a x 3 a x 3 a x 3
a x 1 a x 2
a x 3
( b ) T h e t r a n s f o r m a t i o n e q u a t i o n s A
p r = a x
a x
q s
c a n b e w r i t t e n
a x q
a x s
A
- 2 1
A
- 2 2
A
2 3
'
_
A
3 1 A 3 2 A 3 3
E x t e n s i o n s o f t h e s e r e s u l t s c a n b e m a d e f o r N > 3 .
F o r h i g h e r r a n k t e n s o r s , h o w e v e r , t h e m a t r i x n o t a -
t i o n f a i l s .
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T E N S O R A N A L Y S I S
1 8 7
T H E L I N E E L E M E N T A N D M E T R I C T E N S O R .
2 9 . I f d s 2 =
g j
d x k d x k i s a n i n v a r i a n t , s h o w t h a t
9 j k
i s a s y m m e t r i c c o v a r i a n t t e n s o r o f r a n k t w o .
B y P r o b l e m 2 5 , 4 ) = d s 2 , A J = d x a n d
A k
= d x k ; i t f o l l o w s t h a t 9 , j k c a n b e c h o s e n s y m m e t r i c . A l s o
s i n c e d s 2 i s a n i n v a r i a n t ,
g d x ' d x q
=
g , k d x k
d x k
=
g k
a x
p d T P
a x e
d -
=
g . k
a x e a x k
d 0 d x ` q
p q
9 J a x
a x
9 a x
a x
T h e n g ` p q = g j k
a
a x k
a n d
g j k
i s a s y m m e t r i c c o v a r i a n t t e n s o r o f r a n k t w o , c a l l e d t h e m e t r i c t e n s o r .
a x a x
3 0 .
D e t e r m i n e t h e m e t r i c t e n s o r i n ( a ) c y l i n d r i c a l a n d ( b ) s p h e r i c a l c o o r d i n a t e s .
( a ) A s i n P r o b l e m 7 , C h a p t e r 7 ,
d s 2 = d p 2 + p 2 d 0 2 + d z 2 .
I f x = p , x 2 = 0 , x 3 = z t h e n
g 1 1 = 1 ' g 2 2 _ p 2
g 3 3 - I ' g 1 2J g 2 1 - 0 ' "2 3 - g 3 2 ^ O ' g 3 1 = g 1 3 = O .
9 1 1 g 1 2
g 1 3
I n m a t r i x f o r m t h e m e t r i c t e n s o r c a n b e w r i t t e n
9 2 1 9 2 2
g 2 3
\ g 2 1
g 3 2
9 3 3
( b ) A s i n P r o b l e m 8 ( a ) , C h a p t e r 7 ,
d s 2 = d r 2 + r 2 d 8 2 + r 2 s i n 2 8 d c 2 .
1 0 0
0
p 2
0
0 0
1
f 1 0
0
I f x 1 = r , x 2 = 8 , x 3 =
t h e m e t r i c t e n s o r c a n b e w r i t t e n
0
r 2
0
0 0
r 2 s i n 2 8
I n g e n e r a l f o r o r t h o g o n a l c o o r d i n a t e s , g k = 0 f o r i t k .
j
3 1 . ( a ) E x p r e s s t h e d e t e r m i n a n t g =
g 1 1
g 1 2
g 1 3
g 2 1 g 2 2 g 2 3
g 3 1
g 3 2
g 3 3
i n t e r m s o f t h e e l e m e n t s i n t h e s e c o n d r o w a n d
t h e i r c o r r e s p o n d i n g c o f a c t o r s .
( b ) S h o w t h a t
g j k
G ( j , k ) = g w h e r e G ( j , k ) i s t h e c o f a c t o r o f
g k i n g a n d w h e r e s u m m a t i o n i s o v e r k o n l y .
j
( a ) T h e c o f a c t o r o f
g j k
i s t h e d e t e r m i n a n t o b t a i n e d f r o m g b y ( 1 ) d e l e t i n g t h e r o w a n d c o l u m n i n w h i c h
g k a p p e a r s a n d ( 2 ) a s s o c i a t i n g t h e s i g n ( - 1 ) j + k t o t h i s d e t e r m i n a n t .
T h u s ,
C o f a c t o r o f g
=
( - 1 ) 2 + 1
g 1 2 g 1 2
,
C o f a c t o r o f g
=
( - - 1 ) 2 + 2 g 1 1 g 1 3
,
2 1
g 3 2 g 3 3
g 3 1 g d 3
C o f a c t o r o f g
_
( - 1 ) 2 + 3
9 1 1 9 1 2
g 3 1 g 3 2
D e n o t e t h e s e c o f a c t o r s b y G ( 2 , 1 ) , G ( 2 , 2 ) a n d G ( 2 , 3 ) r e s p e c t i v e l y . T h e n b y a n e l e m e n t a r y p r i n c i p l e
o f d e t e r m i n a n t s
g 2 1 G ( 2 , 1 ) + g 2 2 G ( 2 , 2 ) + g m G ( 2 , 3 ) =
g
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1 8 8
T E N S O R A N A L Y S I S
( b ) B y a p p l y i n g t h e r e s u l t o f ( a ) t o a n y r o w o r c o l u m n , w e h a v e g j k G ( j , k ) = g w h e r e t h e s u m m a t i o n i s
o v e r k o n l y . T h e s e r e s u l t s h o l d w h e r e g =
I g j k
I
i s a n N t h o r d e r d e t e r m i n a n t .
3 2 . ( a ) P r o v e t h a t
g 2 1
G ( 3 , l ) +
g 2 2
G ( 3 , 2 ) +
g 2 3
G ( 3 , 3 )
=
0 .
( b ) P r o v e t h a t
g 3 k G ( p , k ) = 0
i f
j A p .
( a ) C o n s i d e r t h e d e t e r m i n a n t
g 1 1
g 1 2 g 1 3
g 2 1
g 2 2
g 2 3
g 2 1 g 2 2
g 2 3
w h i c h i s z e r o s i n c e i t s l a s t t w o r o w s a r e i d e n t i c a l . E x -
p a n d i n g a c c o r d i n g t o e l e m e n t s o f t h e l a s t r o w w e h a v e
g 2 1
G ( 3 , 1 ) + g 2 2 G ( 3 , 2 ) + g 2 3 G ( 3 , 3 )
= 0
( b ) B y s e t t i n g t h e c o r r e s p o n d i n g e l e m e n t s o f a n y t w o r o w s ( o r c o l u m n s ) e q u a l w e c a n s h o w , a s i n p a r t ( a ) ,
t h a t g , j k G ( p , k ) = 0 i f j
p . T h i s r e s u l t h o l d s f o r N t h o r d e r d e t e r m i n a n t s a s w e l l .
3 3 . D e f i n e
g 7 k
=
G ( , k )
w h e r e G ( j , k ) i s t h e c o f a c t o r o f
g j k
i n t h e d e t e r m i n a n t g =
g j k
P r o v e t h a t g - k
9 O k
=
8 ' .
B y P r o b l e m 3 1 ,
i g k
G ( g k )
=
1
o r
j g k
g j k
= 1 , w h e r e s u m m a t i o n i s o v e r k o n l y .
B y P r o b l e m 3 2 ,
g k G ( g
k )
= 0
o r
g P k
= 0
i f p
j .
A 0 .
T h e n
g , k g P k ( = 1 i f p = j , a n d 0 i f p J j ) = b .
W e h a v e u s e d t h e n o t a t i o n
g i k
a l t h o u g h w e h a v e n o t y e t s h o w n t h a t t h e n o t a t i o n i s w a r r a n t e d , i . e .
t h a t g j k i s a c o n t r a v a r i a n t t e n s o r o f r a n k t w o . T h i s i s e s t a b l i s h e d i n P r o b l e m 3 4 . N o t e t h a t t h e c o f a c t o r
h a s b e e n w r i t t e n G ( j , k ) a n d n o t
G j k
s i n c e w e c a n s h o w t h a t i t i s n o t a t e n s o r i n t h e u s u a l s e n s e . H o w -
e v e r , i t c a n b e s h o w n t o b e a r e l a t i v e t e n s o r o f w e i g h t t w o w h i c h i s c o n t r a v a r i a n t , a n d w i t h t h i s e x t e n s i o n
o f t h e t e n s o r c o n c e p t t h e n o t a t i o n G j k c a n b e j u s t i f i e d ( s e e S u p p l e m e n t a r y P r o b l e m 1 5 2 ) .
3 4 . P r o v e t h a t
g j k
i s a s y m m e t r i c c o n t r a v a r i a n t t e n s o r o f r a n k t w o .
S i n c e g
3 . k
i s s y m m e t r i c , G ( j , k ) i s s y m m e t r i c a n d s o
g j k
= G ( j , k ) / g i s s y m m e t r i c .
I f
B 0
i s a n a r b i t r a r y c o n t r a v a r i a n t v e c t o r , B q = g
q
B 0 i s a n a r b i t r a r y c o v a r i a n t v e c t o r . M u l t i p l y i n g
b y g j q ,
g J q B q =
g J q
g
B '
=
8 i B O
=
B i
o r
g J q B q =
B , 9
q
0
S i n c e B q i s a n a r b i t r a r y v e c t o r , g I q i s a c o n t r a v a r i a n t t e n s o r o f r a n k t w o , b y a p p l i c a t i o n o f t h e q u o t i e n t
l a w . T h e t e n s o r g j k i s c a l l e d t h e c o n j u g a t e m e t r i c t e n s o r .
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T E N S O R A N A L Y S I S
3 5 . D e t e r m i n e t h e c o n j u g a t e m e t r i c t e n s o r i n (
( a ) F r o m P r o b l e m 3 0 ( a ) ,
S i m i l a r l y
g
3 k
= 0
9 1 1
g
9 2 2
_
1
0
0
0
p 2 0
0
0
1
y l i n d r i c a l a n d ( b ) s p h e r i c a l c o o r d i n a t e s .
p 2
c o f a c t o r o f g 1 1
1 p 2 0
g p 2 0
1
c o f a c t o r o f g 2 2 1
g
p 2
c o f a c t o r o f g
1
9 W
= g
=
2
c o f a c t o r o f g 1 2
1
g 1 2 =
g
p 2
1
0
0 p 2
0 0
0
1
1
P
2
= 0
i f j
k .
I n m a t r i x f o r m t h e c o n j u g a t e m e t r i c t e n s o r c a n b e r e p r e s e n t e d b y
1
0
0
0
1 / p 2 0
0
0
1
( b ) F r o m P r o b l e m 3 0 ( b ) ,
g
A s i n p a r t ( a ) , w e f i n d
t h i s c a n b e w r i t t e n
r 2
r 4 s i n 2 &
1 8 9
1
a n d g j k = 0 f o r j
k , a n d i n m a t r i x f o r m
r 2 s i n 2 6
1
0
0
0
1 / r 2
0
0
0
1 / r 2 s i n 2 6
3 6 . F i n d ( a ) g a n d ( b )
g j k
c o r r e s p o n d i n g t o d s 2 = 5 ( d x 1 ) 2 + 3 ( d x 2 ) 2 + 4 ( d x 3 ) 2 - 6 d x 1 d x 2 + 4 d x 2 d x 3
.
( a ) g 1 1 = 5 ,
g =
5 - 3 0
- 3 3
2
0 . 2 4
= 4 .
( b ) T h e c o f a c t o r s G ( j , k ) o f g J k a r e
G ( 1 , 1 ) = 8 , G ( 2 , 2 ) = 2 0 , G ( 3 , 3 ) = 6 , G ( 1 , 2 ) = G ( 2 , 1 ) = 1 2 , G ( 2 , 3 ) = G ( 3 , 2 ) = - 1 0 , G ( 1 , 3 ) = G ( 3 , 1 ) = - 6
T h e n g 1 1 = 2 , g 2 2 = 5 , g 3 3 = 3 / 2 , g 1 2 = g 2 . = 3 , e = g 2 2 = - 5 / 2 , g 1 3 = g 3 1 = - 3 / 2
N o t e t h a t t h e p r o d u c t o f t h e m a t r i c e s ( g J . k ) a n d ( g j k ) i s t h e u n i t m a t r i x I , i . e .
5
- 3
0 2 3
- 3 / 2 1
0
0
- 3
3
2
3 5 - 5 / 2
=
0
1
0
0
2
4 - 3 / 2 - 5 / 2 3 / 2
0 0
1
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1 9 0
T E N S O R A N A L Y S I S
A S S O C I A T E D T E N S O R S .
3 7 . I f A =
g j k
A k ,
s h o w t h a t
A k
= g i k A
.
M u l t i p l y A j = g J , k A k b y g " q
T h e n
g J q
A j = g j g g j k A " - = 8 q A k =
A q ,
i . e .
A q
=
g J q
A o r A k = g j k A j .
T h e t e n s o r s o f r a n k o n e , A j a n d A k , a r e c a l l e d a s s o c i a t e d . T h e y r e p r e s e n t t h e c o v a r i a n t a n d c o n t r a -
v a r i a n t c o m p o n e n t s o f a v e c t o r .
3 8 . ( a ) S h o w t h a t L 2 = g p q A P A q
i s a n i n v a r i a n t .
( b ) S h o w t h a t L 2 = g p g A P A q .
( a ) L e t A j a n d A k
b e t h e c o v a r i a n t a n d c o n t r a v a r i a n t c o m p o n e n t s o f a v e c t o r . T h e n
A p = a X A 1 ,
a n d A p
A
p
=
a x k a z p A . A k
a x ' p a x k
q
a x g A k
a x k
= k A A k
=
A . A l
s o t h a t A j A I
i s a n i n v a r i a n t w h i c h w e c a l l L 2 . T h e n w e c a n w r i t e
L 2
=
A j A ' l
=
g j k
A k A I
g p q
A 0 A q
( b ) F r o m ( a ) ,
L 2 = A . A I = A I g k j A k = g j k A j A k = g , q A p A q .
T h e s c a l a r o r i n v a r i a n t q u a n t i t y L =
A P
A P i s c a l l e d t h e m a g n i t u d e o r l e n g t h o f t h e v e c t o r w i t h
c o v a r i a n t c o m p o n e n t s A p a n d c o n t r a v a r i a n t c o m p o n e n t s A p .
3 9 .
( a )
I f
A P
a n d B q a r e v e c t o r s , s h o w t h a t g p q A P B q i s a n i n v a r i a n t .
A p B 4
( b ) S h o w t h a t 9 p g
i s a n i n v a r i a n t .
( A p A
p ) ( B g B q )
( a ) B y P r o b l e m 3 8 , A
0
B
0
=
A P g p q B g = g p q
A I B '
i s a n i n v a r i a n t .
( b ) S i n c e A p A P a n d B g B q a r e i n v a r i a n t s
( A p A p ) ( B g B q ) i s a n i n v a r i a n t a n d
i n v a r i a n t .
W e d e f i n e
c o s 6
g p g A P B g
s o
g p g A A B q
( A ' A p ) ( B q B q )
i s a n
( A P A p ) ( B g B q )
a s t h e c o s i n e o f t h e a n g l e b e t w e e n v e c t o r s A p a n d B q .
I f
g p q A 1 B ' =
A 1 B 1
= 0 , t h e v e c t o r s a r e
c a l l e d o r t h o g o n a l .
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T E N S O R A N A L Y S I S
4 0 . E x p r e s s t h e r e l a t i o n s h i p b e t w e e n t h e a s s o c i a t e d t e n s o r s :
( a )
A j k l a n d
A 0 g r
,
( b )
a n d
A g k r
( c ) A .
q . .
a n d
A j q . s l
t
k
( a )
A j k l
=
g j P g k g g l r
A p q r
o r
A p q r =
g .
g k q g 1 r A
( b )
A ' k l
g j q g l r
A g k r
o r
A g k r =
g j g 9 1 r A ' k l
j k l
P . r s .
j r k
. . . s l
. . . s l
t i
5 . r s .
o r
A
=
g
g
g
c ) A
9
A
q . . t
j g k
p j
l
r k 9
. q . . t ,
g k
1 9 1
4 1 . P r o v e t h a t t h e a n g l e s 8 1 2 , 0 2 3 a n d 6 3 1 b e t w e e n t h e c o o r d i n a t e c u r v e s i n a t h r e e d i m e n s i o n a l c o -
o r d i n a t e s y s t e m a r e g i v e n b y
c o s 6
= g 1 2
c o s 6
=
9 2 3
c o s 6
=
9 3 1
1 2
1
g 3
V " g 1 1 g 2 2
g
- -
2
3 3
A l o n g t h e x 1 c o o r d i n a t e c u r v e , x 2 = c o n s t a n t a n d x 3 = c o n s t a n t .
T h e n f r o m t h e m e t r i c f o r m ,
d s 2 = g 1 1 ( d x i 2
o r
d x 1
=
1
d s
7 9 1 ,
T h u s a u n i t t a n g e n t v e c t o r a l o n g t h e x 1 c u r v e i s A l =
6 1 . S i m i l a r l y , u n i t t a n g e n t v e c t o r s a l o n g
t h e x 2 a n d x 3 c o o r d i n a t e c u r v e s a r e A r -
a n d A r =
b r
2 -
; 7 1 8 2
3 - " r
3
3 3
T h e c o s i n e o f t h e a n g l e 6 1 2 b e t w e e n A l a n d A 2 i s g i v e n b y
p q
1
1
q
g 1 2
c o s 6 1 2 =
g p q A l A 2
=
p q V I ' g - 1 1
V I - g ;
8 1 2
g 1 1 g 2 2
S i m i l a r l y w e o b t a i n t h e o t h e r r e s u l t s .
4 2 . P r o v e t h a t f o r a n o r t h o g o n a l c o o r d i n a t e s y s t e m , g 1 2
= g 2 3 = g 3 1 = 0 .
T h i s f o l l o w s a t o n c e f r o m P r o b l e m 4 1 b y p l a c i n g 6
1
2 = 0 1 = 9 6 0 .
6
F r o m t h e f a c t t h a t 9 1 q = g q j
i t a l s o f o l l o w s t h a t
g 2 1 = g 3 2 = g 1 3 = D .
4 3 . P r o v e t h a t f o r a n o r t h o g o n a l c o o r d i n a t e s y s t e m , g 1 1 =
g 1 1 '
9 2 2
9 2 2 '
9 3 3
g 3 3
F r o m P r o b l e m 3 3 , g P r 9
r q
b q .
I f p = q = 1 ,
g 1 r 9 r 1 = 1
o r
T h e n u s i n g P r o b l e m 4 2 ,
9 1 1
1 1
9
9 1 1
+ g 1 2 g 2 1
+ 9 1 3 9 3 1
1
=
g 1 1 .
1 .
S i m i l a r l y i f p = q = 2 , g =
g 1
; a n d i f p = q = 3 , 9 3 8 = 1 3
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1 9 2
C H R I S T O F F E L ' S S Y M B O L S .
4 4 . P r o v e ( a ) [ p q , r ] _ [ q p , r ] ,
( a ) [ p q , r ]
_
T E N S O R A N A L Y S I S
( b )
P q
=
4 P
,
( c ) [ p q , r ] = g r s { ; q }
i ( a g P r +
a g g r
_
i a g g r +
. . _ P
-
a g g p )
_
[ q P , r ] .
a x q a x p
a x r
a x p a x q
a x r
( b )
p q
= g " [ P q , r ] =
g s r [ q p , r ]
=
f S
q P
( c )
g k s P q
=
g k s g s r [ p q , r ]
=
8 k [ p q , r ] = [ P q , k ]
s
o r
[ P q , k ] = g k s
p q
r s
p q , r ]
g
P q
i . e .
