Mathematical Methods for Physics and Engineering - Matematica.NET

LINE, SURFACE AND VOLUME INTEGRALSydVSRUcTCa b xFigure 11.3 A simply connected region R bounded by the curve C.These ideas can be extended to regions that are not planar, such as generalthree-dimensional surfaces and volumes. The same criteria concerning the shrinkingof closed curves to a point also apply when deciding the connectivity of suchregions. In these cases, however, the curves must lie in the surface or volumein question. For example, the interior of a torus is not simply connected, sincethere exist closed curves in the interior that cannot be shrunk to a point withoutleaving the torus. The region between two concentric spheres of different radii issimply connected.11.3 Green’s theorem in a planeIn subsection 11.1.1 we considered (amongst other things) the evaluation of lineintegrals for which the path C is closed and lies entirely in the xy-plane. Sincethe path is closed it will enclose a region R of the plane. We now discuss how toexpress the line integral around the loop as a double integral over the enclosedregion R.Suppose the functions P (x, y), Q(x, y) and their partial derivatives are singlevalued,finite and continuous inside and on the boundary C of some simplyconnected region R in the xy-plane. Green’s theorem in a plane (sometimes calledthe divergence theorem in two dimensions) then states∮∫∫ ( ∂Q(P dx+ Qdy)=CR ∂x − ∂P )dx dy, (11.4)∂yand so relates the line integral around C to a double integral over the enclosedregion R. This theorem may be proved straightforwardly in the following way.Consider the simply connected region R in figure 11.3, and let y = y 1 (x) and384