1. The gravitational force field created by the mass M and acting on the mass m is F = -GMm 3, where r(x, y, z) = (x, y, z). In the following calculations, assume you are using units where GMm = 1. Find the flux of F through a horizontal plane z = a. Surprisingly, the answer is inde- pendent of a. Hint: parameterize the plane in terms of u and v and then evaluate the flux integral by using polar coordinates in the uv plane. 2. Suppose S is a non-closed surface with a closed boundary curve C and let F(x, y, z) be a vector field defined on S. Stokes' Theorem says that the line integral of F over the closed curve C is equal to the flux integral of V x F over the surface S. The equation for this relation is OF . dr = (V x F) . as. Let S represent the half of the unit sphere where z 2 0 and let F = (z - y, x - z, y - x). Verify Stokes' Theorem for this case by computing each side of the equation and showing the results are equal 3. A torus is a donut-shaped geometrical object with an inner radius r and an outer radius R. They can be parameterized as: r(u, v) = ((R + rcosu) cosv, (R + rcosu) sinv, r sinu) OSUS2T, OSUS 2T a) Show that the normal vector with outward orientation is ar Or n = av = r(R + r cos u) (cos u cos v, cos u sin v, sin u). b) Use the result from part (a) to calculate (and greatly simplify) the magnitude of n and use it derive the formula for the surface area of a torus: SA = 472 Rr. c) Suppose r = 1 and R = 3. Using the parameterization and the normal vector of the torus, manually compute the flux of the vector field F = (x, y, z) through the torus. d) Compute the flux again from part (c) using the Divergence Theorem. Make sure the answers agree! Hint: the volume of a torus is V = 272 Rr2