N o t e t h a t m u l t i p l y i n g
[ p q , r ]
b y g s r h a s t h e e f f e c t o f r e p l a c i n g r b y s , r a i s i n g t h i s i n d e x a n d r e -
p l a c i n g s q u a r e b r a c k e t s b y b r a c e s t o y i e l d
P q
{ ; }
.
S i m i l a r l y , m u l t i p l y i n g
F q
s
b y g r s o r g s r h a s t h e
e f f e c t o f r e p l a c i n g s b y r , l o w e r i n g t h i s i n d e x a n d r e p l a c i n g b r a c e s b y s q u a r e b r a c k e t s t o y i e l d [ p q , r ] .
4 5 . P r o v e ( a )
a g p q
= [ P m , q ] +
a x
[ q m , P ]
P n
( b )
a g t
=
q
l n V
c )
P
= a
g
q n p
a x m
m a
g
m n
( a ) ]
=
] +
[
m
2
a g
( a g + a g m
_
a g g r o ) =
a g m g
_ a g P K ) +
a g f i g
, pq
q
a x m
Z
a x p
a x q
a x m a x q
a x p
' a x " '
k
J k
b )
g i j )
a x m
( b i )
x m ( g
=
0 .
T h e n
p k a g J
+
a g
. .
a x
a x i
g i
a x m
- 9
t i r g i k ( [ i r n , j ]
+ [ j m , i ] )
_ i t
k
j k
r
g
i m
g
j m
a n d t h e r e s u l t f o l l o w s o n r e p l a c i n g r , k , i , j b y p , q , n , n r e s p e c t i v e l y .
0
o r
j
k
j k
a g i
a g
g i
a x m
g
a x m
i r
. .
a g ' k
i t g j k a g t i j
g g 2 J
a x m $ a x m
- 1
k
8 r 1 9 -
=
j a x
a g r k
( c ) F r o m P r o b l e m 3 1 , g = g j k G ( j , k )
( s u m o v e r k o n l y ) .
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T E N S O R A N A L Y S I S
a g .
a g .
a g
=
a g
r =
G ( j , r )
j r
a x ' s
a g x m
a x l n
j r
a g .
= 9 9
a r
a z j
= 9 9
j r
[ r m , j ] )
g
T h u s
i a g
2 g a x q
j m
T h e r e s u l t f o l l o w s o n r e p l a c i n g j b y p a n d m b y q .
1 9 3
4 6 . D e r i v e t r a n s f o r m a t i o n l a w s f o r t h e C h r i s t o f f e l s y m b o l s o f ( a ) t h e f i r s t k i n d , ( b ) t h e s e c o n d k i n d .
( a ) S i n c e
( 1 )
( 2 )
( 3 )
S i n c e G ( j , k ) d o e s n o t c o n t a i n g j k e x p l i c i t l y ,
a
= G ( j , r ) .
T h e n , s u m m i n g o v e r j a n d r ,
9 j - r
a x p - 6 . q
g j k
' 2
a z k a x k
g p q
a g j k
=
a x p a x q
a g p q
a x r
+
a x j '
a 2 x q g
+
a 2 x j ' a x q
a x q
a x j a ` x k a x r a x j a x k a x q a x k p q
a x q a z j a x k
g p 9
B y c y c l i c p e r m u t a t i o n o f i n d i c e s j , k , m a n d p , q , r
a g k l n
= a x q a x r a g q r a x p
+
a x q
a 2 x r
a x q a x k a x m a x q a z j
a x k a x q a x q
-
X 1
a x k
, a x - r a x k a g r j i a x q
+
a x r
a 2 x P
a ' m a z j a x q a x k
a x q ' a r k a x j
q r
a 2 x q
a x r
a x k a z k a x 1 a
'
q r
a 2 x r
a x 7 5
g
a z k - 3 x 1 4 a ' x j
r r
S u b t r a c t i n g ( 1 ) f r o m t h e s u m o f ( 2 ) a n d ( 3 ) a n d m u l t i p l y i n g b y 2 , w e o b t a i n o n u s i n g t h e d e f i n i t i o n
o f t h e C h r i s t o f f e l s y m b o l s o f t h e f i r s t k i n d ,
( 4 )
a x a x q a x r
a 2 X P
a x q
' 3 V a x k - a - %
a x j a x k a z j
g p q
( b ) M u l t i p l y ( 4 ) b y
- n m =
a x n a z
a x s a x t
t o o b t a i n
_
a x , a x q a x r a x - n a z j '
s t
a 2 x q
a x q a x n a z j
s t
n m [
a x j a x k a x m a x s a x t
g
[ p q , r ] +
a z j a x ' k a z m a x s a x t
g
g p q
8/19/2019 M. R. Spiegel, Vector Analysis, Schaum's Series
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1 9 4
T h e n
j n
j k
}
T E N S O R A N A L Y S I S
a x p a x q a z n
r
g
s t
7 2 x 0
a z n
q S t
a z n a x k a x s
t
[ P q . r ]
- 6 1 i a x k a x s
b g
g p g
a x p a x q a z n j s
+
7 2 x 1 '
a z n
a x k a x k a x s
p q
a x k a z k a x p
4 7 . P r o v e
s i n c e
s
g s t
[ p q , r ]
7 2 x 2
a x q a x k
-
F r o m P r o b l e m 4 6 ( b ) ,
g s r [ p q , r ]
n
j k
n
j k
s
s t
s
1 ' q
a n d b g
g p q
= g
g g p q
=
b
s
P
.
a x ' s
a x p a x Q
m
a x n ` a x k a x k
p q
}
=
a x p a x q a x n
s
a x k a x k a x s
P q
+
a 2 x p
a x n
- a y j a x k a x p .
M u l t i p l y i n g b y
a x n
k
a x
=
a x p a x k
S S
s
a x
1
a x
d x j a x
P q
_
a x p a x q ` m
a z n a x k
P q
2
S o l v i n g f o r a x
,
t h e r e s u l t f o l l o w s .
a x q a x k
+
a 2 x p
a x k a z k
+
a 2
x
a x a x k
4 8 . E v a l u a t e t h e C h r i s t o f f e l s y m b o l s o f ( a ) t h e f i r s t k i n d , ( b ) t h e s e c o n d k i n d , f o r s p a c e s w h e r e
g 1 ' 9 = 0 i f p , - q .
( a )
I f P = q = r .
[ P q r ]
I f P = q t r .
I f P = r X q ,
[ P q , r ]
[ p p , r ]
1
` e g o
+
a g p p
a g p p
2
a x p a x p
a x p
1
( a g p r
+
a g p r _
N g p p )
2
a x p
a x p
a x ' _
1 a g p p
2 a x r
[ p q . p ] = 1
( ' 9 p p
+
a g g p _
a g p g
_
1
a g p p
2
a x q a x p a x p
2 a x q
I f p , q , r a r e d i s t i n c t ,
[ p q , r ] = 0 .
W e h a v e n o t u s e d t h e s u m m a t i o n c o n v e n t i o n h e r e .
( b ) B y P r o b l e m 4 3 , g i i = g
( n o t s u m m e d ) .
T h e n
1 J
p q
=
g s r [ p q , r ]
= 0 i f r # s ,
a n d = g s s [ p q , s ]
_ [
g q ,
s ] ( n o t s u m m e d ) i f
r = s .
s s
B y ( a ) :
8/19/2019 M. R. Spiegel, Vector Analysis, Schaum's Series
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T E N S O R A N A L Y S I S
s
_ P
=
[ P P ' P ]
_
1
a g p p
I n g
P q
=
P P
g p p -
2 g p p
a x p
a x p
p p
s
s
[ P P s ]
1
a g p p
P q
P P
g s s 2 g s s a x s
S
=
P
[ p q , p ]
=
1
a g p p
a
I n g
P q
P q
g p p
2 g p p a x q
2 a x g
p p
I f p , q , s a r e d i s t i n c t ,
j s
P q
4 9 . D e t e r m i n e t h e C h r i s t o f f e l s y m b o l s o f t h e s e c o n d k i n d i n ( a ) r e c t a n g u l a r , ( b ) c y l i n d r i c a l , a n d
( c ) s p h e r i c a l c o o r d i n a t e s .
W e c a n u s e t h e r e s u l t s o f P r o b l e m 4 8 , s i n c e f o r o r t h o g o n a l c o o r d i n a t e s g p q = 0 i f p
q .
( a ) I n r e c t a n g u l a r c o o r d i n a t e s , g p p = 1 s o t h a t
i P q
= 0 .
( b ) I n c y l i n d r i c a l c o o r d i n a t e s , x 1 = p , x 2 = 0 , x 3 = z , w e h a v e b y P r o b l e m 3 0 ( a ) ,
g 1 1 =
1 , g 2 2 = p 2 , g 3 3 = 1 .
T h e o n l y n o n - z e r o C h r i s t o f f e l s y m b o l s o f t h e s e c o n d k i n d c a n o c c u r w h e r e p = 2 .
T h e s e a r e
1
1
2 2
2 g 1 1
( C )
T h e o n l y n o n - z e r o C h r i s t o f f e l s y m b o l s o f t h e s e c o n d k i n d c a n o c c u r w h e r e p = 2 o r 3 .
T h e s e
= 0 .
- a a
g 2 2
a x ,
-
2 ( p 2 )
p
2 2
g 2 2
1
a
2
( p
)
-
1
2 1
1 2
2 8 2 , a x l
2 ) 0 2 a p
p
I n s p h e r i c a l c o o r d i n a t e s , x 1 = r , x 2 = 6 , x 3 = 0 , w e h a v e b y P r o b . 3 0 ( b ) , g 1 1 = 1 ,
g 2 2 = r 2 , g , = r 2
s i n 2 6 .
1
2 2
1
g 2 2
2 g 1 1
a x 1
=
2
_ 2
_
1
2 1 1 2
2 g 2 2
2 8 2 2 a x e
=
2
d r
2 r 2 8 ( r 2 s i n
2 0 )
- s i n 6 c o s 0
3 3
_ _
1
a g
1
a
1
-
( r 2 s i n 2 6 )
_
3 1
1 3
2 8 3 3
a x I
2 r 2 s i n 2 6 a r
r
3
1 3
1
a g 3 3
_ 1 a
2
2
( r s i n 6 )
=
c o t 6
3 2 2 3
2 8 3 3 a x e
2 r 2 s i n 2 6 a 6
-
2
a ( r 2 )
_
- r
2
_
1
a ( r 2 )
=
1
a x 1
2 r 2 a r
r
1
a
( r 2 s i n 2 6 )
r s i n e 6
a r e
8/19/2019 M. R. Spiegel, Vector Analysis, Schaum's Series
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1 9 6
T E N S O R A N A L Y S I S
G E O D E S I C S .
t
2 F ( t , x , z ) d t b e a n e x t r e m u m ( m a x i m u m o r m i n -0 . P r o v e t h a t a n e c e s s a r y c o n d i t i o n t h a t I = f t
1
i m u m ) i s t h a t
a F
_ d U
a x
d t
( a x )
L e t t h e c u r v e w h i c h m a k e s I a n e x t r e m u m b e x = X ( t ) , t 1 < t < t 2 .
T h e n x = X ( t ) + E 7 ] ( t ) , w h e r e E i s
i n d e p e n d e n t o f t , i s a n e i g h b o r i n g c u r v e t h r o u g h t 1 a n d t 2 s o t h a t
7 7 ( t 1 ) = 7 ) ( t 2 ) = 0 .
T h e v a l u e o f I f o r t h e
n e i g h b o r i n g c u r v e i s
1 ( E )
= f t
t 2 F ( t , X + E 7 7 , X + E 7 ] ) d t
1
T h i s i s a n e x t r e m u m f o r E = 0 . A n e c e s s a r y c o n d i t i o n t h a t t h i s b e s o i s t h a t
d l I
= 0 .
B u t b y d i f f e r -
e n t i a t i o n u n d e r t h e i n t e g r a l s i g n , a s s u m i n g t h i s v a l i d ,
a = o
( e ?
+
-
7 7 ) d t
0
d E
f t
E = 1
w h i c h c a n b e w r i t t e n a s
J t 2
7 1 d t
1
a x
t 2
t 2 , 7 7 d
+ a z
t 1
f
d t
( a x )
d t
=
1
S i n c e 7 7 i s a r b i t r a r y , t h e i n t e g r a n d
a F
_
d ( - F )
= 0 .
a x
d t
a x
t
2
f t
t 2
1
7 7
a F
d
a F
d t
a x
d t
( a x )
F ( t , x 1 , z 1 , X 2 , 1 2 , . . . , x X a c ' ) d t
h e r e s u l t i s e a s i l y e x t e n d e d t o t h e i n t e g r a l
J
t
a n d y i e l d s
1
a F
d ( a F )
_
a x k
d t a z k
0
c a l l e d E u l e r ' s o r L a g r a n g e ' s e q u a t i o n s . ( S e e a l s o P r o b l e m 7 3 . )
0
2 r
p
q
5 1 . S h o w t h a t t h e g e o d e s i c s i n a R i e m a n n i a n s p a c e a r e g i v e n b y
d d
s 2 +
r ) d x
d
=
d s
d s
0
p q
r t 2
W e m u s t d e t e r m i n e t h e e x t r e m u m o f g
. 1 p
x
d t u s i n g E u l e r ' s e q u a t i o n s ( P r o b l e m 5 0 ) w i t h
F = g p q z p z q .
W e h a v e
t 1
p q
a F
=
1 ( g
z p z q ) - 1 / 2
a g p q
z p z q
a x k
2
p q
a x k
a z
2
( g p q
x p x q ) - 1 / 2
2 g P k
U s i n g
d t
= g p g
z p z q
, E u l e r ' s e q u a t i o n s c a n b e w r i t t e n
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T E N S O R A N A L Y S I S
. p
g x
d
d t
S
o r
W r i t i n g
a g p k
a x q
g p k
x p
z p
+
a g p k
x p z q
a x q
a
a
1
- 3 9 p q
x p z q
=
0
2 s
- a x k
a g p q p g p k z
s
q
=
x
2
a x k
t h i s e q u a t i o n b e c o m e s
g p k x
. p . .
g k x s
z p q
[ p q ,
x =
S
s
I f w e u s e a r e l e n g t h a s p a r a m e t e r , 1 , I S ' = 0 a n d t h e e q u a t i o n b e c o m e s
g
a d _ x - 0
+
[ p q , k ]
d x p d x q
= 0
p k d s 2
d s
d s
M u l t i p l y i n g b y g r k , w e o b t a i n
d
2
x
r
+ r
d x p d x q
d s 2
P q
d s d s
T H E C O V A R I A N T D E R I V A T I V E
.
a A
5 2 . I f A p a n d
A P a r e t e n s o r s s h o w t h a t ( a )
a x q
s
A s
a n d ( b ) A p q = a
- p
+
4 S
A s a r e t e n s o r s .
( a ) S i n c e A .
=
x
A r
a - x I
a A
a x r
a A r
a x t
( 1 )
_
+
F r o m P r o b l e m 4 7 ,
a x k
- a x a x t - a x ,
a 2 x r
a x j a z k
n
j k
S u b s t i t u t i n g i n ( 1 ) ,
A .
a x r a x t
a A r
g p k
+
q = 1 (
g g k ) z p x q
2
a x q a x p
a x k
2
a x r
a z k a x k
A r
a x r
_ a x z a x l
r
a z n
a z j a x k ( i t
n
j k
a r
A
}
j
a x n
r
a x a x
z j a z k a t
a x p a x q a A p
a x p a z k a x q
n
a x p a x q
s
1 k
A n
w a x k a z k
P q
A s
o r
a A j
n
a x p a x q
a A p
_ _
A n
a x k
1 k
a x
a z k
a x g
a x i a x l
r
A .
1 9 7
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1 9 8
T E N S O R A N A L Y S I S
a n d
n d
-
s
A
i s a c o v a r i a n t t e n s o r o f s e c o n d r a n k , c a l l e d t h e c o v a r i a n t d e r i v a t i v e o f A w i t h
a x q
p q
s
r e s p e c t t o x q a n d w r i t t e n
( b ) S i n c e A j =
a x J
r
A r ,
a x
( 2 )
a A J
=
a z j a A r a x t
+
a 7 1
a x t A r
a x k
a x r a x t o x k
a x r o x t a x k
F r o m P r o b l e m 4 7 , i n t e r c h a n g i n g x a n d x c o o r d i n a t e s ,
S u b s t i t u t i n g i n ( 2 ) ,
a A
J
_
a x ` s a x t a A r
a x k a x r a z k a x t
a x k a x t a A r
a x r a x k a x t
a i d a x q a A P
a x p a x k a x q
o r
n
a z j
r t
a x n
a n t i o x l
j
a x r a x t
i l
+
n
a x k a x t A r
r t
a x n a x k
+ n a x j a x t
A r
r t
a x n a x k
s
+ { p } i ,
q a x k a x k
5 3 . W r i t e t h e c o v a r i a n t d e r i v a t i v e w i t h r e s p e c t t o x q o f e a c h o f t h e f o l l o w i n g t e n s o r s :
( a ) A j k , ( b ) A J k , ( c ) A k , ( d ) A k l ,
( a ) A j k , q
( b ) A
q
a x q
j
s
A s k -
1 9
/ A
s k +
q s
( c ) A j - a A k
- s
A j
k , q
a x q k q }
s
( d )
A j
_ a A k l
_ { : q } 4 i
k l , q
a x q
A
J S
a n d
a A q
+
p
A
s
i s a m i x e d t e n s o r o f s e c o n d r a n k , c a l l e d t h e c o v a r i a n t d e r i v a t i v e o f A " w i t h
s
r e s p e c t t o x q a n d w r i t t e n A , q .
A j s
a A k
{ - T } A
i
=
a z a x
( a A
+
p
A s
a x
k i
a x p a x a x q q s
- 6 . 4 1 k
a x q
j k
a A k
S
k q
{
k
q s
I I
q s
s
l q
a x e o x l o x t
1
a x r a x t a x k
i t
' a 7
r
a x r
k t i t
}
A r
A s
k
A i
+ 1 j
k s
q s
}
S
k l
A
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T E N S O R A N A L Y S I S
1 9 9
j k l
j k l
a A m n
( e ) A m n , q
a x q
s
A j k l
_
s
A j k l +
j
A s k l
+ j
k
A j s l +
I
A j k s
m q
s n
n q m s
q s
m n
q s
s n
q s m n
5 4 . P r o v e t h a t t h e c o v a r i a n t d e r i v a t i v e s o f ( a )
( b ) g j k , ( c ) 8 k a r e z e r o .
( a )
g 1 k , q
a g j k s
s
a x q
j q J g s k -
k q
i s
a g e
9
a x q
( b )
g j k
q
( c ) 8 k , q
-
[ j q , k ]
-
[ k q , j ] = 0 b y P r o b l e m 4 5 ( a ) .
a g j k
+
{ / } g s k
q s
x q
k
i s
+
q s
g
a b k
_
s
j
+
a x q
k { q } 6 s
=
0
b y P r o b l e m 4 5 ( b ) .
0
5 5 . F i n d t h e c o v a r i a n t d e r i v a t i v e o f
A k B u m w i t h r e s p e c t t o x q .
