LINE, SURFACE AND VOLUME INTEGRALS
S
S
V
C
dS
dS(a)(b)Figure 11.5 (a) A closed surface and (b) an open surface. In each case a
normal to the surface is shown:dS=nˆdS.All the above integrals are taken over some surfaceS, which may be either
open or closed, and are therefore, in general, double integrals. Following the
notation for line integrals, for surface integrals over a closed surface
∫
Sis replaced
by
∮
S.
The vector differentialdSin (11.9) represents a vector area element of thesurfaceS. It may also be writtendS=nˆdS,whereˆnis a unit normal to the
surface at the position of the element anddSis the scalar area of the element used
in (11.8). The convention for the direction of the normalnˆto a surface depends
on whether the surface is open or closed. A closed surface, see figure 11.5(a),
does not have to be simply connected (for example, the surface of a torus is not),
but it does have to enclose a volumeV, which may be of infinite extent. The
direction ofnˆis taken to point outwards from the enclosed volume as shown.
An open surface, see figure 11.5(b), spans some perimeter curveC. The direction
ofnˆis then given by the right-hand sense with respect to the direction in which
the perimeter is traversed, i.e. follows the right-hand screw rule discussed in
subsection 7.6.2. An open surface does not have to be simply connected but for
our purposes it must be two-sided (a M ̈obius strip is an example of a one-sided
surface).
The formal definition of a surface integral is very similar to that of a lineintegral. We divide the surfaceSintoNelements of area ∆Sp,p=1, 2 ,...,N,
each with a unit normalnˆp.If(xp,yp,zp) is any point in ∆Spthen the second type
of surface integral in (11.9), for example, is defined as
∫Sa·dS= lim
N→∞∑Np=1a(xp,yp,zp)·nˆp∆Sp,where it is required that all ∆Sp→0asN→∞.