Spherical Coordinates

The spherical coordinate system is defined with respect to the Cartesian system in Figure 4.4.1. The spherical system uses

, the distance measured from the origin;¹

, the angle measured from the

axis toward the

plane; and

, the angle measured in a plane of constant

, identical to

in the cylindrical system.

Figure 4.4.1: Spherical coordinate system and associated basis vectors. Image used with permission (CC BY SA 4.0; K. Kikkeri).

Spherical coordinates are preferred over Cartesian and cylindrical coordinates when the geometry of the problem exhibits spherical symmetry. For example, in the Cartesian coordinate system, the surface of a sphere concentric with the origin requires all three coordinates (

, and

) to describe. However, this surface can be described using a single constant parameter – the radius

– in the spherical coordinate system. This leads to a dramatic simplification in the mathematics in certain applications.

The basis vectors in the spherical system are

, and

. As always, the dot product of like basis vectors is equal to one, and the dot product of unlike basis vectors is equal to zero. For the cross-products, we find:

A useful diagram that summarizes these relationships is shown in Figure 4.4.2.

Figure 4.4.2: Cross products among basis vectors in the spherical system. (See Figure 4.1.10 for instructions on the use of this diagram.)Image used with permission (CC BY SA 4.0; K. Kikkeri).

Like the cylindrical system, the spherical system is often less useful than the Cartesian system for identifying absolute and relative positions. The reason is the same: Basis directions in the spherical system depend on position. For example,

is directed radially outward from the origin, so

for locations along the

-axis but

for locations along the

axis and

for locations along the

axis. Similarly, the directions of

and

vary as a function of position. To overcome this awkwardness, it is common to begin a problem in spherical coordinates, and then to convert to Cartesian coordinates at some later point in the analysis. Here are the conversions:

The conversion from Cartesian to spherical coordinates is as follows:

where

is the four-quadrant inverse tangent function.²

Dot products between basis vectors in the spherical and Cartesian systems are summarized in Table 4.4.1. This information can be used to convert between basis vectors in the spherical and Cartesian systems, in the same manner described in Section 4.3; e.g.

and so on.

$.$	$\hat{\bf r}$	$\hat{\bf \theta}$	$\hat{\bf \phi}$
$\hat{\bf x}$	$\sin\theta\cos\phi$	$\cos\theta\cos\phi$	$-\sin\phi$
$\hat{\bf y}$	$\sin\theta\sin\phi$	$\cos\theta\sin\phi$	$\cos\phi$
$\hat{\bf z}$	$\cos\theta$	$-\sin\theta$	$0$

Table 4.4.1: Dot products between basis vectors in the spherical and Cartesian coordinate systems.

Exercise

EXAMPLE 4.4.1: CARTESIAN TO SPHERICAL CONVERSION

A vector field

. Develop an expression for

in spherical coordinates.

Simply substitute expressions in terms of spherical coordinates for expressions in terms of Cartesian coordinates. Use Table 4.4.1 and Equations 4.4.4- 4.4.6. Making these substitutions and applying a bit of mathematical clean-up afterward, one obtains

Integration Over Length

A differential-length segment of a curve in the spherical system is

Note that

is an angle, as opposed to a distance. The associated distance is

in the

direction. Note also that in the

direction, distance is

in the

plane and less by the factor

for

As always, the integral of a vector field

over a curve

To demonstrate line integration in the spherical system, imagine a sphere of radius

centered at the origin with “poles” at

and

. Let us calculate the integral of

, where

is the arc drawn directly from pole to pole along the surface of the sphere, as shown in Figure 4.4.3. In this example,

since

and

(which could be any value) are both constant along

. Subsequently,

and the above integral is

i.e., half the circumference of the sphere, as expected.

Figure 4.4.3: Example in spherical coordinates: Poleto-pole distance on a sphere. Image used with permission (CC BY SA 4.0; K. Kikkeri).

Note that the spherical system is an appropriate choice for this example because the problem can be expressed with the minimum number of varying coordinates in the spherical system. If we had attempted this problem in the Cartesian system, we would find that both

and either

(or all three) vary over

and in a relatively complex way.

Integration Over Area

Now we ask the question, what is the integral of some vector field

over the surface

of a sphere of radius

centered on the origin? This is shown in Figure 4.4.4. The differential surface vector in this case is

Figure 4.4.4: Example in spherical coordinates: The area of a sphere. Image used with permission (CC BY SA 4.0; K. Kikkeri).

As always, the direction is normal to the surface and in the direction associated with positive flux. The quantities in parentheses are the distances associated with varying

and

, respectively. In general, the integral over a surface is

In this case, let’s consider

; in this case

and the integral becomes

which we recognize as the area of the sphere, as expected. The corresponding calculation in the Cartesian or cylindrical systems is quite difficult in comparison.

Integration Over Volume

The differential volume element in the spherical system is

For example, if

and the volume

is a sphere of radius

centered on the origin, then

which is the volume of a sphere.

Footnotes

1
Note that some textbooks use “R” in lieu of r for this coordinate.
2
Note that this function is available in MATLAB and Octave as
$\text{atan}2\left ( \text{y,x} \right )$
.