I
I n
( A k B n ) q
I
I s
a ( A k B n )
_
s A j B l n m
_
s
A j
B l m
n q
k s
q s
q
x
+
A s B l m
+ l
A j B s m +
r n
A j B l s
q s
k n
q s
k n
q s
k n
+
A k
I
s
A j
k q
s
+
j
, 4 s
B l n m
q s
k
a B l m
n
_
s
B l m s
+
l
B s m
a x q
n q q s
n
=
A
B l i n
+
A I B l i n
k , q n
k
n , q
+ m
B l s
q s
n
T h i s i l l u s t r a t e s t h e f a c t t h a t t h e c o v a r i a n t d e r i v a t i v e s o f a p r o d u c t o f t e n s o r s o b e y r u l e s l i k e t h o s e
o f o r d i n a r y d e r i v a t i v e s o f p r o d u c t s i n e l e m e n t a r y c a l c u l u s .
5 6 . P r o v e
k m
( g j k A n ) , q
_
g j k
A k i n
n q
k m
k m
(
j k
A n m
) , q
s i n c e g j k
q
= 0 b y P r o b . 5 4 ( a ) .
J
g k , q ` 4
+ g k A n , q
k A n , q
I n c o v a r i a n t d i f f e r e n t i a t i o n , g j k , g s k a n d
b j c a n b e t r e a t e d a s c o n s t a n t s .
k m
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2 0 0
A P
=
' 3 A
A
a x k
p k
G R A D I E N T , D I V E R G E N C E A N D C U R L I N T E N S O R F O R M .
5 7 . P r o v e t h a t
d i v
A P
=
1 a ( / A " ) .
a x k
T h e d i v e r g e n c e o f
A P i s t h e c o n t r a c t i o n o f t h e c o v a r i a n t d e r i v a t i v e o f A P , i . e . t h e c o n t r a c t i o n o f
A 0 , q
o r
A 0 , p .
T h e n , u s i n g P r o b l e m 4 5 ( c ) ,
d i v
A 0
T E N S O R A N A L Y S I S
a A k
a
k
a A k
1 - a v g -
k
1
a
k
a
}
A
x k
+
I n ) A
=
k
+
(
,
a x k
- ) A
=
, a x k
x k
a x
5 8 . P r o v e t h a t
V 2 < p
=
1
a
V " g g k r
a x k
a x
T h e g r a d i e n t o f ( D i s
g r a d
_ V 4 ) = a x r
a c o v a r i a n t t e n s o r o f r a n k o n e ( s e e P r o b l e m 6 ( b ) ) d e -
f i n e d a s t h e c o v a r i a n t d e r i v a t i v e o f ( 1 ) , w r i t t e n ( D , , r . T h e c o n t r a v a r i a n t t e n s o r o f r a n k o n e a s s o c i a t e d w i t h
k
= g k r
a
r
i s A
a x r
. T h e n f r o m P r o b l e m 5 7 ,
d i v ( g k r
)
a x
5 9 . P r o v e t h a t
A p p q - A q , p
A p , q
A q . j , =
a A P
a A q
L A ,
_
s
A
a x q
p q S
T h i s t e n s o r o f r a n k t w o i s d e f i n e d t o b e t h e c u r l o f A P .
a k (
V ' g
- g k r
a x
a x
L A
( _ { : } A S )
a x q
r a x p
6 0 . E x p r e s s t h e d i v e r g e n c e o f a v e c t o r
A P i n t e r m s o f i t s p h y s i c a l c o m p o n e n t s f o r ( a ) c y l i n d r i c a l ,
( b ) s p h e r i c a l c o o r d i n a t e s .
( a ) , F o r c y l i n d r i c a l c o o r d i n a t e s x 1 = P . x 2 = 0 , x 3 = z ,
g
1
0
0
0 p 2 0
0 0
1
p 2
a n d ' = p
( s e e P r o b l e m 3 0 ( a ) )
T h e p h y s i c a l c o m p o n e n t s , d e n o t e d b y A . A o , A . a r e g i v e n b y
A p = V g _ A l = A l .
A =
g - A 2 = p A 2 ,
A z = V A 3 = A 3
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T E N S O R A N A L Y S I S
T h e n
( b ) F o r s p h e r i c a l c o o r d i n a t e s
g
1 0
0
0
r 2
0
0
r 4 s i n 2 8 a n d
I r g = r 2 s i n 8
( s e e P r o b l e m 3 0 ( b ) )
d i v
A 0
_
a x k
( 4 A k )
0
r 2 s i n 2 8
T h e p h y s i c a l c o m p o n e n t s , d e n o t e d b y A r , A 8 , A 4 ) a r e g i v e n b y
A r =
u A 1 = A I
,
T h e n
A e = V - g - A 2 = r A 2 ,
A O =
g 3 3 A 3 = r s i n 8 A 3
2 2
d i v A
=
1
- 6
A k )
- v l g
a x k
( v
( r s i n 8 A B ) +
( r A , ) ]
r 2 s i n e
- 6 r
( r 2 s i n 8 A r ) +
- 6 6
1
a ( r 2 A r )
+
1
a
( s i n 8 A ) +
1
r 2 a r
r s i n e a 8
e
r s i n e
6 1 . E x p r e s s t h e L a p l a c i a n o f 4 > , V 2 c
, i n ( a ) c y l i n d r i c a l c o o r d i n a t e s , ( b ) s p h e r i c a l c o o r d i n a t e s .
( a ) I n c y l i n d r i c a l c o o r d i n a t e s g 1 1 = 1 , g 2 2 = 1 / P 2 , g = 1 ( s e e P r o b l e m 3 5 ( a ) ) . T h e n f r o m P r o b l e m 5 8 ,
v 2 4 5
( , r g - g k r
V g - -
a x k a
l
a a
a 1 a
a a
P C a P c P a P ) + a o c P
a z
c P a z ) ]
1 a
a ( f )
1 D 2
A )
P a P c P a P ' +
p 2
2 2
+
a z 2
( b ) I n s p h e r i c a l c o o r d i n a t e s g 1 1 = 1 , g 2 2 = 1 / r 2 , g
= 1 / r 2 s i n 2 8 ( s e e P r o b l e m 3 5 ( b ) ) . T h e n
v % )
_
7 a x k ' ' g k r a
1
a 2
M )
a
a
1
a 4 )
r 2 s i n
8
- 6 r
( r s i n 8
a r ) + a e ( S i n
8
a e ) + V ( s i n 4 ]
x 1 = r , x 2 = 8 , x 3 = 0 ,
2 0 1
t
a r a
+
1
a
B
)
+
1
a 2
r 2 a r ( 2 r '
r 2 s i n e a e
( s i n
a 8 2 s i n 2 8
- 4 2
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2 0 2
T E N S O R A N A L Y S I S
I N T R I N S I C D E R I V A T I V E S .
6 2 . C a l c u l a t e t h e i n t r i n s i c d e r i v a t i v e s o f e a c h o f t h e f o l l o w i n g t e n s o r s , a s s u m e d t o b e d i f f e r e n t i a b l e
f u n c t i o n s o f t : ( a ) a n i n v a r i a n t ( l ) , ( b ) A i , ( c ) A k
,
( d ) A l k n .
4 d
q
a
q
d 4 )
t h
d i
d
i t i
( a ) b
, q =
t
a x
d t
=
,
e o r
n a r y
v a
r
v e .
( b )
8 A J
= A
d x q
a A i +
1
A s
d x q
=
a A i d x q +
1
A s d x q
8 t
q d t
a x q
q s
d t
a x q
d t
q s
d t
( c )
S A k
S t
A j
d x q
k , q
d t
d A I
+
1
A s d x q
d t
q s
d t
{ ; q } 4 + { } A : )
q
d t
d A k j
A j
d z
+
A s d x q
d t
k q
s d t
q s
k d t
j k
j k
j k
S
.
A l m n
,
j k
d x q
l a A l n n
s
A
l q
n
m q
l s n
s
A j k +
{ i }
A s k +
k A j s
n q
I n s
q s
I n n
q s
l m n
d A A
d x q
d t
l m n
I s
A j k
d x q _ s A j k d x q
s A j k
_
x
d t l q s 1 n n d t
l m q f l s n
d t
n q
i n s
d t
+ j A s k
d x q +
£ k A J S d x q
q s
l m n d t
q s
I n n
d t
6 3 . P r o v e t h e i n t r i n s i c d e r i v a t i v e s o f g j k ,
g j k a n d
a r e
r e z e r o .
8 9 j k
= ( g
k
)
d x q
= 0 ,
8 t
, q
d t
R E L A T I V E T E N S O R S .
8 g l k
=
g j k d x q
= 0 ,
S t ' q d t
S b k
- 8
d x q
= 0
b t
k , q
d t
b y P r o b l e m 5 4 .
6 4 . I f A q a n d B t s a r e r e l a t i v e t e n s o r s o f w e i g h t s w 1 a n d w 2 r e s p e c t i v e l y , s h o w t h a t t h e i r i n n e r a n d
o u t e r p r o d u c t s a r e r e l a t i v e t e n s o r s o f w e i g h t w 1 +
w 2 -
B y h y p o t h e s i s ,
A
k
j w 1 a z a x q A
T h e o u t e r p r o d u c t i s
A k B n m
B
l m _
j w 2 1 3 F - ' a z m a x t B r s
a x P a x k
q ' n
a x r a x s a x n
t
j w , w 2 a x q a x q a x l a x m a x t r s
a x p a z k a x r a x s a z n
A q B t
q
a r e l a t i v e t e n s o r o f w e i g h t w 1 + w 2 .
A n y i n n e r p r o d u c t , w h i c h i s a c o n t r a c t i o n o f t h e o u t e r p r o d u c t , i s a l s o
a r e l a t i v e t e n s o r o f w e i g h t w 1 + w 2 .
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T E N S O R A N A L Y S I S
2 0 3
6 5 . P r o v e t h a t V g - i s a r e l a t i v e t e n s o r o f w e i g h t o n e , i . e . a t e n s o r d e n s i t y .
T h e e l e m e n t s o f d e t e r m i n a n t g g i v e n b y g
t r a n s f o r m a c c o r d i n g t o g j
=
- 3 . p - 3 . . q
g
q
k
a x e a x k o q
T a k i n g d e t e r m i n a n t s o f b o t h s i d e s , g =
a x e
a x e
g =
J 2 g
o r
J V ,
w h i c h s h o w s
a z j
- 3 - x
6 6 . P r o v e t h a t d V = V r g - A l d x 2 . . . d x 1 i s a n i n v a r i a n t .
B y P r o b l e m 6 5 ,
d V =
V g - d x 1 d x
. . .
d x y =
v r g - J d x 1 d x ` 2
. . .
d x ' A
_ v g
a x
d z 1 d x . . . d x N = d x 1 d x 2 . . . d x " = d V
a )
F r o m t h i s i t f o l l o w s t h a t i f
f . . . f d v
i s a n i n v a r i a n t , t h e n
=
f . . . f d v
V
V
f o r a n y c o o r d i n a t e s y s t e m s w h e r e t h e i n t e g r a t i o n i s p e r f o r m e d o v e r a v o l u m e i n N d i m e n s i o n a l s p a c e . A
s i m i l a r s t a t e m e n t c a n b e m a d e f o r s u r f a c e i n t e g r a l s .
M I S C E L L A N E O U S A P P L I C A T I O N S .
6 7 . E x p r e s s i n t e n s o r f o r m ( a ) t h e v e l o c i t y a n d ( b ) t h e a c c e l e r a t i o n o f a p a r t i c l e .
( a ) I f t h e p a r t i c l e m o v e s a l o n g a c u r v e x k = x k ( t ) w h e r e t i s t h e p a r a m e t e r t i m e , t h e n v k =
k
d t
i s i t s v e -
l o c i t y a n d i s a c o n t r a v a r i a n t t e n s o r o f r a n k o n e ( s e e P r o b l e m 9 ) .
k
2 k
( b ) T h e q u a n t i t y d t
= t 2
i s n o t i n g e n e r a l a t e n s o r a n d s o c a n n o t r e p r e s e n t t h e p h y s i c a l q u a n t i t y
a c c e l e r a t i o n i n a l l c o o r d i n a t e s y s t e m s . W e d e f i n e t h e a c c e l e r a t i o n a k a s t h e i n t r i n s i c d e r i v a t i v e o f
t h e v e l o c i t y , i . e . a k = S t k
w h i c h i s a c o n t r a v a r i a n t t e n s o r o f r a n k o n e .
6 8 . W r i t e N e w t o n ' s l a w i n t e n s o r f o r m .
A s s u m e t h e m a s s M o f t h e p a r t i c l e t o b e a n i n v a r i a n t i n d e p e n d e n t o f t i m e t .
T h e n M a k = F k a
c o n t r a v a r i a n t t e n s o r o f r a n k o n e i s c a l l e d t h e f o r c e o n t h e p a r t i c l e . T h u s N e w t o n ' s l a w c a n b e w r i t t e n
F k = M a k = M
S k k
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2 0 4
2
k
6 9 . P r o v e t h a t
a k =
6 v k = d x
S t
d t 2
T E N S O R A N A L Y S I S
k
d x 0 d x q
p q
d t
d t
'
S i n c e v k i s a c o n t r a v a r i a n t t e n s o r , w e h a v e b y P r o b l e m 6 2 ( b )
8 v k
=
d v k
S t
d t
F r o m P r o b l e m 6 7 ( a ) , t h e c o n t r a v a r i a n t c o m p o n e n t s o f t h e v e l o c i t y a r e
_
d
2
x
k
+ k
d x p d x q
d - i
p q
d t
d t
d
2
x
k
+
k
V P
d x q
d t 2
q p
d t
7 0 . F i n d t h e p h y s i c a l c o m p o n e n t s o f ( a ) t h e v e l o c i t y a n d ( b ) t h e a c c e l e r a t i o n o f a p a r t i c l e i n c y l i n -
d r i c a l c o o r d i n a t e s .
( a )
d x 1 d p d x 2 d o
d x 3
d t
d t
'
d t
d t
a n d
d t
d z
d t
T h e n t h e p h y s i c a l c o m p o n e n t s o f t h e v e l o c i t y a r e
v
d x 1
=
d p d x 2
d o
a n d
v
d x 3
=
d z
9 1 1
d t d t 9 2 2 d t
' d t
9 3 3
d t
d t
u s i n g
+
k
v s d x q
q s
d t
9 1 1 = 1 , g 2 2 = p 2 . g 3 3 = 1 .
( b ) F r o m P r o b l e m s 6 9 a n d 4 9 ( b ) , t h e c o n t r a v a r i a n t c o m p o n e n t s o f t h e a c c e l e r a t i o n a r e
l
d 2 x 1 +
d x 2 d x 2
d 2 p
d o
2
d t 2 2 2
_
d t
d t
d t 2
_ p (
d t
)
2
a
d 2 x 2
+
2
d x l d x 2
+
2
d x 2 d x l d o
+
2 d p d o
d t 2
1 2
d t
d t
2 1
d t
d t
d t 2
p d t d t
d 2 x 3
d 2 z
a n d
3
a
_
d t 2 d t 2
T h e n t h e p h y s i c a l c o m p o n e n t s o f t h e a c c e l e r a t i o n a r e
a l = 0 - - p
1 1
2 2 a 2
= p
+ 2 p q 5
w h e r e d o t s d e n o t e d i f f e r e n t i a t i o n s w i t h r e s p e c t t o t i m e .
3
a n d
3 3 a = z
7 1 . I f t h e k i n e t i c e n e r g y T o f a p a r t i c l e o f c o n s t a n t m a s s M m o v i n g w i t h v e l o c i t y h a v i n g m a g n i t u d e v
i s g i v e n b y T = 2 M v 2 = 2 M g p q 0 x q , p r o v e t h a t
d ( a T )
_
a T
=
M a
d t a x k
a x k
k
w h e r e a k d e n o t e s t h e c o v a r i a n t c o m p o n e n t s o f t h e a c c e l e r a t i o n .
S i n c e T =
2 M g p q z p z q , w e h a v e
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T E N S O R A N A L Y S I S
a T
a x k
T h e n
I
a g p q
z p z q ,
a T
= M g
z q
a n d
a x k ' 3 1 k
k q
d ( a T )
-
a T
=
M
( g k q z q
+
d t
a z k
a x k
a g k
q
a x k
+ j q , ,
( ' 3 T
=
M ( g k 9
z q
a g
d t a x
a x
z j z q
-
1
a g p q
z P z q
2
a x k
M
z q + 1
( a g k q
+
a g k p
- a g p q )
x p
( g k q
2
a x p a x q
a x k
z q
+ [ p q , k ]
x p x q )
( g k q
x q
=
M g
x r
+
r
z j z q
=
M g
a r
=
M a k
k r p q
k r
u s i n g P r o b l e m 6 9 . T h e r e s u l t c a n b e u s e d t o e x p r e s s t h e a c c e l e r a t i o n i n d i f f e r e n t c o o r d i n a t e s y s t e m s .
2 0 5
7 2 . U s e P r o b l e m 7 1 t o f i n d t h e p h y s i c a l c o m p o n e n t s o f t h e a c c e l e r a t i o n o f a p a r t i c l e i n c y l i n d r i c a l
c o o r d i n a t e s .
S i n c e
d s 2 = d p 2 + , 0 2 d c a 2 + d z 2 ,
v 2 = ( ) 2
=
2 +
z 2
a n d
T = 2 M v 2 =
e ( ; ; ?
+ Z 2 ) ,
F r o m P r o b l e m 7 1 w i t h x 1 = p , x 2 = 0 , x 3 = z
w e f i n d
a 1 = P
a 2 = d t
a 3 z
T h e n t h e p h y s i c a l c o m p o n e n t s a r e g i v e n b y
a ,
a 2
a g
1 V , g j - l
2 2
3 3
s i n c e
g 1 1 = 1 , g 2 2 = p 2 ,
g 3 3 = 1 . C o m p a r e w i t h P r o b l e m 7 0 .
7 3 . I f t h e c o v a r i a n t f o r c e a c t i n g o n a p a r t i c l e i s g i v e n b y F k = - a
k
w h e r e V ( x 1 . . . . . x j ' ) i s t h e
p o t e n t i a l e n e r g y , s h o w t h a t d t ( a L k ) - a L k = 0 w h e r e L = T - - V .
F r o m L = T - - V ,
a L
r
_
a T
s i n c e V i s i n d e p e n d e n t o f z k . T h e n f r o m P r o b l e m 7 1 ,
a x k
a x k
d
a T a T
_
d t
( a z k
a z k
M a k
=
F k
=
- -
a V
a n d
d ( a L ) -
a L
a x k d t
a z k
a x k
0
T h e f u n c t i o n L i s c a l l e d t h e L a g r a n g e a n . T h e e q u a t i o n s i n v o l v i n g L , c a l l e d L a g r a n g e ' s e q u a t i o n s ,
a r e i m p o r t a n t i n m e c h a n i c s . B y P r o b l e m 5 0 i t f o l l o w s t h a t t h e r e s u l t s o f t h i s p r o b l e m a r e e q u i v a l e n t t o t h e
s t a t e m e n t t h a t a p a r t i c l e m o v e s i n s u c h a w a y t h a t f L d t i s a n e x t r e m u m . T h i s i s c a l l e d H a m i l t o n ' s
t 1
p r i n c i p l e .
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2 0 6
T E N S O R A N A L Y S I S
7 4 . E x p r e s s t h e d i v e r g e n c e t h e o r e m i n t e n s o r f o r m .
L e t A k d e f i n e a t e n s o r f i e l d o f r a n k o n e a n d l e t v k d e n o t e t h e o u t w a r d d r a w n u n i t n o r m a l t o a n y p o i n t
o f a c l o s e d s u r f a c e S b o u n d i n g a v o l u m e V . T h e n t h e d i v e r g e n c e t h e o r e m s t a t e s t h a t
f f f A
k k d V
I I
V
S
A k v k d S
F o r N d i m e n s i o n a l s p a c e t h e t r i p l e i n t e g r a l i s r e p l a c e d b y a n N t u p l e i n t e g r a l , a n d t h e d o u b l e i n t e g r a l b y
a n N - 1 t u p l e i n t e g r a l .
