Review of Coordinate Systems
A good understanding of coordinate systems can be very helpful in solving problems
related to Maxwell’s Equations. The three most common coordinate systems are
rectangular (x, y, z), cylindrical (r,  , z), and spherical ( r ,  ,  ).
Unit vectors in rectangular, cylindrical, and spherical coordinates
In rectangular coordinates a point P is specified by x,
y, and z, where these values are all measured from the
origin (see figure at right). A vector at the point P is
specified in terms of three mutually perpendicular
ˆ (also called
components with unit vectors ˆi, ˆj, and k
ˆ form a rightˆx, ˆy, and ˆz ). The unit vectors ˆi, ˆj, and k
handed set; that is, if you push ˆi into ˆj with your right
hand, your right thumb will point along k̂ direction.
In cylindrical coordinates a point P is specified by
r,  , z, where  is measured from the x axis (or x-z
plane) (see figure at right). A vector at the point P
is specified in terms of three mutually perpendicular
components with unit vectors r̂ perpendicular to
the cylinder of radius r, φ̂ perpendicular to the
plane through the z axis at angle  , and ẑ
perpendicular to the x-y plane at distance z. The
unit vectors r̂ , φ̂ , ẑ form a right-handed set.
In spherical coordinates a point P is specified by
r ,  ,  , where r is measured from the origin,  is
measured from the z axis, and  is measured from
the x axis (or x-z plane) (see figure at right). With
z axis up,  is sometimes called the zenith angle
and  the azimuth angle. A vector at the point P
is specified in terms of three mutually
perpendicular components with unit vectors r̂
θ̂
perpendicular to the sphere of radius r,
perpendicular to the cone of angle  , and φ̂
perpendicular to the plane through the z axis at
angle  . The unit vectors rˆ , θˆ , φˆ form a righthanded set.
Infinitesimal lengths and volumes
An infinitesimal length in the rectangular system is given by
dL =
d x2 + d y2 + d z2
(1)
and an infinitesimal volume by
dv = dx dy dz
(2)
In the cylindrical system the corresponding quantities are
dL =
d r 2 + r 2 d 2 + d z 2
d v = d r r d d z
and
(3)
(4)
In the spherical system we have
dL =
and
d r 2 + r 2 d 2 + r 2 sin 2  d 2
d v = dr r d r sin  d
(5)
(6)
Direction cosines and coordinate-system transformation
As shown in the figure on the right, the
projection x of the scalar distance r on the x axis
is given by r cos  where  is the angle
between r and the x axis. The projection of r on
the y axis is given by r cos  , and the
projection on the z axis by r cos  . Note that
 =  so cos  = cos  .
The quantities cos  , cos  , and cos  are
called the direction cosines. From the theorem
of Pythagoras,
cos 2  + cos 2  + cos 2  = 1
(7)
The scalar distance r of a spherical coordinate
system transforms into rectangular coordinate
distance
x = r cos  = r sin  cos 
y = r cos  = r sin  sin 
z = r cos  = r cos 
from which
cos  = sin  cos 
cos  = sin  sin 
cos  = cos 
(8)
(9)
(10)
direction cosines
(11)
(12)
(13)
As the converse of (8), (9), and (10), the spherical coordinate values ( r ,  ,  ) may be
expressed in terms of rectangular coordinate distances as follows:
r =
x2 + y2 + z2
z
 = cos −1
 = tan −1
r0
x +y +z
2
y
x
2
2
(14)
(0     )
(15)
(16)
From these and similar coordinate transformations of spherical to rectangular and
rectangular to spherical coordinates, we may express a vector A at some point P with
spherical components Ar , A , A as the rectangular components Ax , Ay , and Az , where
Ax = Ar sin  cos  + A cos  cos  − A sin 
(17)
Ay = Ar sin  sin  + A cos  sin  + A cos 
(18)
Az = Ar cos  − A sin 
(19)
Note that the direction cosines are simply the dot products of the spherical unit vector r̂
with the rectangular unit vectors ˆx, ˆy, and ˆz :
rˆ  xˆ = sin  cos  = cos α
rˆ  yˆ = sin  sin  = cos β
rˆ  zˆ = cos  = cos Υ
(20)
(21)
(22)
These and other dot product combinations are listed in the following table:
Spherical
ŷ
ẑ
r̂
φ̂
ẑ
r̂
θ̂
φ̂
x̂
1
0
0
cos 
− sin 
0
sin  cos 
cos  cos 
− sin 
ŷ
0
1
0
sin 
cos 
0
sin  sin 
cos  sin 
cos 
ẑ
0
0
1
0
0
1
cos 
− sin 
0
r̂
cos 
sin 
0
1
0
0
sin 
cos 
0
φ̂
− sin 
cos 
0
0
1
0
0
0
1
ẑ
0
0
1
0
0
1
cos 
− sin 
0
r̂
sin  cos 
sin  sin 
cos 
sin 
0
cos 
1
0
0
θ̂
cos  cos 
cos  sin 
− sin 
cos 
0
− sin 
0
1
0
φ̂
− sin 
cos 
0
0
1
0
0
0
1
Spherical
x̂
Rectangular
Cylindrical
Cylindrical
Rectangular
Note that the unit vectors r̂ in the cylindrical and spherical systems are not the same.
For example,
Spherical
rˆ  xˆ = sin  cos 
rˆ  yˆ = sin  sin 
rˆ  zˆ = cos 
Cylindrical
rˆ  xˆ = cos 
rˆ  yˆ = sin 
rˆ  zˆ = 0
In addition to rectangular, cylindrical, and spherical coordinate systems, there are many
other systems such as the elliptical, spheroidal (both prolate and oblate), and paraboloidal
systems. Although the number of possible systems is infinite, all of them can be treated
in terms of a generalized curvilinear coordinate system.
The fundamental parameters of the rectangular, cylindrical, and spherical coordinate
systems are summarized in the following table:
Coordinate
system
Rectangular
Coordinates
Range
x
y
−  to + 
−  to + 
−  to + 
z
r
Cylindrical

