MATH 223 Final, Fall 2005

Math 223–Final Exam (Fall 05) B. Shayya Problem 1 Let K be the convex set spanned by the points p0 = (0, 0, 1), p1 = (6, 0, 1), p2 = (0, 6, 1), and p3 = (6, 0, 2) in R3. Also, let F : R3 → R3 be the vector field given by F(x, y, z) = (x + y, x + y + z, z − 1). (i) (4 pts) Show that K has a positively oriented boundary. (ii) (18 pts) Find � ∂K F (a) directly, and (b) using the divergence theorem. Problem 2 (18 pts) Show that the system of equations y1 sin y2 + ey2 + 7x2 1 − x2 + x2x3 = 38 ey1−1 + y2 + x1y2 + 2x2 − x3 = −3 can be solved for (y1, y2) in terms of (x1, x2, x3) in a neighborhood of the point (2, −3, −2, 1, 0). Find an explicit approximate solution. Problem 3 (i) (7 pts) Suppose K ⊂ Rn is compact and has measure zero. Show that K has content zero. (ii) (5 pts) Does the result of part (i) remain true if we remove the assumption that K is com- pact? Problem 4 (i) (7 pts) Suppose Ω ⊂ Rn is open, f : Rn → R is a C∞ function, and n > 1. Show that f is not one to one. (ii) (5 pts) Suppose that Ω ⊂ Rn is open, f : Rn → Rk is a C∞ function, and n > k. Show that f is not one to one. Problem 5 (16 pts) Suppose Ω ⊂ Rn is open, f : Ω → R is C1, p ∈ Ω, and ∇f(p) ̸= 0. Prove that there are neighborhoods U and V of 0 and p respectively and a C1 diffeomorphism G : U → V with G(0) = p and f ◦ G(x) = f(p) + xn for all x = (x1, x2, . . . , xn) ∈ U. Problem 6 Let U = {(u, v, r, s) ∈ R4 : 0 < v < u < π 2 and 0 < s < r}. For (u, v, r, s) ∈ U, define G(u, v, r, s) = � r cos u + s cos v, r sin u + s sin v, r2 + s2, arctan r s � . (i) (4 pts) Prove that G is 1-1. (ii) (4 pts) Find JG. (iii) (4 pts) Let X = G(U). Show that X is an open subset of R4 and that G : U → X is a C1 diffeomorphism. (v) (8 pts) Suppose ∆ ⊂ U is compact, the topological boundaries of ∆ and G(∆) have content zero, f, F : R4 → R are continuous functions, and � G(∆) F(x, y, z, w) dxdydzdw = � ∆ f ◦ G(u, v, r, s) dudvdrds. Find F(x, y, z, w) in terms of f(x, y, z, w). Hint. If (x, y, z, w) = G(u, v, r, s), then x2 + y2 = r2 + s2 + 2rs cos(u − v), r = √z sin w, and s = √z cos w. Also, if 0 ≤ θ ≤ π/2, then sin θ = � (1 − cos θ)(1 + cos θ).