lecture 1, 2

7/18/2019 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

7/18/2019 Lecture 1, 2


Laws of Vectors Algebra

7/18/2019 Lecture 1, 2


7/18/2019 Lecture 1, 2


Vector omponents and *nit Vectors

7/18/2019 Lecture 1, 2


'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.

7/18/2019 Lecture 1, 2


'he ross 7roduct

!ample

A

-−

#

:= 3

−

−

/

:=

A 3×

#-−

#−

#1−

=

A 3×

a!

A!

ay

Ay

a

A

A ! 3 " a %4A4434 sin θA3

7/18/2019 Lecture 1, 2


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.⋅

⋅ ⋅ ⋅

7/18/2019 Lecture 1, 2


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( )

7/18/2019 Lecture 1, 2


'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

!

7/18/2019 Lecture 1, 2


'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⋅

(⋅ ⋅ ⋅ ⋅

7/18/2019 Lecture 1, 2


;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.

lecture 1, 2

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