z
Spherical
r


Unit
vectors
ˆx or ˆi
ˆy or ˆj
ˆ
ˆz or k
0 to 
0 to 2
−  to + 
r̂
φ̂
ẑ
0 to 
0 to 
0 to 2
r̂
Length
elements
dx
dy
dz
Plane
dr
r d
dz
Cylinder r=constant
Plane
 =constant
Plane
z=constant
dr
r dθ
r sin  d
θ̂
φ̂
Coordinate
surfaces
Plane
x=constant
Plane
y=constant
Sphere
Cone
Plane
z=constant
r=constant
θ=constant
 =constant
The following two tables give the unit vector dot products in rectangular coordinates for
both rectangular-cylindrical and rectangular-spherical coordinates.
x̂
x
r̂
φ̂
ẑ
ŷ
y
ẑ
x2 + y2
−y
x2 + y2
x
0
x2 + y2
x2 + y2
0
0
0
1
Rectangular-cylindrical product in rectangular coordinates
Example: φˆ  yˆ = cos  =
x
x2 + y2
·
ŷ
y
x̂
r̂
x
θ̂
x2 + y2 + z 2
xz
x2 + y2 + z 2
yz
x2 + y 2 x2 + y 2 + z 2
φ̂
x +y
2
x2 + y2 + z 2
x2 + y 2 x2 + y 2 + z 2
y
−
ẑ
z
−
x2 + y 2
x2 + y2 + z 2
x
0
x + y2
2
2
Rectangular-spherical product in rectangular coordinates
x
Example: xˆ  rˆ = sin  cos  =
x + y2 + z2
2
Here are the transformations of vector components between coordinate systems:
Rectangular to cylindrical
A = − Ax
y
x +y
2
2
+ Ay
Cylindrical to rectangular
x
Ay = Ar sin φ + Aφ cos φ
x + y2
2
Az = Az
Az = Az
Rectangular to spherical
x
Ar = Ax
+ Ay
2
x + y2 + z2
A = Ax
xz
y
x +y +z
2
2
+ Ay
x2 + y 2 x2 + y 2 + z 2
y
x
A = − Ax
+ Ay
x2 + y 2
x2 + y 2
2
+ Az
x + y2 + z2
yz
x2 + y 2 x2 + y 2 + z 2
Spherical to rectangular
Ax = Ar sin cos + A cos cos  − A sin 
Ay = Ar sin  sin  + A cos sin  + A cos 
Az = Ar cos − A sin
z
2
− Az
x2 + y 2
x2 + y 2 + z 2
And here are expressions for the gradient, divergence, and curl in all three coordinate
systems:
Rectangular coordinates
f
f
f
+ yˆ
+ zˆ
x
y
z
A Ay Az
•A = x +
+
x
y
z
f = xˆ
xˆ
 Az Ay   Ax Az   Ay Ax 

 + yˆ 
 =
  A = xˆ 
−
−
−
 + zˆ 
z   z
x   x
y  x
 y
Ax
yˆ

y
Ay
zˆ

z
Az
Cylindrical coordinates
f
1 f
f
+ φˆ
+ zˆ
r
r 
z
1 
1 A Az
•A =
rAr +
+
r r
r 
z
f = rˆ
1
r
 1 Az A 
1 
Ar 

 Ar Az 
 + φˆ 
=
  A = rˆ 
−
−
 + zˆ  rA −
z 
r 
r  r
  r
 z
 r 
Ar
rˆ
φˆ


rA
1
r

z
Az
zˆ
Spherical coordinates
f ˆ 1 f
1 f
+θ
+ φˆ
r
r 
r sin  
1 
1

1 A
 • A = 2 r 2 Ar +
( A sin  ) +
r r
r sin  
r sin  

1  
A 
1  1 Ar 
1 
A 
 ( A sin  ) −   + θˆ 
  A = rˆ
− rA  + φˆ  rA − r 
r sin   
 
r  sin   r
r  r
 

f = rˆ