lecture 1, 2

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Scalars and Vectors

Scalar Fields (temperature)Vector Fields (gravitational, magnetic)

 Vector: A quantity with both magnitude and direction.

 !ample: Force to the east i.e. F"#$ %

Scalar: A quantity that does not posses direction, &eal or comple!. 'emperature i.e. '" $

Vector Algebra

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Laws of Vectors Algebra

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Vector omponents and *nit Vectors

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'he +ot product

'he Vector Field

!ample

! ,−:=y -:=

/:=F

$., y !⋅−( )⋅

$

$$

! !⋅   y y⋅+   ⋅+−

:=

a) F /.01=

b)F

F

$.2

$

$.12-−

=

A.3 " 4A4434 cos θA3

3 in the direction o5 A

6ou need to normalie a

 be5ore the dot product.

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'he ross 7roduct

!ample

A

-−

#

:= 3

−

−

/

:=

A 3×

#-−

#−

#1−

=

A 3×

a!

A!

ay

Ay

a

A

A ! 3 " a %4A4434 sin θA3

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ircular ylindrical oordinate System

# 8 *nit Vector 

 vary with

'he coordinate φSince direction

changes

9 +ot 7roduct

ρ   dρ⋅   dφ⋅

ρ   d.⋅

ρ   dφ⋅   d.⋅

⋅ ⋅ ⋅

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ircular ylindrical oordinate System

!   ρ cos  φ( )⋅

y   ρ sin φ( )⋅

ρ !(

y(+ ρ $≥

φ atany

!

A A! a!⋅   Ay ay⋅+   A a⋅+

A Aρ  aρ⋅   Aφ aφ⋅+   A a⋅+

Aρ   A aρ⋅   Aφ   A aφ⋅   A A

Aρ   A!a!⋅   Ay ay⋅+   A a⋅+( ) aρ⋅   A!a!⋅   aρ⋅   Ay ay⋅   aρ⋅+

Aφ   A!a!⋅   Ay ay⋅+   A a⋅+( ) aφ⋅   A!a!⋅   aφ⋅   Ay ay⋅   aφ⋅+

A A! a!⋅   Ay ay⋅+   A a⋅+( ) a⋅   A a⋅   a⋅   A

a aρ⋅   a φ⋅   $

+ot 7roduct

a! aρ⋅   cos  φ( )   ay aρ⋅   sin  φ( )   a a⋅   #

a! aφ⋅   sin φ( )−   ay aφ⋅   cos  φ( )

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'he Spherical oordinate System! r s in   θ( )⋅   cos   φ( )⋅

y r sin   θ   sin   φ( )⋅( )⋅

. r cos   θ( )⋅

r !(

y(+

(+ r $≥

θ acos

!(

y(+

(+

 $   θ≤ #$≤

φ atany

!

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'he Spherical oordinate System

! r s in   θ( )⋅   cos   φ( )⋅

y r sin   θ   sin   φ( )⋅( )⋅

. r cos   θ( )⋅

r dr ⋅ dθ⋅

r s in  θ( )⋅ dr ⋅ dφ⋅

r 

sin θ( )⋅ dθ⋅ dφ⋅

(⋅ ⋅ ⋅ ⋅

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;omewor< hapter #

Sur5ace=Area #.1$0=

Sur5ace=Area r#=sur5ace r=sur5ace+ θ#=sur5ace+ θ=sur5ace+ φ =sur5ace+:=

θ=sur5ace

φ#

φ

φr#

r

r r sin  θ( )⋅⌠ ⌡

d⌠ ⌡

d:=φ =sur5ace .$0=

θ#=sur5aceφ#

φ

φr#

r

r r sin  θ#( )⋅⌠ ⌡ d

⌠ ⌡ d:=

φ =sur5ace

r#

r

r 

θ#

θ

θr ⌠ ⌡

d⌠ ⌡

d:=

r=sur5ace .0=r#=sur5ace $.1-=

r=sur5ace

φ#

φ

φθ#

θ

θr

sin θ( )⋅⌠ ⌡

d⌠ ⌡

d:=r#=sur5ace

φ#

φ

φθ#

θ

θr#

sin θ( )⋅⌠ ⌡

d⌠ ⌡

d:=

 Again, the surface area consists of six sides: 2 different surfaces where θ is constant, 2

identical surfaces where φ  is constant, and 2 different surfaces where r is constant.

b)

Volume .0$0=Volume

r#

r

r 

φ#

φ

φθ#

θ

θr 

sin θ( )⋅⌠ ⌡

d⌠ ⌡

d⌠ ⌡

d:=a)

For the volume enclosed within these spherical coordinates, find a) the enclosed volume b) the

surface area c) the total length of the twelve edges of the surface d) the length of the longest straightline that lies within the volume.

φ 1$   π⋅-1$

⋅:=φ# $   π⋅-1$

⋅:=θ /$   π⋅-1$

⋅:=θ# -$   π⋅-1$

⋅:=r :=r# :=2.