MATH 223 Final, Fall 2000

Math 223—Final Exam (Fall 01) B. Shayya 1. (25 pts) State and prove Fubini’s theorem in the case n = 2. 2. (25 pts) Suppose 0 <6 < 1. Vind the surface area of the spherical cap Cs = {8 = (9.02.63) © RB’: [6] = 1 and # > 1-5}. 3. Suppose /.: R S R is a positive C? function and define a mapping T : R? + R? by T(u,u) = (u, vh{u)). Let Qy {(w,y) ER 0S" <land0<y< (1 —--x)h(z) OQ = {(yieP:0<4< land (l-a)h(2) <y<h and Q = 2; UQ». Also consider the 2-chain W = To, + To in R?, where o; = [0,e;,e2] and o = [e, + 2, €2,e;] (of course, o; and g2 are the oriented affine 2-simplexes in R? given by a, (u,v) = uey + veg = (u,v) and Gn(e) =; + ey — wey — veg = (L—u, 1 —2).) (i) (5 pts) Prove that the C? mapping T is one-to-one and that Jp(u.v) > 0 for all (u,v)'e R’. (it) (6 pts) Prove that Im(To,) = ©, and Im(Toy) = Q». (ii) (9 pts) Lf f 2B? S R is continnous, prove that flr y)de Ady = I, Flay) dr dy. (iv) (10 pts) Find ow, (v) Find | yd + 22 dy JOY (a) (12 pts) directly. and ()) (8 pts) using Stokes’ theorem.