T h e i n v a r i a n t A k k i s t h e d i v e r g e n c e o f A k ( s e e P r o b l e m 5 7 ) .
T h e i n v a r i a n t
A k v k i s t h e s c a l a r p r o d u c t o f A k a n d v k , a n a l o g o u s t o A
n i n t h e v e c t o r n o t a t i o n o f C h a p t e r 2 .
W e h a v e b e e n a b l e t o e x p r e s s t h e t h e o r e m i n t e n s o r f o r m ; h e n c e i t i s t r u e f o r a l l c o o r d i n a t e s y s t e m s
s i n c e i t i s t r u e f o r r e c t a n g u l a r s y s t e m s ( s e e C h a p t e r 6 ) . A l s o s e e P r o b l e m 6 6 .
7 5 . E x p r e s s M a x w e l l ' s e q u a t i o n s ( a ) d i v B = 0 , ( b ) d i v D = 4 7 r p , ( c ) V x E = -
a B ,
( d ) V x H = 4 Z f
i n t e n s o r f o r m .
D e f i n e t h e t e n s o r s
c a n b e w r i t t e n
( a ) B k k 0
( b ) D k k = 4 7 T p
( c )
- - -
E J k q E k , q
( d ) - E j k q
H k
, 4
B k , D k , E k , H k , 1 k a n d
s u p p o s e t h a t p a n d c a r e i n v a r i a n t s . T h e n t h e e q u a t i o n s
1 a B j
C
a t
4 7 r 1
J
c
o r
E J W q E k . q
o r
E j k q
H k ,
q
1 a B j
c
T h e s e e q u a t i o n s f o r m t h e b a s i s f o r e l e c t r o m a g n e t i c t h e o r y .
7 6 . ( a ) P r o v e t h a t
A l , g r
A 0 , r q =
R n g r A n w h e r e A 0 i s a n a r b i t r a r y c o v a r i a n t t e n s o r o f r a n k
o n e .
( b ) P r o v e t h a t R
q r
i s a t e n s o r .
( c ) P r o v e t h a t
R p g r s
g n s
R '
i s a t e n s o r .
( a ) A p , g r =
( A O . q ) r
a A M
-
1
A .
- { ' } A
x r
{ P r
J . q
q r
O 9
a A
J , _
i
i
( B A .
j
k
P S
l
A . -
-
{ } A k )
-
P q
P r
q r
a x q
P 1
a 2 A
A j
k
A .
. -
- -
+
A
; ; i q }
P q - a x ,
P r
a x q
P r
I q
k
1
a A P
+
1
l
A
q r
a x q
q r
P 1
B y i n t e r c h a n g i n g q a n d r a n d s u b t r a c t i n g , w e f i n d
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T E N S O R A N A L Y S I S
A p , q r
_
A
i k
p r q
p r
j q
A k
w h e r e
R q r
-
p
r
k q
A j
=
R p J g r
A
I
a
j
J A .
-
k
a x r
P 4
I
{ p J q } { J } A
r
k
a ;
A
x r
P q
9
a
_
k j
p q
k r
A i
=
{ : r } { q } _
- { : q } { L r } +
i
x r P 4
a x q
p r
R e p l a c e j b y n a n d t h e r e s u l t f o l l o w s .
2 0 7
( b ) S i n c e A p , g r - A p , r q i s a t e n s o r , R q r A n i s a t e n s o r ; a n d s i n c e A n i s a n a r b i t r a r y t e n s o r , R q r i s
a t e n s o r b y t h e q u o t i e n t l a w . T h i s t e n s o r i s c a l l e d t h e R i e m a n n - C h r i s t o f f e l t e n s o r , a n d i s s o m e t i m e s
n
n
n
w r i t t e n R , p q r , R p q r ,
o r s i m p l y R p q r
( c ) R p g r s = g n s R p g r i s a n a s s o c i a t e d t e n s o r o f R p g r a n d t h u s i s a t e n s o r .
I t i s c a l l e d t h e c o v a r i a n t
c u r v a t u r e t e n s o r a n d i s o f f u n d a m e n t a l i m p o r t a n c e i n E i n s t e i n ' s g e n e r a l t h e o r y o f r e l a t i v i t y .
S U P P L E M E N T A R Y P R O B L E M S
A n s w e r s t o t h e S u p p l e m e n t a r y P r o b l e m s a r e g i v e n a t t h e e n d o f t h i s C h a p t e r .
7 7 . W r i t e e a c h o f t h e f o l l o w i n g u s i n g t h e s u m m a t i o n c o n v e n t i o n .
( a ) a 1 x 1 x 3 + a 2 x 2 x 3 + . . . + a ) x N x 3
( b )
A 2 1
B 1 + A 2 2 B 2 + A ' B 3 +
+
` 4 2 I
B y
( e ) B 1 1 1 + B
1 2 2
1 2
( c ) A l B 1 + A 2 B 2
+ A 3 B 3 +
( d ) g 2 1
g 1 1 + g 2 2 g 2 1
+
8 2 2 1 + 8 2 2 2
2 1
2 2
7 8 . W r i t e t h e t e r m s i n e a c h o f t h e f o l l o w i n g i n d i c a t e d s u m s .
k
( a )
a x k ( i A k ) , N = 3
( b )
B p
C
, N = 2
( c )
a z , 7
a '
k
+ g 2 3 g
3 1
. . . +
A j
B N
+ g 2 4 g
4 1
7 9 . W h a t l o c u s i s r e p r e s e n t e d b y a k x k x k = 1 w h e r e z k , k = 1 , 2 , . . . , N a r e r e c t a n g u l a r c o o r d i n a t e s , a k a r e
p o s i t i v e c o n s t a n t s a n d N = 2 , 3 o r 4 ?
8 0 . I f N = 2 , w r i t e t h e s y s t e m o f e q u a t i o n s r e p r e s e n t e d b y a p q x q = b p .
k
8 1 . W r i t e t h e l a w o f t r a n s f o r m a t i o n f o r t h e t e n s o r s ( a ) A k ,
( b ) B ,
( c ) C a n ,
( d ) A n .
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2 0 8
T E N S O R A N A L Y S I S
8 2 . D e t e r m i n e w h e t h e r t h e q u a n t i t i e s B ( j , k , m ) a n d C ( j , k , m , n ) w h i c h t r a n s f o r m f r o m a c o o r d i n a t e s y s t e m
x t i t o a n o t h e r x t i a c c o r d i n g t o t h e r u l e s
k a x r
a x p a x q a x q a x s
( a ) B ( p . q , r ) =
a x
a x
B ( j , k , m )
( b ) C ( p , q , r . s ) =
C ( j , k , m , n )
a x p a x q a x x
a x q a z k a z r a x n
a r e t e n s o r s . I f s o , w r i t e t h e t e n s o r s i n s u i t a b l e n o t a t i o n a n d g i v e t h e r a n k a n d t h e c o v a r i a n t a n d c o n t r a -
v a r i a n t o r d e r s .
8 3 . H o w m a n y c o m p o n e n t s d o e s a t e n s o r o f r a n k 5 h a v e i n a s p a c e o f 4 d i m e n s i o n s ?
8 4 . P r o v e t h a t i f t h e c o m p o n e n t s o f a t e n s o r a r e z e r o i n o n e c o o r d i n a t e s y s t e m t h e y a r e z e r o i n a l l c o o r d i n a t e
s y s t e m s .
8 5 . P r o v e t h a t i f t h e c o m p o n e n t s o f t w o t e n s o r s a r e e q u a l i n o n e c o o r d i n a t e s y s t e m t h e y a r e e q u a l i n a l l c o -
o r d i n a t e s y s t e m s .
k
k
8 6 . S h o w t h a t t h e v e l o c i t y t
=
v k o f
a f l u i d i s a t e n s o r , b u t t h a t v i s n o t a t e n s o r .
8 7 . F i n d t h e c o v a r i a n t a n d c o n t r a v a r i a n t c o m p o n e n t s o f a t e n s o r i n
( a ) c y l i n d r i c a l c o o r d i n a t e s p , 0 , z ,
( b ) s p h e r i c a l c o o r d i n a t e s r , 6 ,
i f i t s c o v a r i a n t c o m p o n e n t s i n r e c t a n g u l a r c o o r d i n a t e s a r e 2 x - - z , x 2 y ,
y z .
8 8 . T h e c o n t r a v a r i a n t c o m p o n e n t s o f a t e n s o r i n r e c t a n g u l a r c o o r d i n a t e s a r e y z , 3 , 2 x + y . F i n d i t s c o v a r i a n t
c o m p o n e n t s i n p a r a b o l i c c y l i n d r i c a l c o o r d i n a t e s .
8 9 . E v a l u a t e ( a ) 8 q B a s , ( b ) S q
S r A q s ,
( c )
8 p 8
q
8 s
, ( d ) 8 q 8 r 8 s
8 s
.
9 0 . I f
A r q
i s a t e n s o r , s h o w t h a t A
r
r
i s a c o n t r a v a r i a n t t e n s o r o f r a n k o n e .
9 1 . S h o w t h a t
1 j = k
0
j # k
i s n o t a c o v a r i a n t t e n s o r a s t h e n o t a t i o n m i g h t i n d i c a t e .
9 2 .
I f A 0 = a A q p r o v e t h a t A q =
- a x q
A P
9 3 .
I f A
=
a z p a x s
' I S p r o v e t h a t A s =
a x q a x r
.
a x q a z r
s s
a x p a x s
A r
9 4 . I f ( D i s a n i n v a r i a n t , d e t e r m i n e w h e t h e r
a i s a t e n s o r .
a x p a x q
9 5 .
I f A q a n d B r a r e t e n s o r s , p r o v e t h a t
A 0 B r
a n d A q B q a r e t e n s o r s a n d d e t e r m i n e t h e r a n k o f e a c h .
9 6 . S h o w t h a t i f A r s i s a t e n s o r , t h e n
P q
+ A S S i s a s y m m e t r i c t e n s o r a n d A r s - A s r
i s a s k e w - s y m m e t r i c
t e n s o r .
9 7 .
I f
A p q
a n d B r s a r e s k e w - s y m m e t r i c t e n s o r s , s h o w t h a t
C p s =
A p 4
B r s i s s y m m e t r i c .
9 8 . I f a t e n s o r i s s y m m e t r i c ( s k e w - s y m m e t r i c ) , a r e r e p e a t e d c o n t r a c t i o n s o f t h e t e n s o r a l s o s y m m e t r i c ( s k e w -
s y m m e t r i c ) ?
9 9 . P r o v e t h a t A p q x p x q = 0 i f A p q i s a s k e w - s y m m e t r i c t e n s o r .
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T E N S O R A N A L Y S I S
2 0 9
1 0 0 . W h a t i s t h e l a r g e s t n u m b e r o f d i f f e r e n t c o m p o n e n t s w h i c h a s y m m e t r i c c o n t r a v a r i a n t t e n s o r o f r a n k t w o
c a n h a v e i f ( a ) N = 4 . ( b ) N = 6 ?
W h a t i s t h e n u m b e r f o r a n y v a l u e o f N ?
1 0 1 . H o w m a n y d i s t i n c t n o n - z e r o c o m p o n e n t s , a p a r t f r o m a d i f f e r e n c e i n s i g n , d o e s a s k e w - s y m m e t r i c c o v a r i a n t
t e n s o r o f t h e t h i r d r a n k h a v e
1 0 2 .
I f A I r s i s a t e n s o r , p r o v e t h a t a d o u b l e c o n t r a c t i o n y i e l d s a n i n v a r i a n t .
1 0 3 . P r o v e t h a t a n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n t h a t a t e n s o r o f r a n k R b e c o m e a n i n v a r i a n t b y r e p e a t e d
c o n t r a c t i o n i s t h a t R b e e v e n a n d t h a t t h e n u m b e r o f c o v a r i a n t a n d c o n t r a v a r i a n t i n d i c e s b e e q u a l t o R / 2 .
1 0 4 .
I f A p q a n d
B r s
a r e t e n s o r s , s h o w t h a t t h e o u t e r p r o d u c t i s a t e n s o r o f r a n k f o u r a n d t h a t t w o i n n e r p r o d -
u c t s c a n b e f o r m e d o f r a n k t w o a n d z e r o r e s p e c t i v e l y .
1 0 5 .
I f A ( p , q ) B q = C
w h e r e B q i s a n a r b i t r a r y c o v a r i a n t t e n s o r o f r a n k o n e a n d C
i s a c o n t r a v a r i a n t t e n s o r
o f r a n k o n e , s h o w t h a t A ( p , q ) m u s t b e a c o n t r a v a r i a n t t e n s o r o f r a n k t w o .
1 0 6 . L e t
A P
a n d B q b e a r b i t r a r y t e n s o r s . S h o w t h a t i f A P R q C ( p , q ) i s a n i n v a r i a n t t h e n C ( p , q ) i s a t e n s o r
w h i c h c a n b e w r i t t e n C C .
1 0 7 . F i n d t h e s u m S = A + B , d i f f e r e n c e D = A - B , a n d p r o d u c t s P = A B a n d Q = B A , w h e r e A a n d B a r e
t h e
m a t r i c e s
3 - 1
4 3
( a ) A =
2
4
'
B
_
-
2 - 1
2
0
1 1
- i
2
( b ) A - 1
(
- 2
2
B =
3
2
- 4
_ 1
3 - 1
- i - 2
2
1 0 8 . F i n d ( 3 A - 2 B ) ( 2 A - B ) , w h e r e A a n d B a r e t h e m a t r i c e s i n t h e p r e c e d i n g p r o b l e m .
1 0 9 .
( a ) V e r i f y t h a t d e t ( A B ) = { d e t A } { d e t B }
f o r t h e m a t r i c e s i n P r o b l e m 1 0 7 .
( b ) I s
d e t ( A B ) = d e t ( B A ) ?
1 1 1 1 .
L e t
A =
I
- 3 2 - 1
B =
1
3
- 2
2
1
2
S h o w t h a t ( a ) A B i s d e f i n e d a n d f i n d i t , ( b ) B A a n d A + B a r e n o t d e f i n e d .
2
- 1 3
x
1
1 1 1 . F i n d x , y a n d z s u c h t h a t
1
2
- 4
y
=
- . 3
- 1
3 - 2 z
6
1 1 2 . T h e i n v e r s e o f a s q u a r e m a t r i x A , w r i t t e n
A ' 1
i s d e f i n e d b y t h e e q u a t i o n A A - 1 = 1 ,
w h e r e 1 i s t h e u n i t
m a t r i x h a v i n g o n e s d o w n t h e m a i n d i a g o n a l a n d z e r o s e l s e w h e r e .
_
( 1 2
- 1
1
F i n d A - ' i f ( a ) A = ( _ 5
4
( b ) A =
1
- 1
.
1 - 1
2
I s
A - 1
A = 1 i n t h e s e c a s e s ?
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2 1 0
T E N S O R A N A L Y S I S
2
1 - 2
1 1 3 . P r o v e t h a t
A =
1
- 2
3
h a s n o i n v e r s e .
4 - 3
4
1 1 4 . P r o v e t h a t ( A B ) 7 1 - = B - 1 A - 1 , w h e r e A a n d B a r e n o n - s i n g u l a r s q u a r e m a t r i c e s .
1 1 5 . E x p r e s s i n m a t r i x n o t a t i o n t h e t r a n s f o r m a t i o n e q u a t i o n s f o r
( a ) a c o n t r a v a r i a n t v e c t o r ( b ) a c o v a r i a n t t e n s o r o f r a n k t w o ( c ) a m i x e d t e n s o r o f r a n k t w o .
1 1 6 . D e t e r m i n e t h e v a l u e s o f t h e c o n s t a n t X s u c h t h a t A X = I N X ,
w h e r e A =
- 3
1
a n d X i s a n a r b i -
t r a r y m a t r i x . T h e s e v a l u e s o f X a r e c a l l e d c h a r a c t e r i s t i c v a l u e s o r e i g e n v a l u e s o f t h e m a t r i x A .
1 1 7 . T h e e q u a t i o n F ( X ) = 0 o f t h e p r e v i o u s p r o b l e m f o r d e t e r m i n i n g t h e c h a r a c t e r i s t i c v a l u e s o f a m a t r i x A i s
c a l l e d t h e c h a r a c t e r i s t i c e q u a t i o n f o r A . S h o w t h a t F ( A ) = 0 , w h e r e F ( A ) i s t h e m a t r i x o b t a i n e d b y r e -
p l a c i n g A . b y A i n t h e c h a r a c t e r i s t i c e q u a t i o n a n d w h e r e t h e c o n s t a n t t e r m c i s r e p l a c e d b y t h e m a t r i x c l ,
a n d 0 i s a m a t r i x w h o s e e l e m e n t s a r e z e r o ( c a l l e d t h e n u l l m a t r i x ) . T h e r e s u l t i s a s p e c i a l c a s e o f t h e
H a m i l t o n - C a y l e y t h e o r e m w h i c h s t a t e s t h a t a m a t r i x s a t i s f i e s i t s o w n c h a r a c t e r i s t i c e q u a t i o n .
1 1 8 . P r o v e t h a t ( A B )
= B T A _ T
.
1 1 9 . D e t e r m i n e t h e m e t r i c t e n s o r a n d c o n j u g a t e m e t r i c t e n s o r i n
( a ) p a r a b o l i c c y l i n d r i c a l a n d ( b ) e l l i p t i c c y l i n d r i c a l c o o r d i n a t e s .
1 2 0 . P r o v e t h a t u n d e r t h e a f f i n e t r a n s f o r m a t i o n
- ' r
= a s x p + b r , w h e r e a p a n d
b r a r e c o n s t a n t s s u c h t h a t
a p a q
= b q , t h e r e i s n o d i s t i n c t i o n b e t w e e n t h e c o v a r i a n t a n d c o n t r a v a r i a n t c o m p o n e n t s o f a t e n s o r .
I n
t h e s p e c i a l c a s e w h e r e t h e t r a n s f o r m a t i o n s a r e f r o m o n e r e c t a n g u l a r c o o r d i n a t e s y s t e m t o a n o t h e r , t h e
t e n s o r s a r e c a l l e d c a r t e s i a n t e n s o r s .
1 2 1 . F i n d g a n d g j k c o r r e s p o n d i n g t o
d s 2 = 3 ( d x 1 ) 2 + 2 ( d x 2 ) 2
1 2 2 .
I f A k = g i k A i , s h o w t h a t A J = g
J , k
A k a n d c o n v e r s e l y .
1 2 3 . E x p r e s s t h e r e l a t i o n s h i p b e t w e e n t h e a s s o c i a t e d t e n s o r s
( a ) A p q a n d q , ( b ) A q r a n d A j ' q l , ( c ) A p g r a n d A . . , ,
+ 4 ( d x 3 ) 2 - 6 d x 1 d x 3 .
1 2 4 . S h o w t h a t ( a ) A P q
B .
s
=
A 1 ' g B p r s
,
( b )
B 7 r = A A g r B p r =
B
.
H e n c e d e m o n s t r a t e t h e g e n -
e r a l r e s u l t t h a t a d u m m y s y m b o l i n a t e r m m a y b e l o w e r e d f r o m i t s u p p e r p o s i t i o n a n d r a i s e d f r o m i t s
l o w e r p o s i t i o n w i t h o u t c h a n g i n g t h e v a l u e o f t h e t e r m .
