Advanced Calculus fi..

336 Advanced Calculus, Fifth EditionDivergence theorem. One can also define a degree 6 for the correspondence ofboundaries, and a formula analogous to (5.107) then holds without any assumptionabout one-to-one correspondence on boundaries:s///F(x, y, z)dxdy dz = J/JFIX(U, U, w), ...I a(x' y'z) dudvdw. (5.109)acu, V , W)R,,:R,,, ,I,The degree 6 measures the effective number of times the bounding surface S,,; istraced by (x, y, z) as the point (u, v, w) traces the bounding surface S,,, of R,,,,,.The surfaces S,,, and S,,, are both considered oriented, the normal being the outernormal for both; in measuring S, one counts negatively the parts of S,,, that aremapped into S,,, with reversal of orientation. Thus for the mappingof the sphere u2 + v2 + w2 = 1 onto the sphere x2 + y2 + z2 = I, one has 6 = -1.Formula (5.109) can also be extended to the case in which R,,, and R,,; are eachbounded by several surfaces, as in two dimensions. In the case in which R,,, is simplyconnected and is bounded by a single surface S,,, (which has then the structure of asphere), the degree S can be computed, by analogy with the planar case, by referenceto a "solid angle" (see Problem 6).PROBLEMS1. Transform the integrals, using the substitution given:' v 2a) JoJ6(x +y2)d~dy,u=y,v=x; 3b) JIR,, (X -y)dx dy, where R,?, is the region^*+^^ 5 1, andx = u +(1 - u2 - v2),y = v + (1- u2 - v2); (Hint: Use as R,, the region u2 + v2 ( 1.) 4c) JJR,, xy dx dy, where R,, is the region x2 + y2 5 1 and x = u2 - v2, y = 2uv.[Hint: Choose R,, as in (b).]2. Let x = f (u, v), v = g(u, v) be given as in Theorem I1 and let C,, enclose the origin 0.Show that the degree S can be evaluated by the formulawhere x, y , dx, dy are expressed in terms of u, v:(Hint: The line integral measures the change in 8 = arctan y/x, as shown in Section 5.6.)3. Apply the formula of Problem 2 to evaluate the degree for the following mappings of thecircle u2 + v2 = 1 into the circle x2 + y2 = 1:3u + 4v 4u - 3va)x=-, Y = ~ -b) x = u2 - v2, y = 2uvc) X = u3 - uv2, y = 3u2v - v3