Mathematical Methods for Physics and Engineering : A Comprehensive Guide

LINE, SURFACE AND VOLUME INTEGRALS

y

d

c

a b x

S R

T

C

U

V

Figure 11.3 A simply connected regionRbounded by the curveC.

These ideas can be extended to regions that are not planar, such as general

three-dimensional surfaces and volumes. The same criteria concerning the shrink-

ing of closed curves to a point also apply when deciding the connectivity of such

regions. In these cases, however, the curves must lie in the surface or volume

in question. For example, the interior of a torus is not simply connected, since

there exist closed curves in the interior that cannot be shrunk to a point without

leaving the torus. The region between two concentric spheres of different radii is

simply connected.

11.3 Green’s theorem in a plane

In subsection 11.1.1 we considered (amongst other things) the evaluation of line

integrals for which the pathCis closed and lies entirely in thexy-plane. Since

the path is closed it will enclose a regionRof the plane. We now discuss how to

express the line integral around the loop as a double integral over the enclosed

regionR.

Suppose the functionsP(x, y),Q(x, y) and their partial derivatives are single-

valued, finite and continuous inside and on the boundaryCof some simply

connected regionRin thexy-plane.Green’s theorem in a plane(sometimes called

the divergence theorem in two dimensions) then states

∮

C

(Pdx+Qdy)=

∫∫

R

(

∂Q

∂x

−

∂P

∂y

)

dx dy, (11.4)

and so relates the line integral aroundCto a double integral over the enclosed

regionR. This theorem may be proved straightforwardly in the following way.

Consider the simply connected regionRin figure 11.3, and lety=y 1 (x)and