1 2 5 . S h o w t h a t i f
A
B
C r
A ;
q r
= B ;
q
a f r e e i n d e x i n a t e n s o r e q u a t i o n m a y b e r a i s e d o r l o w e r e d w i t h o u t a f f e c t i n g t h e v a l i d i t y o f t h e e q u a -
t i o n .
1 2 6 . S h o w t h a t t h e t e n s o r s g
p q '
g p q a n d 8 9 a r e a s s o c i a t e d t e n s o r s .
1 2 7 . P r o v e
( a ) E j k
a x
_ g p q
a x Q k
, ( b ) g d k
a x p
=
g p q a x e
a x p a x
a x
a x
1 2 8 . I f
A P
i s a v e c t o r f i e l d , f i n d t h e c o r r e s p o n d i n g u n i t v e c t o r .
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2 1 2
T E N S O R A N A L Y S I S
1 4 8 . S h o w t h a t i f n o e x t e r n a l f o r c e a c t s , a m o v i n g p a r t i c l e o f c o n s t a n t m a s s t r a v e l s a l o n g a g e o d e s i c g i v e n b y
p
a s ( d s ) =
a .
1 4 9 . P r o v e t h a t t h e s u m a n d d i f f e r e n c e o f t w o r e l a t i v e t e n s o r s o f t h e s a m e w e i g h t a n d t y p e i s a l s o a r e l a t i v e
t e n s o r o f t h e s a m e w e i g h t a n d t y p e .
1 5 0 .
i f
A p q i s
a r e l a t i v e t e n s o r o f w e i g h t w , p r o v e t h a t g - V / 2
A p q . i s
a n a b s o l u t e t e n s o r .
1 5 1 .
I f A ( p , q ) B q s =
w h e r e B r
i s a n a r b i t r a r y r e l a t i v e t e n s o r o f w e i g h t w l a n d C p r i s a k n o w n r e l a t i v e
t e n s o r o f w e i g h t w 2 , p r o v e t h a t A ( p , q ) i s a r e l a t i v e t e n s o r o f w e i g h t w 2 - - - w 1 . T h i s i s a n e x a m p l e o f
t h e q u o t i e n t l a w f o r r e l a t i v e t e n s o r s .
1 5 2 . S h o w t h a t t h e q u a n t i t y G ( j , k ) o f S o l v e d P r o b l e m 3 1 i s a r e l a t i v e t e n s o r o f w e i g h t t w o .
1 5 3 . F i n d t h e p h y s i c a l c o m p o n e n t s o f ( a ) t h e v e l o c i t y a n d ( b ) t h e a c c e l e r a t i o n o f a p a r t i c l e i n s p h e r i c a l c o -
o r d i n a t e s .
1 5 4 . L e t A r a n d B r b e t w o v e c t o r s i n t h r e e d i m e n s i o n a l s p a c e .
S h o w t h a t i f , \ a n d , i a r e c o n s t a n t s , t h e n
C r = X A r + L R r i s a v e c t o r l y i n g i n t h e p l a n e o f A r a n d B r . W h a t i s t h e i n t e r p r e t a t i o n i n h i g h e r d i m e n -
s i o n a l s p a c e ?
1 5 5 . S h o w t h a t a v e c t o r n o r m a l t o t h e s u r f a c e 0 ( x i , x 2 , x 3 ) = c o n s t a n t i s g i v e n b y
A O
= 9 a .
F i n d t h e
c o r r e s p o n d i n g u n i t n o r m a l .
a s
1 5 6 . T h e e q u a t i o n o f c o n t i n u i t y i s g i v e n b y V ( 0 - V ) +
a c
= 0 w h e r e c r i s t h e d e n s i t y a n d v i s t h e v e l o c i t y o f
a f l u i d . E x p r e s s t h e e q u a t i o n i n t e n s o r f o r m .
1 5 7 . E x p r e s s t h e c o n t i n u i t y e q u a t i o n i n ( a ) c y l i n d r i c a l a n d ( b ) s p h e r i c a l c o o r d i n a t e s .
1 5 8 . E x p r e s s S t o k e s ' t h e o r e m i n t e n s o r f o r m .
1 5 9 . P r o v e t h a t t h e c o v a r i a n t c u r v a t u r e t e n s o r R p q r s i s s k e w - s y m m e t r i c i n ( a ) p a n d q , ( b ) r a n d s , ( c ) q a n d s .
1 6 0 . P r o v e R p q r s = R r s j i q
1 6 1 . P r o v e
( a ) R j , g r s + R p s q r + R j i r s q
=
0 ,
0 .
( b ) R ¢ g r s + R r g p s + R r s p q + R ¢ s r q =
1 6 2 . P r o v e t h a t c o v a r i a n t d i f f e r e n t i a t i o n i n a E u c l i d e a n s p a c e i s c o m m u t a t i v e . T h u s s h o w t h a t t h e R i e m a n n -
C h r i s t o f f e l t e n s o r a n d c u r v a t u r e t e n s o r a r e z e r o i n a E u c l i d e a n s p a c e .
1 6 3 . L e t T
0
=
d s P b e t h e t a n g e n t v e c t o r t o c u r v e C w h o s e e q u a t i o n i s x P = x ' ( s ) w h e r e s i s t h e a r c l e n g t h .
( a ) S h o w t h a t g , g T P T q - 1 . ( b ) P r o v e t h a t g i g T O T
q
= 0 a n d t h u s s h o w t h a t N q =
K & s q
i s a u n i t
n o r m a l t o C f o r s u i t a b l e K . ( c ) P r o v e t h a t
N q
i s o r t h o g o n a l t o N q
a s
1 6 4 . W i t h t h e n o t a t i o n o f t h e p r e v i o u s p r o b l e m , p r o v e :
( a ) g i g
T ' N q
= 0
( b ) g i g
T O S N q
= - K o r g p q T
( S N q
+ K T q ) = 0 .
r
H e n c e s h o w t h a t B r =
I ( 6 N
+ K T r ) i s a u n i t v e c t o r f o r s u i t a b l e T o r t h o g o n a l t o b o t h
a n d N q .
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T E N S O R A N A L Y S I S
2 1 3
1 6 5 . P r o v e t h e F r e n e t - S e r r e t f o r m u l a s
$ s
K
N p ,
S =
T B 1 ' -
K T 1 ' ,
s B
= -
T N 1 '
w h e r e T N 1 ' a n d B P a r e t h e u n i t t a n g e n t , u n i t n o r m a l a n d u n i t b i n o r m a l v e c t o r s t o C , a n d K a n d T a r e
t h e c u r v a t u r e a n d t o r s i o n o f C .
1 6 6 . S h o w t h a t d s 2 = c 2 ( d x 4 ) 2 - d x k d x k ( N = 3 ) i s i n v a r i a n t u n d e r t h e l i n e a r ( a f f i n e ) t r a n s f o r m a t i o n
x 1 = y ( x 1 - v x 4 )
,
x 2
= x 2 ,
x 3
= x 3 ,
z 4
= y ( x 4 - -
1 8
x 1 )
2
w h e r e ' y , , 8 , c a n d v a r e c o n s t a n t s , 8 = v / c a n d y = ( 1 - , 8 ) 7 1 / 2
T h i s i s t h e L o r e n t z t r a n s f o r m a t i o n
o f s p e c i a l r e l a t i v i t y .
P h y s i c a l l y , a n o b s e r v e r a t t h e o r i g i n o f t h e x i s y s t e m s e e s a n e v e n t o c c u r r i n g a t
p o s i t i o n x 1 , x 2 , x 3 a t t i m e x 4 w h i l e a n o b s e r v e r a t t h e o r i g i n o f t h e ' x i s y s t e m s e e s t h e s a m e e v e n t o c c u r -
r i n g a t p o s i t i o n 3 F 1 , ` x 2 , ` x 3 a t t i m e z 4 .
I t i s a s s u m e d t h a t ( 1 ) t h e t w o s y s t e m s h a v e t h e x 1 a n d Z 1 a x e s
c o i n c i d e n t , ( 2 ) t h e p o s i t i v e x 2 a n d x 3 a x e s a r e p a r a l l e l r e s p e c t i v e l y t o t h e p o s i t i v e x 2 a n d x 3 a x e s ,
( 3 ) t h e x i s y s t e m m o v e s w i t h v e l o c i t y v r e l a t i v e t o t h e x i s y s t e m , a n d ( 4 ) t h e v e l o c i t y o f l i g h t c i s a
c o n s t a n t .
1 6 7 . S h o w t h a t t o a n o b s e r v e r f i x e d i n t h e x i ( ( i ) s y s t e m , a r o d f i x e d i n t h e x i ( x i ) s y s t e m l y i n g p a r a l l e l t o
t h e x 1 ( x i ) a x i s a n d o f l e n g t h L i n t h i s s y s t e m a p p e a r s t o h a v e t h e r e d u c e d l e n g t h L A -
T h i s
p h e n o m e n a i s c a l l e d t h e L o r e n t z - F i t z g e r a l d c o n t r a c t i o n .
A N S W E R S T O S U P P L E M E N T A R Y P R O B L E M S .
7 7 .
( a ) a k x k x 3
( b ) A 2 3 B
( c ) A k B k
( d ) g 2 q g q 1 , N = 4
( e )
B a r , N =
2
7 8 . ( a ) 2
1
( v g A 1 )
+
a x 2 ( v g A 2 ) +
2 x 3 ( V - g A 3 )
( c )
a z j a x 1
+
a x i a x e
+
. . .
+
A
a x a x
( b ) A l l B P C ,
+
A 2 '
B 1 ' C 2
+ A 1 2
B 2 C ,
+ A 2 2
B 2 C 2
a x i
a x - ' I R
a x 2 a z m
- a x l - a - - n
7 9 . E l l i p s e f o r N = 2 , e l l i p s o i d f o r N j = 3 , h y p e r e l l i p s o i d f o r N = 4 .
8 0 .
x
8 1 .
( a ) A -
A
r
a x i - a x , a v r
k
( b ) B - 1 ' g r =
a x p a z q a x a x ' s B i j k
S
a x i a x i a x k a T s
a x x a x n
( c ) C
q
a x 1 ' a x q
( d ) A
=
a x i
A l l
a x i
C x n
8 2 . ( a ) B ( / , k , m ) i s a t e n s o r o f r a n k t h r e e a n d i s c o v a r i a n t o f o r d e r t w o a n d c o n t r a v a r i a n t o f o r d e r o n e .
I t c a n
b e w r i t t e n
( b ) C ( j , k , m , n ) i s n o t a t e n s o r .
8 3 . 4 5 = 1 0 2 4
a 1 1 x 1 + a 1 2 x 2
=
b 1
a 2 1 x 1 + a 2 2 x 2
=
b 2
k
i j
p q
a
- p a x q a
8 7 . ( a ) 2 p c o s 2 c -
z c o s 0 + p 3 s i n 2 0 c o s 2 o ,
2
- - 2 p s i n 0 c o s 0 + p z s i n 0 + p 4 s i n 0 c o s 3
p z s i n 0 .
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2 1 4
T E N S O R A N A L Y S I S
( b ) 2 r s i n 2 6 c o s 2 0 - r s i n 6 c o s 6 c o s ( p
+
r 3 s i n 4 6 s i n 2 0 c o s 2 0 +
r 2 s i n 6 c o s 2 6 s i n k ,
2 r 2 s i n 6 c o s 6 c o s 2 0 - r 2 c o s 2 6 c o s 4 +
r 4 s i n 3 6 c o s ( 9 s i n 2 O c o s 2 O
- r 3 s i n 2 6 c o s 6 s i n k ,
- 2 r
2 s i n 2 6 s i n
c o s ) +
r 2 s i n e c o s 6 s i n
+ r 4 s i n 4 6 s i n o c o s 3 o
s
P
8 8 . u 2 v z + 3 v ,
3 u - u v 2 z ,
u 2 + u v - v 2 8 9 .
( a ) B q
r ,
( b )
( c ) b s ,
( d ) N
9 4 .
I t i s n o t a t e n s o r .
9 5 . R a n k 3 a n d r a n k 1 r e s p e c t i v e l y .
9 8 . Y e s .
1 0 0 .
( a ) 1 0 ,
( b ) 2 1 ,
( c ) N ( N + 1 ) / 2 1 0 1 . N ( N - 1 ) ( N - 2 ) / 6
7
2
- 1 - 4
1 4
1 0 1 8
8
1 0 7 . ( a ) S
0
3 '
D
4
5
P
0 2
Q
- 8 - 2
3
- 1
3 1 1 - 1
1 - 4
6
1
8 - 3
( b ) S =
2 0 - 2 ,
D =
- 4 - 4 6
,
p =
- 9 - 7
1 0
, Q = 8 - 1 6 1 1
- 2
1
1
0
5 - 3
9
9 - 1 6
- 2
1 0 - 7
1 0 8 . ( a )
3 - 1 6
2 0
5 2
)
0 4
- 8 6
( b )
9
1 6 3 - 1 3 6
1 1 0 . - 4
- 6 1 - 1 3 5
1 0 4
1 3 2
1 1 1 . x = - 1 , y = 3 , z = 2
( b )
A 3 1 A 3 2 A 3 3
( c )
3 3
3
A l A 2
A 3
a x 1 a x 2
a x a x
a x 1
a x 1
a x 1
a x 2
a x
a x 2
a x 1
a x 2
a x 3 a x
a x 1
a x 2
5 3
1 7 - 2
1
2
1
1 1 2 .
( a )
5 / 2 3 / 2
1 1 6 . \ . = 4 , - 1
1 1 9 . ( a )
x 1
a x 3
- a x '
A 1 2 A 1 3
1 / 3
1 / 3 0
( b )
5 / 3 1 / 3 1 . Y e s
- 1
0
1
a x 1 a x 1
a x 1
a x 1
a a G 2
a z 3
A
A
a x 2
a x 2 a x 2
2 2
2 3
1
-
-
A A
a x
a x 3
a x 2
a x 3
a z 3
' 3 X 3
3 2
3 3
a x 1 a x 2
a x 3
2
2
v
0 0
2 + v 2
'2
0 0
0
2
+
\
0
0
1
v
1
,
2
2
u + v
0
0 1 0
0
1
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T E N S O R A N A L Y S I S
2 1 5
a 2 ( s i n h 2 u + s i n 2 v )
0
0
1
0
0
( a 2 ( s i n h 2 u + s i n 2 v )
( b )
0
a 2 ( s i n h 2 u + s i n 2 v ) 0
0
1
0
a 2 ( s i n h 2 u + s i n 2 v )
0
0
1
0
0
1
4 / 3
0
1
1 2 1 . g = 6 , 0
1 / 2 0
1
0
1
1 2 3 . ( a )
A p 9
_ g P i A 9 ,
( b ) A .
r
A
A
1 2 8 . o r
A Y ,
g i g
A P A q
0 .
1 3 0 .
( a ) T h e y a r e a l l z e r o .
( b )
[ 2 2 , 1 ] = - p ,
[ 1 2 , 2 ] _ [ 2 1 , 2 ] = p .
A l l o t h e r s a r e z e r o .
( c )
[ 2 2 , 1 ] _ - r ,
[ 3 3 . 1 ]
r s i n 2 8 ,
[ 3 3 , 2 ] _
- r 2 s i n 8 c o s 8
[ 2 1 , 2 ] [ 1 2 , 2 ] = r ,
[ 3 1 , 3 ] = [ 1 3 , 3 ] = r s i n 2 8
[ 3 2 , 3 1 _ [ 2 3 , 3 ] = r 2 s i n 8 c o s e .
A l l o t h e r s a r e z e r o .
1 3 1 .
( a )
[ i i , 1 ] = u , [ 2 2 , 2 ] = v , [ 1 1 , 2 ] _
- v ,
[ 2 2 , 1 ]
- u ,
[ 1 2 , 1 ] = [ 2 1 , 1 ] = v , [ 2 1 , 2 ] = [ 1 2 , 2 ] = u .
1
_ u
2
v
1 - u
2
- v
1 1 u 2 + v 2 '
2 2 u 2 + v 2 '
2 2
u 2 + v 2
1 1
u 2 + v 2
1 _
1
v
2
1 2
u
A l l o t h e r s a r e z e r o .
2 1
1 2
u 2 + v 2
2 1
1 2
u 2 + v 2
( b )
[ 1 1 , 1 ] = 2 a 2 s i n h u C o s h u
,
[ 2 2 , 2 ] = 2 a 2 s i n e c o s v
,
[ 1 1 , 2 ] _ - 2 a 2 s i n v c o s v
[ 2 2 . 1 ] = - 2 a 2 s i n h u c o s h u , [ 1 2 , 1 ] _ [ 2 1 , 1 ] = 2 a 2 s i n v c o s v ,
[ 2 1 , 2 ] = [ 1 2 , 2 ] = 2 a 2 s i n h u c o s h u
5 1
s i n h u c o s h u
2 _ s i n v c o s v
1 - s i n h u c o s h u
1 1
s i n h 2 u + s i n 2 v '
2 2
s i n h 2 u + s i n 2 v '
1 2 2 5 - s i n h 2 u + s i n 2 v
5 2
- s i n v c o s v
1
1
s i n v c o s v
1 1 1
s i n h 2 u + s i n 2 v
2 1
1 2
s i n h 2 u + s i n 2 v '
j 2
=
2 s i n h u c o s h u
.
A l l o t h e r s a r e z e r o .
2 1
1 2
y s i n h 2 u + s i n 2 v
1 3 2 .
( a )
d d s P
- p (
d 2
d 2 0
+ 2
d o d q
d s 2 p d s d s
. . r
r l
J k
( c )
A p
q
= g p j g q k g
A . . 1
d 2
0 ,
d s 2 z
= 0
d ' r d < p
{ b )
s 2
_
r ( d 8 ) 2
r s i n g
0 (
d
) 2
=
0
d 2 B
2 d r
L 6
d s 2
r d s d s
d 2 4 ) + 2 d r d 0 +
d s 2
r d s
d s
s i n 6 c o s 8 ( d O ) 2
= 0
d s
2 c o t 8 d 8 d ( k =
0
d s
d s
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2 1 6
1
_
1 2 2
1 3 5 .
J
x
1 2
2 1
2
d 2 x 1
+ x (
d x 2
} 2
= 0 ' d 2 x 2
d s 2 d s
d s 2
1 3 6 .
( a ) A
( b )
j k
a A l
l , q
a x q
j k
A l n , q
) Ac
k l n c , q
a A
j k
i m
a x q
a A
i
j k
T E N S O R A N A L Y S I S
x 1
( x 1 ) 2 - ( x 2 ) 2
+
x 2
( x 2 ) 2 - ( x
1 ) 2
2
1 2 2
2 x 1
d x 1 d x 2
+
x 2 d x 2
)
2
=
0
( X I ) 2 _ ( X 2 ) 2
d s
d s
( x 2 ) 2
- ( x 1 ) 2
d s
s
l q
s
l q
( s
k q
k l m .
A s k
+
j k
A s p
A j
s
m q
j k l
( d )
A , n , q
-
j k
( e ) A l
n , q
a x q
j k
a A l m n
a x q
j k l
A s
+
k A j s
{ q s
l
+
A l l o t h e r s a r e z e r o .
s
j k
{ / } s k
k
j s
m q
A i s
+ q s
A l i n
+ q s
A l i
s
A j
-
s
A j
+ j A s
l q
k s m
m q
k 1 s
q s
k l m
q s
j k
s
j k
A s m n -
m q
A l s n -
1 3 7 . ( a ) g j k A q ,
( b )
A 1 q
B k + A l B k
0
( C )
k j
A
, 9
j , q
1 4 1 . ( a )
u 2 + v 2
a u
A u )
+
a
(
u 2 + v 2 A v )
a v
j s l
l
j k s
q s
s i
n q f
( b )
u v ( u 2 + v 2 )
a u
( u v
u 2 + v 2 A U ) + a v a ( u v
u 2 + v 2 A V )
+
1
a
1
a
a ( 1 )
( a )
4 2 .
