Mathematical Methods for Physics and Engineering : A Comprehensive Guide

11.5 SURFACE INTEGRALS

x

z

y

R dA

α

S

k

dS

Figure 11.6 A surfaceS(or part thereof) projected onto a regionRin the

xy-plane;dSis a surface element.

11.5.1 Evaluating surface integrals

We now consider how to evaluate surface integrals over some general surface. This

involves writing the scalar area elementdSin terms of the coordinate differentials

of our chosen coordinate system. In some particularly simple cases this is very

straightforward. For example, ifSis the surface of a sphere of radiusa(or some

part thereof) then using spherical polar coordinatesθ, φon the sphere we have

dS=a^2 sinθdθdφ. For a general surface, however, it is not usually possible to

represent the surface in a simple way in any particular coordinate system. In such

cases, it is usual to work in Cartesian coordinates and consider the projections of

the surface onto the coordinate planes.

Consider a surface (or part of a surface)Sas in figure 11.6. The surfaceSis

projected onto a regionRof thexy-plane, so that an element of surface areadS

projects onto the area elementdA. From the figure, we see thatdA=|cosα|dS,

whereαis the angle between the unit vectorkin thez-direction and the unit

normalnˆto the surface atP. So, at any given point ofS, we have simply

dS=

dA

|cosα|

=

dA

|nˆ·k|

.

Now, if the surfaceSis given by the equationf(x, y, z) = 0 then, as shown in sub-

section 10.7.1, the unit normal at any point of the surface is given byˆn=∇f/|∇f|

evaluated at that point, cf. (10.32). The scalar element of surface area then becomes

dS=

dA

|nˆ·k|

=

|∇f|dA

∇f·k

=

|∇f|dA

∂f/∂z

, (11.10)