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Spherical Polar Coordinates: X R Sin Cos y R Sin Sin Z R Cos

The document introduces spherical polar coordinates and cylindrical polar coordinates. It provides the equations to express coordinates, line elements, area elements, and volume elements in both coordinate systems. It also gives the expressions for gradient, divergence, and curl operators in terms of the spherical and cylindrical coordinate variables.

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Mosa A Shorman
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163 views6 pages

Spherical Polar Coordinates: X R Sin Cos y R Sin Sin Z R Cos

The document introduces spherical polar coordinates and cylindrical polar coordinates. It provides the equations to express coordinates, line elements, area elements, and volume elements in both coordinate systems. It also gives the expressions for gradient, divergence, and curl operators in terms of the spherical and cylindrical coordinate variables.

Uploaded by

Mosa A Shorman
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Spherical Polar Coordinates

In EM it may be easier to express a problem within a “natural”


symmetry

x=r sin cos 


y=r sin sin 
z=r cos 

r : [0,∞ ]; : [0, ] ; :[ 0,2 ]


Spherical Polar Coordinates
Spherical Line Element

 sin  d  
d l=dr r r d  r 

Spherical Area Element


2
d a=d l  dl  r =r sin d  d  r
Spherical Volume
Element

2
d =dl r d l  dl =r sin  dr d  d 
Spherical Polar Coordinates
All the vector operators can be expressed in other coord
systems

∂T 1 ∂T  1 ∂T 
∇ T= r   
∂r r ∂ r sin  ∂
1 ∂ 2 1 ∂ 1 ∂v 
∇⋅v= 2
r v r  sin v  
r ∂r r sin  ∂ d  r sin  ∂ 

∇ ×v=
1
r sin  ∂[ ∂
sin v −
∂ v
∂ ] [
r 
1
r sin
1

∂ vr
∂
−
∂
∂r ]
r v  


[
1 ∂
r ∂r
r v −
]∂ vr
∂


Cylindrical Polar Coords

x=r cos  2
r= x y 
2

y=r sin  tan =y / x


z=z z=z

r : [0,∞ ]; : [0,2 ]; z :[−∞ , ∞]


Cylindrical Polar Coordinates
Line Element


d l=dr r r d  d z z
Area Element

d a=r d  dz r (wall)

d a=r dr d  z (end)

Volume Element

d =r dr d  d z
Cylindrical Polar Coordinates

1 ∂ T  ∂T
∂T
∇ T= r   z
∂r r ∂ ∂z

1 ∂ 1 ∂ v ∂ vz
∇⋅v= r v r  
r dr r ∂ ∂z

∇ ×v=
[ 1 ∂ vz
r d
−
∂ v
∂z ][
r 
∂ vr
∂z
−
∂ vz
∂ vr ] [


1 ∂
r ∂r
r v −
∂ vr
∂ ] z

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