Transformation of scalar and vector between Cartesian, Spherical and Cylindrical coordinates

1. Course: Electromagnetic Theory
paper code: EI 503
Course Coordinator: Arpan Deyasi
Department of Electronics and Communication Engineering
RCC Institute of Information Technology
Kolkata, India
Topics: Coordinate Transformation
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Arpan Deyasi
Electromagnetic
Theory

2. Transformation based on type of variables:
Scalar Transformation
Vector Transformation
Transformation between coordinates
Cartesian to Cylindrical
Cartesian to Spherical
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Arpan Deyasi
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Theory

3. Four types of transformations
1. Cartesian to Cylindrical (scalar)
2. Cartesian to Cylindrical (vector)
3. Cartesian to Spherical (scalar)
4. Cartesian to Spherical (vector)
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Theory

4. Cartesian ⇄ Cylindrical Transformation (Scalar)
X
Y
Z
P
ˆ
i
ĵ
k̂
̂
ˆ

ẑ
Point (P) at Cartesian
Coordinate (x, y, z)
Point (P) at Cylindrical
Coordinate (ρ, φ, z)
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Theory

5. Cartesian ⇄ Cylindrical Transformation (Scalar)
Boundary conditions for cylindrical coordinates
0 
  
0 2
 
 
z
−   
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Arpan Deyasi
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6. Cartesian ⇄ Cylindrical Transformation (Scalar)
2 2
x y
 = +
1
tan
y
x
 −  
=  
 
z z
= z z
=
( )
cos
x  
=
( )
sin
y  
=
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Arpan Deyasi
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Theory

7. Cartesian ⇄ Cylindrical Transformation (Scalar)
Prob: For the point P (-2,6,3); determine the coordinate in cylindrical systems
2 2
4 36 6.32
x y
 = + = + =
In cylindrical coordinate system
( )
1 1 1
6
tan tan tan 3 108.43
2
y
x
 − − −
   
= = = − = 
   
−
   
3
z =
Soln:
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Arpan Deyasi
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8. Cartesian ⇄ Cylindrical Transformation (Vector)
X
Y
Z R
ˆ
i
ĵ
k̂
̂
ˆ

ẑ
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Arpan Deyasi
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Theory

9. Cartesian ⇄ Cylindrical Transformation (Vector)
Relation between unit vectors
ˆ
ˆ ˆ
cos( ) sin( )
i    
= −
ˆ
ˆ ˆ
sin( ) cos( )
j    
= +
ˆ ˆ
k z
= ˆ
ẑ k
=
ˆ ˆ ˆ
sin( ) cos( )
i j
  
= − +
ˆ ˆ
ˆ cos( ) sin( )
i j
  
= +
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10. Cartesian → Cylindrical Transformation (Vector)
ˆ
ˆ ˆ
x y z
R R i R j R k
= + +
( ) ( )
ˆ ˆ
ˆ ˆ ˆ
cos( ) sin( ) sin( ) cos( )
x y z
R R R R z
       
= − + + +
ˆ
ˆ ˆ
cos( ) sin( ) sin( ) cos( )
x y x y z
R R R R R R z
     
   
= + + − + +
   
ˆ
ˆ ˆ
z
R R R R z
 
 
= + +
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11. Cartesian → Cylindrical Transformation (Vector)
cos( ) sin( ) 0
sin( ) cos( ) 0
0 0 1
x
y
z z
R R
R R
R R


 
 
     
     
= −
     
     
     
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12. Cylindrical → Cartesian Transformation (Vector)
Inverse transformation gives
cos( ) sin( ) 0
sin( ) cos( ) 0
0 0 1
x
y
z z
R R
R R
R R


 
 
−
     
     
=
     
     
     
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13. Cartesian → Cylindrical Transformation (Vector)
Prob:
Find the vector in cylindrical coordinate system
ˆ ˆ
( ) ( )
R y z i x z j
= + + +
Soln:
In cartesian coordinate system
x
R y z
= +
y
R x z
= +
0
z
R =
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Arpan Deyasi
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14. Cartesian → Cylindrical Transformation (Vector)
cos( ) sin( ) 0
sin( ) cos( ) 0
0 0 1 0
z
R y z
R x z
R


 
 
+
     
     
= − +
     
     
     
 
( )cos( ) ( )sin( )
R y z x z
  
= + + +
0
z
R =
 
( )sin( ) ( )cos( )
R y z x z
  
= − + + +
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Arpan Deyasi
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15. Cartesian → Cylindrical Transformation (Vector)
( )
cos
x  
= ( )
sin
y  
=
Substitutions
( ) ( )
( sin )cos( ) ( cos )sin( )
R z z
      
