Advanced Calculus fi..

338 Advanced Calculus, Fifth Editiond) On the basis of the formula of (c), one obtains as solid angle for a general orientedsurface S the integralShow that for surfaces in parametric form, if 0 is the origin,nco.s)=JJR,,,,I x y z lE 2 %azg 21 du dv.(x2 + Y 2 + z2)~This formula permits one to define a solid angle for complicated surfaces that interact,themselves.e) Show that if the normal of S1 is the outer one, then R(0, S)'= 4n.f) Show that if S forms the boundary of a bounded, closed, simply connected region R,then a(0, S), f 4n, when 0 is inside S and R(0, S) = 0 when 0 is outside S.g) If S is a fixed circular disk and 0 is variable, show that -2n 5 R(0, S) 5 2n andthat Q(0, S) jumps by 437 as 0 crosses S.6. (Degree of mapping of one sur$ace into another) Let S,,, and S,,, be surfaces formingthe boundaries of regions R,,, and R, respectively: it is assumed that R,,, and R,,,are bounded and closed and that R,,, is simply connected. Let S, and Sxyz be orientedby the outer normal. Let s, t be parameters for S,,,:u = U(S, t),the normal having the direction ofLetu = U(S, t), w = W(S, t), (a)(u,i + usj + wsk) x (u,i 4 utj + w,k).'x = x(u, v, w), y = y(u, v, w), z = z(u, v, w) (b)be functions defined and having continuous derivatives in a domain containing S,,,, andlet these equations define a mapping of S,,, into S,,,. The degree S of this mapping isdefined as 1/4n times the solid angle Q(0, S) of the image S of S,,, with respect toa point 0 interior to S,,,. If 0 is the origin, the degree is hence given by the integral(Kronecker integral)s = ~ f4n J $ tR,,X Y Z?$ 2 3where x, y, 2 are expressed in terms of s, t by (a) and (b). It can be shown that 8, as thusdefined, is independent of the choice of the interior point 0 , that 6 is a positive or negativeinteger or zero, and that 6 does measure the effective number of times that S,,, is covered.Let S,,, be the sphere u = sins cos r, v = sins sin t, w = cos s, 0 5 s _( n, 0 5t 5 237. Let S,,, be the sphere x2 + y2 + z2 = 1. Evaluate the degree for the followingmappings of S,,, into Sxyz:a)x=v, y=-w, z=ub) x=u2-v2, y =2uv, z=wJ=1 ds dt,(x2 + y2 + Z2)t