( b )
1 4 3 .
0
u V
2 + v 2 a u
w
s
l q
+
e v
+
u 2 + v 2
a v
e z
a z
j k
A
+
I n s
j A s k +
q s
I n n
k
q s
1
a 2 A z
u v
a z 2
{
1
( a ( p
e u +
a i
e v )
+
a
e 2
a s i n h 2 u + s i n 2 v
a u
a v
a z
w h e r e e u , e v a n d e z a r e u n i t v e c t o r s i n t h e d i r e c t i o n s o f i n c r e a s i n g u , v a n d z r e s p e c t i v e i y .
1
a 2 T
+
a
+
( u 2 + v 2 )
u 2 + v 2
a u 2
a v 2
1 4 5 .
( a )
8 A k
A
d x q
=
a A k
_
s
A
d x q
d A k
-
s
A
d x q
=
) -
8 t
k , q
d t
a x q
k q
A S
d t d t k q
A s
d t
( b )
a A j k
d A j k +
j } A S k d X q
+
A
S t
d t
q s
d t q s
d t
k
( c )
8
( A . B k )
= S A C B k
+ A
b B
b t
S t
b t
i s
A l n n n
d A j -
s
B k
+
+
k
B s d x q
{ . } ' 4 s
d t
d t
q s
}
d t
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I n d e x
A b s o l u t e d e r i v a t i v e , 1 7 4
A b s o l u t e m o t i o n , 5 3
A b s o l u t e t e n s o r , 1 7 5
A c c e l e r a t i o n , a l o n g a s p a c e c u r v e , 3 5 , 3 9 , 4 0 , 5 0 , 5 6
c e n t r i p e t a l , 4 3 , 5 0 , 5 3
C o r i o l i s , 5 3
i n c y l i n d r i c a l c o o r d i n a t e s , 1 4 3 , 2 0 4
i n g e n e r a l c o o r d i n a t e s , 2 0 4 , 2 0 5
i n p o l a r c o o r d i n a t e s , 5 6
i n s p h e r i c a l c o o r d i n a t e s , 1 6 0 , 2 1 2
o f a p a r t i c l e , 3 8 , 4 2 , 4 3 , 5 0 , 5 2 , 8 4 , 2 0 3 , 2 0 5
r e l a t i v e t o f i x e d a n d m o v i n g o b s e r v e r s , 5 2 , 5 3
A d d i t i o n , o f m a t r i c e s , 1 7 0
o f t e n s o r s , 1 6 9
A d d i t i o n , o f v e c t o r s , 2 , 4 , 5
a s s o c i a t i v e l a w f o r , 2 , 5
c o m m u t a t i v e l a w f o r , 2 , 5
p a r a l l e l o g r a m l a w f o r , 2 , 4
t r i a n g l e l a w f o r , 4
A e r o d y n a m i c s , 8 2
A f f i n e t r a n s f o r m a t i o n , 5 9 , 2 1 0 , 2 1 3
A l g e b r a , o f m a t r i c e s , 1 7 0
o f v e c t o r s , 1 , 2
A n g l e , b e t w e e n t w o s u r f a c e s , 6 3
b e t w e e n t w o v e c t o r s , 1 9 , 1 7 2 , 1 9 0
s o l i d , 1 2 4 , 1 2 5
A n g u l a r m o m e n t u m , 5 0 , 5 1 , 5 6
A n g u l a r s p e e d a n d v e l o c i t y , 2 6 , 4 3 , 5 2
A r b i t r a r y c o n s t a n t v e c t o r , 8 2
A r c l e n g t h , 3 7 , 5 6 , 1 3 6 , 1 4 8
i n c u r v i l i n e a r c o o r d i n a t e s , 5 6 , 1 4 8
i n o r t h o g o n a l c u r v i l i n e a r c o o r d i n a t e s , 1 3 6
o n a s u r f a c e , 5 6
A r e a l v e l o c i t y , 8 5 , 8 6
A r e a , b o u n d e d b y a s i m p l e c l o s e d c u r v e , 1 1 1
o f e l l i p s e , 1 1 2
o f p a r a l l e l o g r a m , 1 7 , 2 4
o f s u r f a c e , 1 0 4 , 1 0 5 , 1 6 2
o f t r i a n g l e , 2 4 , 2 5
v e c t o r , 2 5 , 8 3
A s s o c i a t e d t e n s o r s , 1 7 1 , 1 9 0 , 1 9 1 , 2 1 0
A s s o c i a t i v e l a w , 2 , 5 , 1 7
B a s e v e c t o r s , 7 , 8 , 1 3 6
u n i t a r y , 1 3 6
B i n o r m a l , 3 8 , 4 5 , 4 7 , 4 8
B i p o l a r c o o r d i n a t e s , 1 4 0 , 1 6 0
B o x p r o d u c t , 1 7
B r a h e , T y c h o , 8 6
C a l c u l u s o f v a r i a t i o n s , 1 7 3
C a r t e s i a n t e n s o r s , 2 1 0
C e n t r a l f o r c e , 5 6 , 8 5
C e n t r i p e t a l a c c e l e r a t i o n , 4 3 , 5 0 , 5 3
C e n t r o i d , 1 5
C h a i n r u l e , 7 7 , 1 7 7 , 1 7 9
C h a r a c t e r i s t i c e q u a t i o n , 2 1 0
C h a r a c t e r i s t i c v a l u e s , 2 1 0
C h a r g e d e n s i t y , 1 2 6
C h r i s t o f f e l ' s s y m b o l s , 1 7 2 , 1 9 2 - 1 9 5 , 2 1 1
t r a n s f o r m a t i o n l a w s o f , 1 7 2 , 1 9 3 , 1 9 4
C i r c u l a t i o n , 8 2 , 1 3 1
C i r c u m c e n t e r , 3 3
C l o c k w i s e d i r e c t i o n , 8 9
C o f a c t o r , 1 7 1 , 1 8 7 , 1 8 8
C o l l i n e a r v e c t o r s , 8 , 9
n o n - , 7 , 8
C o l u m n m a t r i x o r v e c t o r , 1 6 9
C o m m u t a t i v e l a w , 2 , 5 , 1 6 , 1 7
C o m p o n e n t v e c t o r s , 3 , 7 , 8
r e c t a n g u l a r , 3
C o m p o n e n t s , c o n t r a v a r i a n t , 1 3 6 , 1 5 6 , 1 5 7 , 1 6 7 , 1 6 8
c o v a r i a n t , 1 3 6
o f a d y a d , 7 3
o f a t e n s o r , 1 5 7 , 1 6 7 , 1 6 8
o f a v e c t o r , 3 , 1 3 6 , 1 5 6 , 1 5 7 , 1 5 8 , 1 6 7
p h y s i c a l , ( s e e P h y s i c a l c o m p o n e n t s )
C o n d u c t i v i t y , t h e r m a l , 1 2 6
C o n f o r m a b l e m a t r i c e s , 1 7 0
C o n i c s e c t i o n , 8 7
C o n j u g a t e m e t r i c t e n s o r , 1 7 1 , 1 8 8 , 1 8 9
C o n j u g a t e t e n s o r s , 1 7 1
C o n s e r v a t i o n o f e n e r g y , 9 4
C o n s e r v a t i v e f i e l d , 7 3 , 8 3 , 9 0 , 9 1 , 9 3
m o t i o n o f p a r t i c l e i n , 9 3 , 9 4
n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n f o r , 9 0 , 9 1
C o n t i n u i t y , 3 6 , 3 7
e q u a t i o n o f , 6 7 , 1 2 6 , 2 1 2
C o n t r a c t i o n , 1 6 9 , 1 8 1 , 1 8 2
C o n t r a v a r i a n t c o m p o n e n t s , 1 3 6 , 1 5 6 , 1 5 7 , 1 6 7 , 1 6 8
o f a t e n s o r , 1 5 7 , 1 6 7 , 1 6 8
o f a v e c t o r , 1 3 6 , 1 5 6 , 1 5 7 , 1 6 7
C o n t r a v a r i a n t t e n s o r , o f f i r s t r a n k , 1 5 7 , 1 6 7
o f s e c o n d a n d h i g h e r r a n k , 1 6 8
2 1 8
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I N D E X
C o n t r a v a r i a n t v e c t o r , ( s e e C o n t r a v a r i a n t c o m p o -
n e n t s o f a v e c t o r )
C o o r d i n a t e c u r v e s o r l i n e s , 1 3 5
C o o r d i n a t e s , c u r v i l i n e a r , ( s e e C u r v i l i n e a r c o o r d i -
n a t e s )
C o o r d i n a t e s u r f a c e s , 1 3 5
C o o r d i n a t e t r a n s f o r m a t i o n s , 5 8 , 5 9 , 7 6 , 1 3 5 , 1 6 6
C o p l a n a r v e c t o r s , 3
n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n f o r , 2 7
n o n - , 7 , 8
C o r i o l i s a c c e l e r a t i o n , 5 3
C o s i n e s , d i r e c t i o n , 1 1 , 5 8
l a w o f , f o r p l a n e t r i a n g l e s , 2 0
l a w o f , f o r s p h e r i c a l t r i a n g l e s , 3 3
C o u n t e r c l o c k w i s e d i r e c t i o n , 8 9
C o v a r i a n t c o m p o n e n t s , 1 3 6 , 1 5 7 , 1 5 8 , 1 6 7
o f a t e n s o r , 1 6 7 , 1 6 8
o f a v e c t o r , 1 3 6 , 1 5 7 , 1 5 8 , 1 6 7
C o v a r i a n t c u r v a t u r e t e n s o r , 2 0 7
C o v a r i a n t d e r i v a t i v e , 1 7 3 , 1 9 7 - 1 9 9 , 2 1 1
C o v a r i a n t t e n s o r , o f f i r s t r a n k , 1 5 8
C o v a r i a n t v e c t o r , ( s e e C o v a r i a n t c o m p o n e n t s o f a
v e c t o r )
C r o s s - c u t , 1 1 3
C r o s s p r o d u c t , 1 6 , 1 7 , 2 2 - 2 6
c o m m u t a t i v e l a w f a i l u r e f o r , 1 6
d e t e r m i n a n t f o r m f o r , 1 7 , 2 3
d i s t r i b u t i v e l a w f o r , 1 6 , 2 2 , 2 3
C u b i c , t w i s t e d , 5 5
C u r l , 5 7 , 5 8 , 6 7 - 7 2
i n c y l i n d r i c a l c o o r d i n a t e s , 1 5 3 , 1 5 4
i n o r t h o g o n a l c u r v i l i n e a r c o o r d i n a t e s , 1 3 7 , 1 5 0
i n p a r a b o l i c c y l i n d r i c a l c o o r d i n a t e s , 1 6 1
i n s p h e r i c a l c o o r d i n a t e s , 1 5 4
i n t e g r a l d e f i n i t i o n o f , 1 2 3 , 1 5 2 , 1 5 3
i n v a r i a n c e o f , 8 1
o f t h e g r a d i e n t , 5 8 , 6 9 , 2 1 1
p h y s i c a l s i g n i f i c a n c e o f , 7 2 , 1 3 1
t e n s o r f o r m o f , 1 7 4 , 2 0 0
C u r r e n t d e n s i t y , 1 2 6
C u r v a t u r e , 3 8 , 4 5 , 4 7 , 1 1 3
r a d i u s o f , 3 8 , 4 5 , 4 6 , 5 0
R i e m a n n - C h r i s t o f f e l , 2 0 6
t e n s o r , 2 0 7
C u r v e , s p a c e , ( s e e S p a c e c u r v e s )
C u r v i l i n e a r c o o r d i n a t e s , 1 3 5 - 1 6 5
a c c e l e r a t i o n i n , 1 4 3 , 2 0 4 , 2 0 5 , 2 1 2
a r c l e n g t h i n , 5 6 , 1 3 6 , 1 4 8
d e f i n i t i o n o f , 1 3 5
g e n e r a l , 1 4 8 , 1 5 6 - 1 5 9
o r t h o g o n a l , 4 9 , 1 3 5
s u r f a c e , 4 8 , 4 9 , 5 6 , 1 5 5
v o l u m e e l e m e n t s i n , 1 3 6 , 1 3 7 , 1 5 9
C y c l o i d , 1 3 2
C y l i n d r i c a l c o o r d i n a t e s , 1 3 7 , 1 3 8 , 1 4 1 , 1 4 2 , 1 6 0 , 1 6 1
a r c l e n g t h i n , 1 4 3
C h r i s t o f f e l ' s s y m b o l s i n , 1 9 5 , 2 1 1
c o n j u g a t e m e t r i c t e n s o r i n , 1 8 9
2 1 9
C y l i n d r i c a l c o o r d i n a t e s ,
c o n t i n u i t y e q u a t i o n i n , 2 1 2
c u r l i n , 1 5 3 , 1 5 4
d i v e r g e n c e i n , 1 5 3 , 2 0 0 , 2 0 1
e l l i p t i c , ( s e e E l l i p t i c c y l i n d r i c a l c o o r d i n a t e s .
g e o d e s i c s i n , 2 1 1
g r a d i e n t i n , 1 5 3 , 1 5 4
J a c o b i a n i n , 1 6 1
L a p l a c i a n i n , 1 5 3 , 1 5 4 , 2 0 1
m e t r i c t e n s o r i n , 1 8 7
p a r a b o l i c , ( s e e P a r a b o l i c c y l i n d r i c a l c o o r d i n a t e s )
v e l o c i t y a n d a c c e l e r a t i o n i n , 1 4 3 , 2 0 4 , 2 0 5
v o l u m e e l e m e n t i n , 1 4 4 , 1 4 5
V , ( s e e D e l )
V 2 , ( s e e L a p l a c i a n o p e r a t o r )
D e l ( p ) , 5 7 , 5 8 , ( s e e a l s o G r a d i e n t , D i v e r g e n c e a n d
C u r l )
f o r m u l a s i n v o l v i n g , 5 8
i n t e g r a l o p e r a t o r f o r m f o r , 1 0 7 , 1 2 3
i n v a r i a n c e o f , 8 1
D e l t a , K r o n e c k e r , 1 6 8 , 1 7 9 , 1 8 0 , ( s e e a l s o K r o n -
e c k e r ' s s y m b o l )
D e n s i t y , 1 2 6
c h a r g e , 1 2 6
c u r r e n t , 1 2 6
t e n s o r , 1 7 5 , 2 0 3
D e p e n d e n c e , l i n e a r , 1 0 , 1 5
D e r i v a t i v e , a b s o l u t e , 1 7 4
c o v a r i a n t , 1 7 3 , 1 9 7 - 1 9 9 , 2 1 1
d i r e c t i o n a l , 5 7 , 6 1 - 6 3
i n t r i n s i c , 1 7 4 , 2 0 2 , 2 1 1
D e r i v a t i v e s , o f v e c t o r s , 3 5 - 5 6
o r d i n a r y , 3 5 , 3 6 , 3 9 - 4 3
p a r t i a l , 3 6 , 3 7 , 4 4 , 4 5
D e s c a r t e s , f o l i u m o f , 1 3 2
D e t e r m i n a n t , c o f a c t o r o f , 1 7 1 , 1 8 7 , 1 8 8
c r o s s p r o d u c t e x p r e s s e d a s , 1 7 , 2 3
c u r l e x p r e s s e d a s , 5 7 , 5 8
d i f f e r e n t i a t i o n o f , 4 1
J a c o b i a n , ( s e e J a c o b i a n )
o f a m a t r i x , 1 7 0 , 2 0 9
s c a l a r t r i p l e p r o d u c t e x p r e s s e d a s , 1 7 , 2 6 , 2 7
D e t e r m i n a n t s , m u l t i p l i c a t i o n o f , 1 5 9
D e x t r a l s y s t e m , 3
D i a g o n a l o f a s q u a r e m a t r i x , 1 6 9
D i f f e r e n c e , o f m a t r i c e s , 1 7 0
o f t e n s o r s , 1 6 9
o f v e c t o r s , 2
D i f f e r e n t i a b l e , s c a l a r f i e l d , 5 7
v e c t o r f i e l d , 5 7
D i f f e r e n t i a b i l i t y , 3 6 , 3 7
D i f f e r e n t i a l e q u a t i o n s , 5 4 , 1 0 4
D i f f e r e n t i a l g e o m e t r y , 3 7 , 3 8 , 4 5 - 5 0 , 5 4 - 5 6 , 1 6 6 , 2 1 2 - 1 3
D i f f e r e n t i a l s , 3 7
e x a c t , ( s e e E x a c t d i f f e r e n t i a l s )
D i f f e r e n t i a t i o n o f v e c t o r s , 3 5 - 5 6
f o r m u l a s f o r , 3 6 , 3 7 , 4 0 , 4 1
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2 2 0
D i f f e r e n t i a t i o n o f v e c t o r s ,
o r d e r o f , 3 7 , 6 9
o r d i n a r y , 3 5 , 3 6
p a r t i a l , 3 6 , 3 7
D i f f u s i v i t y , 1 2 7
D i r e c t i o n a l d e r i v a t i v e , 5 7 , 6 1 - 6 3
D i r e c t i o n c o s i n e s , 1 1 , 5 8
D i s t a n c e b e t w e e n t w o p o i n t s , 1 1
D i s t r i b u t i v e l a w , 2
f o r c r o s s p r o d u c t s , 1 6 , 2 2 , 2 3
f o r d o t p r o d u c t s , 1 6 , 1 8
f o r d y a d i c s , 7 4
f o r m a t r i c e s , 1 7 0
D i v , ( s e e D i v e r g e n c e )
D i v e r g e n c e , 5 7 , 6 4 - 6 7
i n c u r v i l i n e a r c o o r d i n a t e s , 1 3 7 , 1 5 0
i n c y l i n d r i c a l c o o r d i n a t e s , 1 5 3 , 2 0 0 , 2 0 1
i n p a r a b o l i c c y l i n d r i c a l c o o r d i n a t e s , 1 6 1
i n s p h e r i c a l c o o r d i n a t e s , 1 6 1 , 2 0 0 , 2 0 1
i n v a r i a n c e o f , 8 1
o f t h e c u r l , 5 8 , 6 9 , 7 0 , 2 1 1
o f t h e g r a d i e n t , 5 8 , 6 4
p h y s i c a l s i g n i f i c a n c e o f , 6 6 , 6 7 , 1 1 9 , 1 2 0
t e n s o r f o r m o f , 1 7 4 , 2 0 0 , 2 0 1
t h e o r e m , ( s e e D i v e r g e n c e t h e o r e m )
D i v e r g e n c e t h e o r e m , 1 0 6 , 1 1 0 , 1 1 1 , 1 1 5 - 1 2 7
e x p r e s s e d i n w o r d s , 1 1 5
G r e e n ' s t h e o r e m a s a s p e c i a l c a s e o f , 1 0 6 , 1 1 0 , 1 1 1
p h y s i c a l s i g n i f i c a n c e o f , 1 1 6 , 1 1 7
p r o o f o f , 1 1 7 , 1 1 8
r e c t a n g u l a r f o r m o f , 1 1 6
t e n s o r f o r m o f , 2 0 6
D o t p r o d u c t , 1 6 , 1 8 - 2 1
c o m m u t a t i v e l a w f o r , 1 6 , 1 8
d i s t r i b u t i v e l a w f o r , 1 6 , 1 8
D u m m y i n d e x , 1 6 7
D y a d , 7 3
D y a d i c , 7 3 - 7 5 , 8 1
D y n a m i c s , 3 8 , ( s e e a l s o M e c h a n i c s )
L a g r a n g e ' s e q u a t i o n s i n , 1 9 6 , 2 0 5
N e w t o n ' s l a w i n , ( s e e N e w t o n ' s l a w )
