Mathematical Methods for Physics and Engineering : A Comprehensive Guide

10.9 CYLINDRICAL AND SPHERICAL POLAR COORDINATES

(ii) If the unit vectors vary as the values of the coordinates change (i.e. are

not constant in direction throughout the whole space) then the derivatives

of these vectors appear as contributions to∇^2 a.

Cartesian coordinates are an example of the first case in which each component

satisfies (∇^2 a)i=∇^2 ai. In this case (10.41) can be applied to each component

separately:

[∇×(∇×a)]i=[∇(∇·a)]i−∇^2 ai. (10.42)

However, cylindrical and spherical polar coordinates come in the second class.

For them (10.41) is still true, but the further step to (10.42) cannot be made.

More complicated vector operator relations may be proved using the relations

given above.

Show that

∇·(∇φ×∇ψ)=0,

whereφandψare scalar fields.

From the previous section we have

∇·(a×b)=b·(∇×a)−a·(∇×b).

If we leta=∇φandb=∇ψthen we obtain

∇·(∇φ×∇ψ)=∇ψ·(∇×∇φ)−∇φ·(∇×∇ψ)=0, (10.43)

since∇×∇φ=0=∇×∇ψ, from (10.37).

10.9 Cylindrical and spherical polar coordinates

The operators we have discussed in this chapter, i.e. grad, div, curl and∇^2 ,

have all been defined in terms of Cartesian coordinates, but for many physical

situations other coordinate systems are more natural. For example, many systems,

such as an isolated charge in space, have spherical symmetry and spherical polar

coordinates would be the obvious choice. For axisymmetric systems, such as fluid

flow in a pipe, cylindrical polar coordinates are the natural choice. The physical

laws governing the behaviour of the systems are often expressed in terms of

the vector operators we have been discussing, and so it is necessary to be able

to express these operators in these other, non-Cartesian, coordinates. We first

consider the two most common non-Cartesian coordinate systems, i.e. cylindrical

and spherical polars, and go on to discuss general curvilinear coordinates in the

next section.

10.9.1 Cylindrical polar coordinates

As shown in figure 10.7, the position of a point in spacePhaving Cartesian

coordinatesx, y, zmay be expressed in terms of cylindrical polar coordinates