Mathematical Methods for Physics and Engineering : A Comprehensive Guide

10.9 CYLINDRICAL AND SPHERICAL POLAR COORDINATES

x

y

z

r

i

j

k

O

θ

P

ˆer

ˆeφ

ˆeθ

φ

Figure 10.9 Spherical polar coordinatesr, θ, φ.

Substituting these relations and (10.44) into the expression forawe find

a=zρsinφ(cosφˆeρ−sinφˆeφ)−ρsinφ(sinφˆeρ+cosφˆeφ)+z^2 ρcosφˆez

=(zρsinφcosφ−ρsin^2 φ)ˆeρ−(zρsin^2 φ+ρsinφcosφ)ˆeφ+z^2 ρcosφeˆz.

Substituting into the expression for∇·agiven in table 10.2,

∇·a=2zsinφcosφ−2sin^2 φ− 2 zsinφcosφ−cos^2 φ+sin^2 φ+2zρcosφ
=2zρcosφ− 1.
Alternatively, and much more quickly in this case, we can calculate the divergence
directly in Cartesian coordinates. We obtain

∇·a=

∂ax

∂x

+

∂ay

∂y

+

∂az

∂z

=2zx− 1 ,

which on substitutingx=ρcosφyields the same result as the calculation in cylindrical
polars.

Finally, we note that similar results can be obtained for (two-dimensional)

polar coordinates in a plane by omitting thez-dependence. For example, (ds)^2 =

(dρ)^2 +ρ^2 (dφ)^2 , while the element of volume is replaced by the element of area

dA=ρdρdφ.

10.9.2 Spherical polar coordinates

As shown in figure 10.9, the position of a point in spaceP, with Cartesian

coordinatesx, y, z, may be expressed in terms of spherical polar coordinates

r, θ, φ,where

x=rsinθcosφ, y=rsinθsinφ, z=rcosθ, (10.53)