Let S be the cylinder defined by x + y = 4 and 2 2. (a) Note that as = C C where C (resp. C) is the circle at the top (resp. bottom) of the cylinder. Give parametrizations c and c2 of C and C2 with the orientation corresponding to outwards pointing normal vectors for S. (b) Use part (a) and Stokes' theorem to evaluate 10 (V x F). dS for F(x, y, z) = xi+yj+xyek. Here S is oriented with outwards pointing normal vectors. Let W be the top half of the ball of radius R> 0 centered at (0, 0, 0), i.e., the surface given by x + y + z R and z 0. (a) Describe W as an elementary region in R. (b) Use part (a) and Gauss' divergence theorem to evaluate How F.dS, where F(x, y, z) = (x + e, y z, xy + z) and OW is oriented with outwards pointing normal vectors. 7