Advanced Calculus fi..

, I316 Advanced Calculus, Fifth EditionFigure 5.29Proof of Divergence theorem.where R,, is a bounded closed region in the xy-plane (cf. Fig. 5.29) bounded by asimple closed curve C. The surface S is then composed of three parts:Si: z = f~(x, y),(x, y) in R,,,S2: z = f2(~, Y), (x, y) in R,,,S3: fi(~, y) i z i f2(~, Y), (x, y) on C.S2 forms the "top" of S, S1 forms the "bottom," and S3 forms the "sides" (the portionS3 may degenerate into a curve, as for a sphere). Now by definition (5.77),where y = c(n, k). Along S3, y = n/2, SO that cos y = 0; along S2, y = y' wherey' is the angle between the upper normal and k; along S1, y = n - y'. Sinceda = sec y ' dx dy on S1 and SZ, one has/ / ~ d ~ = d //~dxd~~+ // ~ d + // ~ ~dxdy d ~S SI s2 h1m i f . . =-//Ncosy'secy'dxdy+ Ncosy'secy'dxdy.R,, 11 R,, 4where z = f~(x, y) in the first integral and z = f2(x, y) in the second:1