1. Let S be the lower half of the ellipsoid x2 + y 2 + 4z 2 = 4, z 0. Orient the surface using the outward unit normal. Let F(x, y, z) = xi + 2yj + 3zk. (a) Sketch S. (b) Using a parametrization based on spherical coordinates, determine a normal vector to S. (c) Using part (b), evaluate F dS. S 2. Let C be the boundary of the half-annulus 2 x2 + y 2 4, x 0 in the x-y plane, traversed in the positive direction. Let n be the outward unit normal to C in the x-y plane. (a) Sketch C. (b) Evaluate F n ds C where F(x, y) = (x2 + 5ex + cos6 (2y), 5ex y + x2 sin3 (4x) 3y 2 ). 3. Let S be an open, oriented surface with outward unit normal n. Let v(x, y, z) and w(x, y, z) be C1 1 irrotational vector elds, and f (x, y, z) and g(x, y, z) be C scalar functions. Show that ( f v) dS = (w S g) dS + S (f v + gw) ds S 4. Let S be the surface of the hemisphere x2 + y 2 + z 2 = 2, z 0. (a) Using spherical coordinates, determine a parametrization for S. (b) Using part (a), evaluate z dS. S 5. Using the special case formula for z = f (x, y), evaluate (x + y) dS S where S is the triangle with vertices (2, 0, 0), (0, 1, 0), and (0, 0, 2). 6. Let S be the surface of the open cone z = x2 + y 2 , z 1. (a) Using cylindrical coordinates, determine a parametrization for S. (b) Using part (a), evaluate F dS S where F (x, y, z) = 3xi 2j + y 2 k