= + + +
 
 
( ) ( )
( sin )sin( ) ( cos )cos( )
R z z
      
= − + + +
 
 
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Arpan Deyasi
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16. 2 2
4 36 40
x y
 = + = + =
6
tan( )
2
y
x

   
= =
   
−
   
Cartesian → Cylindrical Transformation (Vector)
6
sin( )
40
 = 2
cos( )
40

−
=
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Arpan Deyasi
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17. Cartesian → Cylindrical Transformation (Vector)
6 2 2 6
40 3 ( 40 3)
40 40 40 40
R
 
− −
 
= + + +
 
 
 
 
6 6 2 2
40 3 ( 40 3)
40 40 40 40
R
 
− −
 
= − + + +
 
 
 
 
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18. Cartesian → Cylindrical Transformation (Vector)
( )
2 6 18 6 12
6 3 ( 2 3)
40 40 40 40 40
R
−
   
= + + − + = − + = −
 
 
   
( )
6 2 54 2 56
6 3 ( 2 3)
40 40 40 40 40
R
−
   
= − + + − + = − − = −
 
 
   
0
z
R =
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19. In cylindrical coordinate system
Cartesian → Cylindrical Transformation (Vector)
12 56 ˆ
ˆ
40 40
R  
= − + −
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Arpan Deyasi
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Theory

20. Cartesian ⇄ Spherical Transformation (Scalar)
X
Y
Z
P
ˆ
i
ĵ
k̂
r̂
ˆ

ˆ

Point (P) at Cartesian
Coordinate (x, y, z) Point (P) at Spherical
Coordinate (r, θ, φ)
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Theory

21. Cartesian ⇄ Spherical Transformation (Scalar)
Boundary conditions for spherical coordinates
0 r
  
0  
 
0 2
 
 
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Arpan Deyasi
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Theory

22. Cartesian ⇄ Spherical Transformation (Scalar)
2 2 2
r x y z
= + +
2 2
1
tan
x y
z
 −
 
+
 
=
 
 
1
tan
y
x
 −  
=  
 
( )
sin( )cos
x r  
=
( )
sin( )sin
y r  
=
cos( )
z r 
=
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23. Cartesian ⇄ Spherical Transformation (Scalar)
Prob: For the point P (-2,6,3); determine the coordinate in spherical systems
Soln: In spherical coordinate system
2 2 2 2 2 2
( 2) 6 3 7
r x y z
= + + = − + + =
2 2 2 2
1 1 ( 2) 6
tan tan 64.63
3
x y
z
 − −
   
+ − +
   
= = = 
   
   
( )
1 1 1
6
tan tan tan 3 108.43
2
y
x
 − − −
   
= = = − = 
   
−
   
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24. Cartesian ⇄ Spherical Transformation (Vector)
X
Y
Z
P
ˆ
i
ĵ
k̂
r̂
ˆ

ˆ

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Arpan Deyasi
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Theory

25. Cartesian ⇄ Spherical Transformation (Vector)
Relation between unit vectors
ˆ ˆ
ˆ ˆ
sin( )cos( ) cos( )cos( ) sin( )
i r
      
= − −
ˆ ˆ
ˆ ˆ
sin( )sin( ) cos( )sin( ) cos( )
j r
      
= + +
ˆ ˆ
ˆ
cos( ) sin( )
k r
  
= −
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26. Cartesian ⇄ Spherical Transformation (Vector)
Relation between unit vectors
ˆ
ˆ ˆ
ˆ sin( )cos( ) sin( )sin( ) cos( )
r i j k
    
= + +
ˆ
ˆ ˆ ˆ
cos( )cos( ) cos( )sin( ) sin( )
i j k
     
= + −
ˆ ˆ ˆ
sin( ) cos( )
i j
  
= − +
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27. ˆ
ˆ ˆ
x y z
R R i R j R k
= + +
Cartesian → Spherical Transformation (Vector)
ˆ ˆ
ˆ
[sin( )cos( ) cos( )cos( ) sin( ) ]
ˆ ˆ
ˆ
[sin( )sin( ) cos( )sin( ) cos( ) ]
ˆ
ˆ
[cos( ) sin( ) ]
x
y
z
R r R
r R
r R
      
      
  
= − − +
+ + +
−
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Theory

28. Cartesian → Spherical Transformation (Vector)
ˆ
[ sin( )cos( ) sin( )sin( ) cos( )]
ˆ ˆ
ˆ
[ cos( )cos( ) cos( )sin( ) sin( )]
ˆ
[ sin( ) cos( )]
x y z
x y z
x y
R R R R r
R r R R
R R
    
      
  
