LINE, SURFACE AND VOLUME INTEGRALS
yxC 1
C 2
R
Figure 11.4 A doubly connected regionRbounded by the curvesC 1 andC 2.to the regionRshown in figure 11.4, the line integrals must be taken over
both boundaries,C 1 andC 2 , in the directions indicated, and the results added
together.
We may also use Green’s theorem in a plane to investigate the path indepen-dence (or not) of line integrals when the paths lie in thexy-plane. Let us consider
the line integral
I=∫BA(Pdx+Qdy).For the line integral fromAtoBto be independent of the path taken, it must
have the same value along any two arbitrary pathsC 1 andC 2 joining the points.
Moreover, if we consider as the path the closed loopCformed byC 1 −C 2 then
the line integral around this loop must be zero. From Green’s theorem in a plane,
(11.4), we see that asufficientcondition forI= 0 is that
∂P
∂y=∂Q
∂x, (11.5)throughout some simply connected regionRcontaining the loop, where we assume
that these partial derivatives are continuous inR.
It may be shown that (11.5) is also anecessarycondition forI= 0 and isequivalent to requiringPdx+Qdyto be an exact differential of some function
φ(x, y) such thatPdx+Qdy=dφ. It follows that
∫B
A(Pdx+Qdy)=φ(B)−φ(A)
and that
∮
C(Pdx+Qdy) around any closed loopCin the regionRis identically
zero. These results are special cases of the general results for paths in three
dimensions, which are discussed in the next section.