Advanced Calculus fi..

Chapter 5 Vector Integral Calculus 327Conversely, if D is simply connected and (5.99) holds, then SX dx + Y dy + Z dzis independent of the path in D; that is, if D is simply connected and curl u = 0 in D,thenIr7u = grad Ffor some F.A domain D of space is called simply connected if every simple closed curvein D forms the boundary of a smooth orientable surface in D. Thus the interiorof a sphere is simply connected whereas the interior of a torus is not. The domainbetween two concentric spheres is simply connected, as is the interior of a sphere witha finite number of points removed. (This definition, which is adequate for practicalapplications, differs from the standard one of advanced mathematics.)In the first part of the theorem we assume that SuT. ds is independent of path.Hence by Theorem I, u = grad F. Accordingly,curl u = curl grad F = 0by the identity of Section 3.6.In the second part of the theorem we assume that D is simply connected andcurl u = 0. To show independence of path, it is sufficient, by Theorem 11, to showthat J, ur ds = 0 on each simple closed curve C in D. By assumption, C formsthe boundary of a piecewise smooth oriented surface S in D. Stokes's theorem isapplicable, and one findsRfor proper direction on C and normal n on S.We remark that the Stokes's theorem can be extended to an arbitrary orientedsurface S whose boundary is formed of distinct simple closed curves C,, . . . , C,. IfBs denotes this boundary, with proper directions, one hasThe proof is like that of Section 5.7. In particular, if curl u E 0 in D,ls uT ds = 0.This can be applied to evaluate line integrals in "multiply connected domains, asin Section 5.7.