= + + +
− + − +
− +
ˆ ˆ
ˆ
r
R R r R R
 
 
= + +
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Arpan Deyasi
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Theory

29. Cartesian → Spherical Transformation (Vector)
sin( )cos( ) sin( )sin( ) cos( )
cos( )cos( ) cos( )sin( ) sin( )
sin( ) cos( ) 0
r x
y
z
R R
R R
R R


    
    
 
     
     
= − −
     
     
−
   
 
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Theory

30. Inverse transformation gives
Spherical → Cartesian Transformation (Vector)
sin( )cos( ) cos( )cos( ) sin( )
sin( )cos( ) cos( )sin( ) cos( )
cos( ) sin( ) 0
x r
y
z
R R
R R
R R


    
    
 
 
−
   
 
   
=  
   
 
   
−
     
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Theory

31. Cartesian → Spherical Transformation (Vector)
Prob:
Find the vector in spherical coordinate system
ˆ ˆ
( ) ( )
R y z i x z j
= + + +
Soln:
In cartesian coordinate system
x
R y z
= +
y
R x z
= +
0
z
R =
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Theory

32. sin( )cos( ) sin( )sin( ) cos( )
cos( )cos( ) cos( )sin( ) sin( )
sin( ) cos( ) 0 0
r
R y z
R x z
R


    
    
 
  +
   
     
= − − +
     
     
−
   
 
Cartesian → Spherical Transformation (Vector)
 
( )sin( )cos( ) ( )sin( )sin( )
r
R y z x z
   
= + + +
 
( )sin( ) ( )cos( )
R y z x z
  
= − + + +
 
( )cos( )cos( ) ( )cos( )sin( )
R y z x z
    
= − + + +
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Theory

33. Cartesian → Spherical Transformation (Vector)
( )
sin( )cos
x r  
= ( )
sin( )sin
y r  
= cos( )
z r 
=
Substitutions
( )
( )
( )
( )
sin( )sin cos( ) sin( )cos( )
sin( )cos cos( ) sin( )sin( )
r
r
R
r
    
    
 
+ +
 
=
+
 
 
( )
( )
( )
( )
sin( )sin cos( ) cos( )cos( )
sin( )cos cos( ) cos( )sin( )
r
R
r

    
    
 
− + +
 
=
+
 
 
( )
( )
( )
( )
sin( )sin cos( ) sin( )
sin( )cos cos( ) cos( )
r
R
r

   
   
 
− + +
 
=
+
 
 
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Theory

34. 2 2 2 2 2 2
( 2) 6 3 7
r x y z
= + + = − + + =
2 2 2 2
( 2) 6 40
tan( )
3 3
x y
z

   
+ − +
   
= = =
   
   
Cartesian → Spherical Transformation (Vector)
40
sin( )
7
 =
3
cos( )
7
 =
6
tan( )
2
y
x

   
= =
   
−
   
6
sin( )
40
 =
2
cos( )
40

−
=
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Theory

35. Cartesian → Spherical Transformation (Vector)
40 6 3 40 2 40 2 3 40 6
7 . . 7 . .
7 7 7 7 7 7
40 40 40 40
r
R
 
     
− −
= + + +
 
     
     
 
     
 
40 6 3 3 2 40 2 3 3 6
7 . . 7 . .
7 7 7 7 7 7
40 40 40 40
R
 
   
− −
   
= − + + +
 
   
   
   
   
 
   
 
40 6 3 6 40 2 3 2
7 . 7 . .
7 7 7 7
40 40 40 40
R
 
   
− −
= − + + +
 
   
   
 
   
 
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Theory

36. Cartesian → Spherical Transformation (Vector)
6 3 2 2 3 6 18 6 12
7 7
7 7 7 7 7 7 7 7 7
r
R
 − − 
       
= + + + = − + = −
     
   
       
 
6 3 3 2 2 3 3 6
7 . 7 .
7 7 7 7 7 7
40 40
54 18 72
7 40 7 40 7 40
R
 
− −
   
   
= − + + +
   
 
   
   
   
 
= + =
6 3 6 2 3 2 54 2 56
7 7 .
7 7 7 7
40 40 40 40 40
R
 − − 
   
= − + + + = − − = −
   
 
   
 
18-09-2021 Arpan Deyasi, Electromagnetic Theory 36
Arpan Deyasi
Electromagnetic
Theory

37. In spherical coordinate system
Cartesian → Spherical Transformation (Vector)
12 72 56
ˆ ˆ
ˆ
7 7 40 40
R r  
= − + −
18-09-2021 Arpan Deyasi, Electromagnetic Theory 37
Arpan Deyasi
Electromagnetic
Theory