Math 202 Practice Midterm 3 Please work out each of the given problems. Credit will be based on the steps that you show towards the final answer. Show your work. Problem 1 Please answer the following true or false. If false, explain why or provide a counter example. If true explain why. A. Let f(x,y,z) be a function with continuous second order partial derivatives and let F(x,y,z) be the gradient of f(x,y,z). If S is the ellipsoid
x^{2} oriented outward, then
B. Let F(x,y) be a conservative vector field, then
Problem 2 Show that
for any closed surface S. Problem 3 A fish starting at the origin swims in a straight path to the point (0.2,0.1,0.3), then changes direction and swims along the circular path through the point (0.5,0.2,0.6) and the point (0.7,0.5,0.8), and finally changes directions heading straight to the point (1.5,1,2). The current can by represented by the vector field F(x,y,z) = (2x + 2z)i + (1  3z)j + (2x  3y + 5)k Find the total work done by the current. Problem 4 Evaluate where F is the vector field
and S is the rectangular solid with vertices (0,0,0), (1,0,0), (1,2,0), (0,2,0), (0,0,3), (1,0,3), (1,2,3), (0,2,3). Problem 5 Find the work done by sailing a ship from the point (2,3) the the point (1,2) against the wind with velocity field F(x,y) = yi + (3x + 2y)j Problem 6 Find the flux of F through the surface S where F(x,y,z) = 3zi  4j + yk and S is the part of the plane x + y + z = 1 in the first octant with upwardly pointing unit normal. SolutionProblem 7 Let F and G be differentiable transformations from R^{2} to R^{2} such that for both of these transformations x(u,v) = f (u) is a function of only u and y(u,v) = g(v) is a function of only v. Prove that J_{FoG} = J_{GoF}
