11.5 SURFACE INTEGRALS
xzyR dAαS
k
dSFigure 11.6 A surfaceS(or part thereof) projected onto a regionRin the
xy-plane;dSis a surface element.11.5.1 Evaluating surface integralsWe 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 surfaceSisprojected 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)