E c c e n t r i c i t y , 8 7
E i g e n v a l u e s , 2 1 0
E i n s t e i n , t h e o r y o f r e l a t i v i t y o f , 1 4 8 , 2 0 7 , 2 1 3
E l e c t r o m a g n e t i c t h e o r y , 5 4 , 7 2 , 2 0 6
E l e m e n t , l i n e , 1 7 0 , 1 8 7 - 1 8 9
v o l u m e , 1 3 6 , 1 3 7 , 1 5 9
E l e m e n t s , o f a m a t r i x , 1 6 9
E l l i p s e , 6 3 , 1 3 9
a r e a o f , 1 1 2
m o t i o n o f p l a n e t i n , 8 6 , 8 7
E l l i p s o i d a l c o o r d i n a t e s , 1 4 0 , 1 6 0
E l l i p t i c c y l i n d r i c a l c o o r d i n a t e s , 1 3 9 , 1 5 5 , 1 6 0 , 1 6 1 ,
2 1 1
E n e r g y , 9 4
c o n s e r v a t i o n o f , 9 4
k i n e t i c , 9 4 , 2 0 4
I N D E X
E n e r g y ,
p o t e n t i a l , 9 4
E q u a l i t y , o f m a t r i c e s , 1 7 0
o f v e c t o r s , 1
E q u i l i b r a n t , 6
E u c l i d e a n s p a c e s , 1 7 0
N d i m e n s i o n a l , 1 7 1
E u l e r ' s e q u a t i o n s , 1 9 6
E x a c t d i f f e r e n t i a l s , 8 3 , 9 3 , 1 1 1
n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n f o r , 9 3
E x t r e m u m , 1 9 6
F i c t i t i o u s f o r c e s , 5 3
F i e l d , ( s e e S c a l a r a n d V e c t o r f i e l d )
c o n s e r v a t i v e , ( s e e C o n s e r v a t i v e f i e l d )
i r r o t a t i o n a l , 7 2 , 7 3 , 9 0
s i n k , 1 3 , ( s e e a l s o S i n k )
s o l e n o i d a l , 6 7 , 7 3 , 1 2 0 , 1 2 6
s o u r c e , 1 3 , ( s e e a l s o S o u r c e )
t e n s o r s , 1 6 8
v o r t e x , 7 2
F i x e d a n d m o v i n g s y s t e m s , o b s e r v e r s i n , 5 1 - 5 3
F l u i d m e c h a n i c s , 8 2
F l u i d m o t i o n , 6 6 , 6 7 , 7 2 , 1 1 6 , 1 1 7 , 1 2 5 , 1 2 6
i n c o m p r e s s i b l e , 6 7 , 1 2 6
F l u x , 8 3 , 1 2 0
F o r c e , c e n t r a l , 5 6 , 8 5
C o r i o l i s , 5 3
m o m e n t o f , 2 5 , 2 6 , 5 0
o n a p a r t i c l e , 2 0 3 , 2 0 5
r e p u l s i v e , 8 5
u n i v e r s a l g r a v i t a t i o n a l , 8 6
F o r c e s , f i c t i t i o u s , 5 3
r e a l , 5 3
r e s u l t a n t o f , 1 1
F r a m e s o f r e f e r e n c e , 5 8 , 1 6 6
F r e e i n d e x , 1 6 7
F r e n e t - S e r r e t f o r m u l a s , 3 8 , 4 5 , 2 1 3
F u n d a m e n t a l q u a d r a t i c f o r m , 1 4 8
F u n d a m e n t a l t e n s o r , 1 7 1
G a u s s ' d i v e r g e n c e t h e o r e m , ( s e e D i v e r g e n c e t h e o r e m )
G a u s s ' l a w , 1 3 4
G a u s s ' t h e o r e m , 1 2 4 , 1 2 5
G e o d e s i c s , 1 7 2 , 1 7 3 , 1 9 6 , 1 9 7 , 2 1 1
G e o m e t r y , d i f f e r e n t i a l , ( s e e D i f f e r e n t i a l g e o m e t r y )
G r a d , ( s e e G r a d i e n t )
G r a d i e n t , 5 7 , 5 8 , 5 9 - 6 3 , 1 7 7
i n c y l i n d r i c a l c o o r d i n a t e s , 1 5 3 , 1 5 4
i n o r t h o g o n a l c u r v i l i n e a r c o o r d i n a t e s , 1 3 7 , 1 4 8 , 1 4 9
i n p a r a b o l i c c y l i n d r i c a l c o o r d i n a t e s , 1 6 1 , 2 1 1
i n s p h e r i c a l c o o r d i n a t e s , 1 6 1
i n t e g r a l d e f i n i t i o n o f , 1 2 2 , 1 2 3
i n v a r i a n c e o f , 7 7
o f a v e c t o r , 7 3
t e n s o r f o r m o f , 1 7 4 , 2 0 0
G r a p h i c a l , a d d i t i o n o f v e c t o r s , 4
r e p r e s e n t a t i o n o f a v e c t o r , 1
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I N D E X
G r a v i t a t i o n , N e w t o n ' s u n i v e r s a l l a w o f , 8 6
G r e e n ' s , f i r s t i d e n t i t y o r t h e o r e m , 1 0 7 , 1 2 1
s e c o n d i d e n t i t y o r s y m m e t r i c a l t h e o r e m , 1 0 7 , 1 2 1
t h e o r e m i n s p a c e , ( s e e D i v e r g e n c e t h e o r e m )
G r e e n ' s t h e o r e m i n t h e p l a n e , 1 0 6 , 1 0 8 - 1 1 5
a s s p e c i a l c a s e o f S t o k e s ' t h e o r e m , 1 0 6 , 1 1 0
a s s p e c i a l c a s e o f t h e d i v e r g e n c e t h e o r e m , 1 0 6 ,
1 1 0 , 1 1 1
f o r m u l t i p l y - c o n n e c t e d r e g i o n s , 1 1 2 - 1 1 4
f o r s i m p l y - c o n n e c t e d r e g i o n s , 1 0 8 - 1 1 0
H a m i l t o n - C a y l e y t h e o r e m , 2 1 0
H a m i l t o n ' s p r i n c i p l e , 2 0 5
H e a t , 1 2 6 , 1 2 7
s p e c i f i c , 1 2 6
H e a t e q u a t i o n , 1 2 6 , 1 2 7 , 1 6 1
i n e l l i p t i c c y l i n d r i c a l c o o r d i n a t e s , 1 5 5
i n s p h e r i c a l c o o r d i n a t e s , 1 6 1
H e a t f l o w , s t e a d y - s t a t e , 1 2 7
H e l i x , c i r c u l a r , 4 5
H y p e r b o l a , 8 7
H y p e r p l a n e , 1 7 6
H y p e r s p h e r e , 1 7 6
H y p e r s u r f a c e , 1 7 6
H y p o c y c l o i d , 1 3 2
I n d e p e n d e n c e , o f o r i g i n , 9
o f p a t h o f i n t e g r a t i o n , 8 3 , 8 9 , 9 0 , 1 1 1 , 1 1 4 , 1 2 9 , 1 3 0
I n d e p e n d e n t , l i n e a r l y , 1 0 , 1 5
I n d e x , d u m m y o r u m b r a l , 1 6 7
f r e e , 1 6 7
I n e r t i a l s y s t e m s , 5 3
I n i t i a l p o i n t o f a v e c t o r , 1
I n n e r m u l t i p l i c a t i o n , 1 6 9 , 1 8 2
I n n e r p r o d u c t , 1 6 9 , 1 8 2
I n t e g r a l o p e r a t o r f o r m f o r V , 1 0 7 , 1 2 3
I n t e g r a l s , o f v e c t o r s , 8 2 - 1 0 5
d e f i n i t e , 8 2
i n d e f i n i t e , 8 2
l i n e , ( s e e L i n e i n t e g r a l s )
o r d i n a r y , 8 2
s u r f a c e , ( s e e S u r f a c e i n t e g r a l s )
t h e o r e m s o n , ( s e e I n t e g r a l t h e o r e m s )
v o l u m e , ( s e e V o l u m e i n t e g r a l s )
I n t e g r a l t h e o r e m s , 1 0 7 , 1 2 0 , 1 2 1 , 1 2 4 , 1 2 5 , 1 3 0 ,
( s e e a l s o S t o k e s ' t h e o r e m a n d D i v e r g e n c e t h e o r e m )
I n t e g r a t i o n , ( s e e I n t e g r a l s , o f v e c t o r s )
I n t r i n s i c d e r i v a t i v e , 1 7 4 , 2 0 2 , 2 1 1
I n v a r i a n c e , 5 8 , 5 9 , 7 6 , 7 7 , 8 1 , ( s e e a l s o I n v a r i a n t )
I n v a r i a n t , 5 9 , 1 6 8 , 1 9 0 , ( s e e a l s o I n v a r i a n c e )
I n v e r s e o f a m a t r i x , 1 7 0
I r r o t a t i o n a l f i e l d , 7 2 , 7 3 , 9 0
J a c o b i a n , 7 9 , 1 3 3 , 1 4 6 , 1 4 7 , 1 4 8 , 1 5 9 , 1 6 1 , 1 6 2 , 1 7 5 , 2 0 2 - 3
K e p l e r ' s l a w s , 8 6 , 8 7 , 1 0 2
K i n e m a t i c s , 3 8 , ( s e e a l s o D y n a m i c s a n d M e c h a n i c s )
K i n e t i c e n e r g y , 9 4 , 2 0 4
K r o n e c k e r d e l t a , 1 6 8 , 1 7 9 , 1 8 0
K r o n e c k e r ' s s y m b o l , 7 7 , 2 0 8
2 2 1
L a g r a n g e a n , 2 0 5
L a g r a n g e ' s e q u a t i o n s , 1 9 6 , 2 0 5
L a p l a c e ' s e q u a t i o n , 6 5 , 1 2 7 , 1 3 4
i n p a r a b o l i c c y l i n d r i c a l c o o r d i n a t e s , 1 5 4 , 1 5 5
L a p l a c e t r a n s f o r m s , 1 6 2
L a p l a c i a n o p e r a t o r ( V 2 ) , 5 8 , 6 4 , 8 1 , 2 0 0
i n c u r v i l i n e a r c o o r d i n a t e s , 1 3 7 , 1 5 0 , 1 5 1
i n c y l i n d r i c a l c o o r d i n a t e s , 1 5 3 , 1 5 4 , 2 0 1
i n p a r a b o l i c c y l i n d r i c a l c o o r d i n a t e s , 1 5 4 , 1 5 5 , 2 1 1
i n s p h e r i c a l c o o r d i n a t e s , 1 5 4 , 2 0 1
i n v a r i a n c e o f , 8 1
t e n s o r f o r m o f , 1 7 4 , 2 0 0
L a w s o f v e c t o r a l g e b r a , 2 , 1 8
L e m n i s c a t e , 1 3 2
L e n g t h , o f a v e c t o r , 1 7 1 , 1 7 2 , 1 9 0
L i g h t r a y s , 6 3
L i g h t , v e l o c i t y o f , 8 1
L i n e a r l y d e p e n d e n t v e c t o r s , 1 0 , 1 5
L i n e e l e m e n t , 1 7 0 , 1 8 7 - 1 8 9
L i n e , e q u a t i o n o f , 9 , 1 2
p a r a m e t r i c e q u a t i o n s o f , 1 2
s i n k , 1 3
s o u r c e , 1 3
s y m m e t r i c f o r m f o r e q u a t i o n o f , 9
L i n e i n t e g r a l s , 8 2 , 8 7 - 9 4 , 1 1 1
c i r c u l a t i o n i n t e r m s o f , 8 2 , 1 3 1
e v a l u a t i o n o f , 8 7 - 8 9 , 1 1 1
G r e e n ' s t h e o r e m a n d e v a l u a t i o n o f , 1 1 2
i n d e p e n d e n c e o f p a t h , 8 3 , 8 9 , 9 0 , 1 1 1 , 1 1 4 , 1 2 9 , 1 3 0
w o r k e x p r e s s e d i n t e r m s o f , 8 2 , 8 8
L o r e n t z - F i t z g e r a l d c o n t r a c t i o n , 2 1 3
L o r e n t z t r a n s f o r m a t i o n , 2 1 3
M a g n i t u d e , o f a v e c t o r , 1
M a i n d i a g o n a l , 1 6 9
M a p p i n g , 1 6 2
M a t r i c e s , 1 6 9 , 1 7 0 , 1 8 5 , 1 8 6 , ( s e e a l s o M a t r i x )
a d d i t i o n o f , 1 7 0
c o n f o r m a b l e , 1 7 0
e q u a l i t y o f , 1 7 0
o p e r a t i o n s w i t h , 1 7 0
M a t r i x , 7 3 , 1 6 9 , ( s e e a l s o M a t r i c e s )
a l g e b r a , 1 7 0
c o l u m n , 1 6 9
d e t e r m i n a n t o f , 1 7 0 , 2 0 9
e l e m e n t s o f , 1 6 9
i n v e r s e o f , 1 7 0 , 2 0 9 , 2 1 0
m a i n o r p r i n c i p a l d i a g o n a l o f , 1 6 9
n u l l , 1 6 9
o r d e r o f , 1 6 9
p r i n c i p a l d i a g o n a l o f , 1 6 9
r o w , 1 6 9
s i n g u l a r , 1 7 0
s q u a r e , 1 6 9
t r a n s p o s e o f , 1 7 0 , 2 1 0
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2 2 2
M a x w e l l ' s e q u a t i o n s , 7 2 , 8 1
i n t e n s o r f o r m , 2 0 6
M e c h a n i c s , 3 8 , 5 6 , ( s e e a l s o D y n a m i c s )
f l u i d , 8 2
M e t r i c c o e f f i c i e n t s , 1 4 8
M e t r i c f o r m , 1 4 8
M e t r i c t e n s o r , 1 7 0 , 1 7 1 , 1 8 7 - 1 8 9
M i x e d t e n s o r , 1 6 7 , 1 6 8
M o e b i u s s t r i p , 9 9
M o m e n t o f f o r c e , 2 5 , 2 6 , . 5 0
M o m e n t u m , 3 8
a n g u l a r , 5 0 , 5 1 , 5 6
M o t i o n , a b s o l u t e , 5 3
M o t i o n , o f f l u i d , ( s e e F l u i d m o t i o n )
o f p l a n e t s , 8 5 - 8 7
M o v i n g a n d f i x e d s y s t e m s , o b s e r v e r s i n , 5 1 - 5 3
M o v i n g t r i h e d r a l , 3 8
M u l t i p l i c a t i o n , ( s e e P r o d u c t )
M u l t i p l y - c o n n e c t e d r e g i o n , 1 1 0 , 1 1 2 - 1 1 4
N a b l a , ( s e e D e l )
N e g a t i v e d i r e c t i o n , 8 9
N e w t o n ' s l a w , 3 8 , 5 0 , 5 3
i n t e n s o r f o r m , 2 0 3
o f u n i v e r s a l g r a v i t a t i o n , 8 6
N o r m a l p l a n e , 3 8 , 4 8
N o r m a l , p r i n c i p a l , 3 8 , 4 5 , 4 7 , 4 8 , 5 0
b i - , 3 8 , 4 5 , 4 7 , 4 8
N o r m a l , t o a s u r f a c e , 4 9 , 5 0 , 5 6 , 6 1
p o s i t i v e o r o u t w a r d d r a w n , 4 9 , 8 3
N u l l m a t r i x , 1 6 9
N u l l v e c t o r , 2
I N D E X
O b l a t e s p h e r o i d a l c o o r d i n a t e s , 1 4 0 , 1 4 5 , 1 6 0 , 1 6 1
O p e r a t i o n s , w i t h t e n s o r s , 1 6 9 , 1 7 9 - 1 8 4
O p e r a t o r , d e l , 5 7 , ( s e e a l s o D e l )
L a p l a c i a n , ( s e e L a p l a c i a n o p e r a t o r )
t i m e d e r i v a t i v e , i n f i x e d a n d m o v i n g s y s t e m s ,
5 1 , 5 2
O r d e r , o f a m a t r i x , 1 6 9
o f a t e n s o r , 1 6 7
O r i e n t a b l e s u r f a c e , 9 9
O r i g i n , o f a v e c t o r , 1
i n d e p e n d e n c e o f v e c t o r e q u a t i o n o n , 9
O r t h o c e n t e r , 3 3
O r t h o g o n a l c o o r d i n a t e s , s p e c i a l , 1 3 7 - 1 4 1
b i p o l a r , 1 4 0 , 1 6 0
c y l i n d r i c a l , 1 3 7 , 1 3 8 , ( s e e C y l i n d r i c a l c o o r d i n a t e s )
e l l i p s o i d a l , 1 4 0 , 1 6 0
e l l i p t i c c y l i n d r i c a l , 1 3 9 , 1 5 5 , 1 6 0 , 1 6 1 , 2 1 1
o b l a t e s p h e r o i d a l , 1 4 0 , 1 4 5 , 1 6 0 , 1 6 1
p a r a b o l i c c y l i n d r i c a l , 1 3 8 , ( s e e a l s o P a r a b o l i c
c y l i n d r i c a l c o o r d i n a t e s )
p a r a b o l o i d a l , 1 3 9 , 1 6 0 , 1 6 1 , 2 1 1
p r o l a t e s p h e r o i d a l , 1 3 9 , 1 6 0 , 1 6 1
s p h e r i c a l , 1 3 7 , 1 3 8 , ( s e e S p h e r i c a l c o o r d i n a t e s )
t o r o i d a l , 1 4 1
O r t h o g o n a l c u r v i l i n e a r c o o r d i n a t e s y s t e m s , 4 9 , 1 3 5 ,
1 9 1
s p e c i a l , 1 3 7 - 1 4 1
O r t h o g o n a l t r a n s f o r m a t i o n , 5 9
O s c u l a t i n g p l a n e , 3 8 , 4 8
O u t e r m u l t i p l i c a t i o n , 1 6 9
O u t e r p r o d u c t , 1 6 9
O u t w a r d d r a w n n o r m a l , 4 9 , 8 3
P a r a b o l a , 8 7 , 1 3 8
P a r a b o l i c c y l i n d r i c a l c o o r d i n a t e s , 1 3 8 , 1 4 4 , 1 4 5 , 1 5 4
1 5 5 , 1 6 0 , 1 6 1 , 2 1 1
a r e l e n g t h i n , 1 4 4
C h r i s t o f f e l ' s s y m b o l s i n , 2 1 1
c u r l i n , 1 6 1
d i v e r g e n c e i n , 1 6 1
g r a d i e n t i n , 1 6 1 , 2 1 1
J a c o b i a n i n , 1 6 1
L a p l a c i a n i n , 1 5 4 , 1 5 5 , 2 1 1
S c h r o e d i n g e r ' s e q u a t i o n i n , 1 6 1
v o l u m e e l e m e n t i n , 1 4 5
P a r a b o l o i d a l c o o r d i n a t e s , 1 3 9 , 1 6 0 , 1 6 1 , 2 1 1
P a r a l l e l o g r a m , a r e a o f , 1 7 , 2 4
P a r a l l e l o g r a m l a w o f v e c t o r a d d i t i o n , 2 , 4
P a r a m e t r i c e q u a t i o n s , o f a c u r v e , 3 9 , 4 0
o f a l i n e , 1 2
o f a s u r f a c e , 4 8 , 4 9
P e r i o d s , o f p l a n e t s , 1 0 2
P e r m u t a t i o n s y m b o l s a n d t e n s o r s , 1 7 3 , 1 7 4 , 2 1 1
