Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

LINE, SURFACE AND VOLUME INTEGRALS


dr

r

O


C


Figure 11.8 The conical surface spanning the perimeterCand having its
vertex at the origin.

The hemispherical shell discussed above is an example of an open surface. For

a closed surface, however, the vector area is always zero. This may be seen by


projecting the surface down onto each Cartesian coordinate plane in turn. For


each projection, every positive element of area on the upper surface is cancelled


by the corresponding negative element on the lower surface. Therefore, each


component ofS=



SdSvanishes.
An important corollary of this result is that the vector area of an open surface

depends only on its perimeter, or boundary curve,C. This may be proved as


follows. If surfacesS 1 andS 2 have the same perimeter thenS 1 −S 2 is a closed


surface, for which



dS=


S 1

dS−


S 2

dS= 0.

HenceS 1 =S 2. Moreover, we may derive an expression for the vector area of


an open surfaceS solely in terms of a line integral around its perimeterC.


Since we may choose any surface with perimeterC, we will consider a cone


with its vertex at the origin (see figure 11.8). The vector area of the elementary


triangular region shown in the figure isdS=^12 r×dr. Therefore, the vector area


of the cone, and hence ofanyopen surface with perimeterC, is given by the line


integral


S=

1
2


C

r×dr.

For a surface confined to thexy-plane, r=xi+yj anddr=dxi+dyj,


and we obtain for this special case that the area of the surface is given by


A=^12



C(xdy−ydx), as we found in section 11.3.
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