P h y s i c a l c o m p o n e n t s , 1 7 2 , 2 0 0 , 2 0 1 , 2 0 5 , 2 1 1
P l a n e , d i s t a n c e f r o m o r i g i n t o , 2 1
e q u a t i o n o f , 1 5 , 2 1 , 2 8
n o r m a l , 3 8 , 4 8
o s c u l a t i n g , 3 8 , 4 8
r e c t i f y i n g , 3 8 , 4 8
t a n g e n t , 4 9 , 5 0 , 6 1
v e c t o r p e r p e n d i c u l a r t o , 2 8
v e c t o r s i n a , ( s e e C o p l a n a r v e c t o r s )
P l a n e t s , m o t i o n o f , 8 5 - 8 7
P o i n t f u n c t i o n , s c a l a r a n d v e c t o r , 3
P o i s s o n ' s e q u a t i o n , 1 3 4
P o l a r c o o r d i n a t e s , 9 8
P o s i t i o n v e c t o r , 3
P o s i t i v e d i r e c t i o n , 8 9 , 1 0 6 , 1 1 3
P o s i t i v e n o r m a l , 8 3
P o t e n t i a l e n e r g y , 9 4
P o t e n t i a l , s c a l a r , 7 3 , 8 1 , 8 3 , 9 1 , 9 2
v e c t o r , 8 1
P r i n c i p a l d i a g o n a l , 1 6 9
P r i n c i p a l n o r m a l , 3 8 , 4 5 , 4 7 , 4 8 , 5 0
P r o d u c t , b o x , 1 7
c r o s s , ( s e e C r o s s p r o d u c t )
d o t , ( s e e D o t p r o d u c t )
i n n e r , 1 6 9 , 1 8 2
o f a v e c t o r b y a s c a l a r , 2
o f d e t e r m i n a n t s , 1 5 9
o f m a t r i c e s , 1 7 0
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I N D E X
2 2 3
P r o d u c t ,
o f t e n s o r s , 1 6 9
o u t e r , 1 6 9 , 1 8 1
s c a l a r , 1 8 2 , ( s e e a l s o D o t p r o d u c t )
v e c t o r , ( s e e C r o s s p r o d u c t )
P r o j e c t i l e , 1 0 2
P r o j e c t i o n , o f a v e c t o r , 1 8 , 2 0
o f s u r f a c e s , 9 5 , 9 6
P r o l a t e s p h e r o i d a l c o o r d i n a t e s , 1 3 9 , 1 6 0 , 1 6 1
P r o p e r v e c t o r , 2
P y t h a g o r e a n t h e o r e m , 1 0
Q u a d r a t i c f o r m , f u n d a m e n t a l , 1 4 8
Q u a n t u m m e c h a n i c s , 1 6 1
Q u o t i e n t l a w , 1 6 9 , 1 8 4
R a d i u s , o f c u r v a t u r e , 3 8 , 4 5 , 4 6 , 5 0
o f t o r s i o n , 3 8 , 4 5
R a d i u s v e c t o r , 3
R a n k , o f a t e n s o r , 1 6 7
R a n k z e r o t e n s o r , 1 6 8
R e a l f o r c e s , 5 3
R e c i p r o c a l s e t s o r s y s t e m s o f v e c t o r s , 1 7 , 3 0 , 3 1
3 4 , 1 3 6 , 1 4 7
R e c i p r o c a l t e n s o r s , 1 7 1
R e c t a n g u l a r c o m p o n e n t v e c t o r s , 3
R e c t a n g u l a r c o o r d i n a t e s y s t e m s , 2
R e c t i f y i n g p l a n e , 3 8 , 4 8
R e g i o n , m u l t i p l y - c o n n e c t e d , 1 1 0 , 1 1 2 - 1 1 4
s i m p l y - c o n n e c t e d , 1 1 0 , 1 1 3 , 1 1 4
R e l a t i v e a c c e l e r a t i o n , 5 3
R e l a t i v e t e n s o r , 1 7 5 , 2 0 2 , 2 0 3 , 2 1 2
R e l a t i v e v e l o c i t y , 5 2
R e l a t i v i t y , t h e o r y o f , 1 4 8 , 2 0 7 , 2 1 3
R e s u l t a n t o f v e c t o r s , 2 , 4 , 5 , 6 , 1 0
R i e m a n n - C h r i s t o f f e l t e n s o r , 2 0 7 , 2 1 2
R i e m a n n i a n s p a c e , 1 7 1 , 1 7 2
g e o d e s i c s i n , 1 7 2 , 1 9 6 , 1 9 7
R i g h t - h a n d e d c o o r d i n a t e s y s t e m s , 2 , 3
l o c a l i z e d , 3 8
R i g i d b o d y , m o t i o n o f , 5 9
v e l o c i t y o f , 2 6 , 3 3
R o t , ( s e e C u r l )
R o t a t i n g c o o r d i n a t e s y s t e m s , 5 1 , 5 2
R o t a t i o n , i n v a r i a n c e u n d e r , ( s e e I n v a r i a n c e )
o f a x e s , 5 8 , 7 6 , 7 7
p u r e , 5 9
R o w m a t r i x o r v e c t o r , 1 6 9
S c a l a r , 1 , 4 , 1 6 8
f i e l d , 3 , 1 2 , 1 6 8
f u n c t i o n o f p o s i t i o n , 3
p o i n t f u n c t i o n , 3
p o t e n t i a l , 7 3 , 8 1 , 8 3 , 9 1 , 9 2
p r o d u c t , 1 8 2 , ( s e e a l s o D o t p r o d u c t )
t r i p l e p r o d u c t s , ( s e e T r i p l e p r o d u c t s )
v a r i a b l e , 3 5
S c a l e f a c t o r s , 1 3 5
S c h r o e d i n g e r ' s e q u a t i o n , 1 6 1
S i m p l e c l o s e d c u r v e , 8 2 , 1 0 6
a r e a b o u n d e d b y , 1 1 1
S i m p l y - c o n n e c t e d r e g i o n , 1 1 0 , 1 1 3 , 1 1 4
S i n e s , l a w o f , f o r p l a n e t r i a n g l e s , 2 5
f o r s p h e r i c a l t r i a n g l e s , 2 9 , 3 0
S i n g u l a r m a t r i x , 1 7 0
S i n g u l a r p o i n t s , 1 4 1
S i n k , 1 3 , 6 7 , 1 2 0
S i n k f i e l d , 1 3 , ( s e e a l s o S i n k )
S o l e n o i d a l f i e l d , 6 7 , 7 3 , 1 2 0 , 1 2 6
S o l i d a n g l e , 1 2 4 , 1 2 5
S o u n d r a y s , 6 3
S o u r c e , 1 3 , 6 7 , 1 2 0
S o u r c e f i e l d , 1 3 , ( s e e a l s o S o u r c e )
S p a c e c u r v e s , 3 5
a c c e l e r a t i o n a l o n g , 3 5 , 3 9 , 4 0 , 5 0 , 5 6
a r c l e n g t h o f , 3 7 , 5 6 , 1 3 6 , 1 4 8
b i n o r m a l o f , 3 8 , 4 5 , 4 7 , 4 8
c u r v a t u r e o f , 3 8 , 4 5 , 4 7 , 1 1 3
p r i n c i p a l n o r m a l o f , 3 8 , 4 5 , 4 7 , 4 8 , 5 0
r a d i u s o f c u r v a t u r e o f , 3 8 , 4 5 , 4 6 , 5 0
r a d i u s o f t o r s i o n o f , 3 8 , 4 5
t a n g e n t t o , 3 7 , 3 8 , 4 0 , 4 5 , 4 7 , 4 8 , 5 0
S p a c e i n t e g r a l s , ( s e e V o l u m e i n t e g r a l s )
S p a c e s , E u c l i d e a n , 1 7 0
R i e m a n n i a n , 1 7 1
S p a c e , N d i m e n s i o n a l , 1 6 6
S p e c i a l t h e o r y o f r e l a t i v i t y , 2 1 3
S p e e d , 4
a n g u l a r , 2 6 , 4 3 , 5 2
S p h e r i c a l c o o r d i n a t e s , 1 3 7 , 1 3 8 , 1 4 1 , 1 4 7 , 1 6 0 , 1 6 1
a r e l e n g t h i n , 1 4 4
C h r i s t o f f e l ' s s y m b o l s i n , 1 9 5 , 2 1 1
c o n j u g a t e m e t r i c t e n s o r i n , 1 8 9
c o n t i n u i t y e q u a t i o n i n , 2 1 2
c o v a r i a n t c o m p o n e n t s i n , 1 7 7 , 1 7 8
c u r l i n , 1 5 4
d i v e r g e n c e i n , 1 6 1 , 2 0 0 , 2 0 1
g e o d e s i c s i n , 2 1 1
g r a d i e n t i n , 1 6 1
h e a t e q u a t i o n i n , 1 6 1
J a c o b i a n i n , 1 6 1
L a p l a c i a n i n , 1 5 4 , 2 0 1
m e t r i c t e n s o r i n , 1 8 7
v e l o c i t y a n d a c c e l e r a t i o n i n , 1 6 0 , 2 1 2
v o l u m e e l e m e n t i n , 1 4 4 , 1 4 5
S p h e r o i d a l c o o r d i n a t e s , o b l a t e , 1 4 0 , 1 4 5 , 1 6 0 , 1 6 1
p r o l a t e , 1 3 9 , 1 6 0 , 1 6 1
S t a t i o n a r y s c a l a r f i e l d , 3
S t a t i o n a r y - s t a t e , ( s e e S t e a d y - s t a t e )
S t e a d y - s t a t e , h e a t f l o w , 1 2 7
s c a l a r f i e l d , 3
v e c t o r f i e l d , 3
S t o k e s ' t h e o r e m , 1 0 6 , 1 1 0 , 1 2 7 - 1 3 1
G r e e n ' s t h e o r e m a s s p e c i a l c a s e o f , 1 1 0
p r o o f o f , 1 2 7 - 1 2 9
t e n s o r f o r m o f , 2 1 2
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2 2 4
I N D E X
S u b t r a c t i o n , o f t e n s o r s , 1 6 9
o f v e c t o r s , 2
S u m m a t i o n c o n v e n t i o n , 1 6 7 , 1 7 5 , 1 7 6 , 2 0 7
S u p e r s c r i p t s , 1 6 6
S u r f a c e , a r e a o f , 1 0 4 , 1 0 5 , 1 6 2
S u r f a c e c u r v i l i n e a r c o o r d i n a t e s , 4 8 , 4 9 , 5 6 , 1 5 5
a r c l e n g t h i n , 5 6 , 1 4 8
S u r f a c e i n t e g r a l s , 8 3 , 9 4 - 9 9
d e f i n e d a s l i m i t o f a s u m , 9 4 , 9 5
e v a l u a t i o n o f , 8 3
S u r f a c e s , 3 7
a n g l e b e t w e e n , 6 3
a r c l e n g t h o n , 5 6
c o o r d i n a t e , 1 3 5
o n e - s i d e d , 9 9
o r i e n t a b l e , 9 9
o u t w a r d d r a w n n o r m a l t o , 8 3
t w o - s i d e d , 8 3
S y m m e t r i c f o r m , o f e q u a t i o n o f a l i n e , 9
T a n g e n t , t o s p a c e c u r v e , 3 7 , 3 8 , 4 0 , 4 5 , 4 7 , 4 8 , 5 0
T a n g e n t p l a n e , 4 9 , 5 0 , 6 1
T e n s o r a n a l y s i s , 7 3 , 1 3 7 , 1 5 8 , 1 6 6 - 2 1 7
T e n s o r , a b s o l u t e , 1 7 5
a s s o c i a t e d , 1 7 1 , 1 9 0 , 1 9 1 , 2 1 0
C a r t e s i a n , 2 1 0
c o n j u g a t e , 1 7 1
c o n t r a v a r i a n t , ( s e e C o n t r a v a r i a n t c o m p o n e n t s )
c o v a r i a n t , ( s e e C o v a r i a n t c o m p o n e n t s )
c u r v a t u r e , 2 0 7
d e n s i t y , 1 7 5 , 2 0 3
f i e l d , 1 6 8
f u n d a m e n t a l , 1 7 1
m e t r i c , 1 7 0
m i x e d , 1 6 7 , 1 6 8
o r d e r o f , 1 6 7
r a n k o f , 1 6 7
r e c i p r o c a l , 1 7 1
r e l a t i v e , 1 7 5 , 2 0 2 , 2 0 3 , 2 1 2
s k e w - s y m m e t r i c , 1 6 8 , 1 6 9
s y m m e t r i c , 1 6 8
T e n s o r s , f u n d a m e n t a l o p e r a t i o n s w i t h , 1 6 9 , 1 7 9 - 1 8 4
T e r m i n a l p o i n t o r T e r m i n u s , 1 , 2 , 5 , 1 1
T h e r m a l c o n d u c t i v i t y , 1 2 6
T o r o i d a l c o o r d i n a t e s , 1 4 1
T o r q u e , 5 0 , 5 1
T o r s i o n , 3 8 , 4 5 , 4 7 , 2 1 3
r a d i u s o f , 3 8 , 4 5
T r a n s f o r m a t i o n , a f f i n e , 5 9 , 2 1 0 , 2 1 3
o f c o o r d i n a t e s , 5 8 , 5 9 , 7 6 , 1 3 5 , 1 6 6
o r t h o g o n a l , 5 9
T r a n s l a t i o n , 5 9
T r a n s p o s e , o f a m a t r i x , 1 7 0 , 2 1 0
T r i a d , 3 8
T r i a d i c , 7 3
T r i a n g l e , a r e a o f , 2 4 , 2 5
T r i a n g l e l a w o f v e c t o r a d d i t i o n , 4
T r i h e d r a l , m o v i n g , 3 8
T r i p l e p r o d u c t s , 1 7 , 2 6 - 3 1
T w i s t e d c u b i c , 5 5
U m b r a l i n d e x , 1 6 7
U n i t d y a d s , 7 3
U n i t m a t r i x , 1 6 9
U n i t v e c t o r s , 2 , 1 1
r e c t a n g u l a r , 2 , 3
V a r i a b l e , 3 5 , 3 6
V e c t o r , a r e a , 2 5 , 8 3
c o l u m n , 1 6 9
e q u a t i o n s , 2 , 9
f i e l d , 3 , 1 2 , 1 3 , 1 6 8
f u n c t i o n o f p o s i t i o n , 3
m a g n i t u d e o f a , 1 , 1 0
n u l l , 2
o p e r a t o r V , ( s e e D e l )
p o i n t f u n c t i o n , 3
p o s i t i o n , 3
p o t e n t i a l , 8 1
p r o d u c t , ( s e e C r o s s p r o d u c t )
r a d i u s , 3
r o w , 1 6 9
t i m e d e r i v a t i v e o f a , 5 1 , 5 2
t r i p l e p r o d u c t , ( s e e T r i p l e p r o d u c t s )
V e c t o r s , 1 , 4
a d d i t i o n o f , 2 , 4
a l g e b r a o f , 1 , 2
a n a l y t i c a l r e p r e s e n t a t i o n o f , 1
a n g l e b e t w e e n , 1 9 , 1 7 2 , 1 9 0
b a s e , 7 , 8 , 1 3 6
c o l l i n e a r , ( s e e C o l l i n e a r v e c t o r s )
c o m p o n e n t , 3 , 7 , 8
c o n t r a v a r i a n t c o m p o n e n t s o f , 1 3 6 , 1 5 6 , 1 5 7 , 1 6 7
c o p l a n a r , ( s e e C o p l a n a r v e c t o r s )
c o v a r i a n t c o m p o n e n t s o f , 1 3 6 , 1 5 7 , 1 5 8 , 1 6 7
d i f f e r e n t i a t i o n o f , 3 5 - 5 6
e q u a l i t y o f , 1
g r a p h i c a l r e p r e s e n t a t i o n o f , 1 , 4
i n i t i a l p o i n t o f , 1
o r i g i n o f , 1
r e c i p r o c a l , 1 7
r e s u l t a n t o f , 2 , 4 , 5 , 6 , 1 0
t e r m i n a l p o i n t o f , 1
t e r m i n u s o f , 1
u n i t , 2
u n i t a r y , 1 3 6
V e l o c i t y , a l o n g a s p a c e c u r v e , 3 5 , 3 9 , 4 0
a n g u l a r , 2 6 , 4 3 , 5 2
a r e a l , 8 5 , 8 6
l i n e a r , 2 6
o f a f l u i d , 1 7 9
o f a p a r t i c l e , 4 2 , 5 2 , 2 0 3 , 2 0 4
o f l i g h t , 8 1
r e l a t i v e t o f i x e d a n d m o v i n g o b s e r v e r s , 5 2 , 5 3
V o l u m e , e l e m e n t s o f , 1 3 6 , 1 3 7 , 1 5 9
i n c u r v i l i n e a r c o o r d i n a t e s , 1 3 6 , 1 3 7
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I N D E X
2 2 5
V o l u m e ,
W a v e e q u a t i o n , 7 2
i n g e n e r a l c o o r d i n a t e s , 1 5 9
W e i g h t , o f a t e n s o r , 1 7 5
o f p a r a l l e l e p i p e d , 1 7 , 2 6
W o r k , 2 1 , 8 2 , 8 8 , 8 9 , 9 0 , 9 1
V o l u m e i n t e g r a l s , 8 3 , 9 9 - 1 0 1
a s a l i n e i n t e g r a l , 8 8 , 8 9 , 9 0 , 9 1
d e f i n e d a s l i m i t o f a s u m , 9 9 , 1 0 0
V o r t e x f i e l d , 7 2
Z e r o v e c t o r , 2
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C a t a l o g
I f y o u a r e i n t e r e s t e d i n a l i s t o f S C H A U M ' S
O U T L I N E S E R I E S s e n d y o u r n a m e
a n d a d d r e s s , r e q u e s t i n g y o u r f r e e c a t a l o g , t o
S C H A U M ' S O U T L I N E S E R I E S , D e p t . C
M c G R A W - H I L L B O O K C O M P A N Y
1 2 2 1 A v e n u e o f A m e r i c a s
N e w Y o r k , N . Y . 1 0 0 2 0
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a s t e r v e c t o r a n a l y s i s w i t h t h i s "
m p l e - t o - u s e s t u d y g u i d e . I t w i l l
e l p y o u c u t s t u d y t i m e , h o n e
r o b l e m - s o l v i n g s k i l l s , a n d a c h i e v e
u r p e r s o n a l b e s t o n e x a m s
e n t s l o v e S c h a u m ' s O u t l i n e s b e c a u s e t h e y p r o d u c e r s s u M S . E a c h y e a r ,
r e d s o f t h o u s a n d s o f s t u d e n t s i m p r o v e t h e i r l e s t s c o r e s a n d f i n a l g r a d e s
h t h e s e i n d i s p e n s a b l e s t u d y g u i d e s .
t h e e d g e o n y o u r c l a s s m a t e s U s e S c h a u m ' s '
u d o n ' t h a v e a l o t o f U r n s b u t w a n t t o e x c e l i n c l a s s . t h i s b o o k h e l p s
R e l a t e d T i t l e s i n
S c h a u m ' s o u t l i n e s
M a t h e m a t i c s a S t a t i s t i c s