1. (Problem Set 32, #1) Let S be the portion of :32 4y2 + 22 = 0 between y = 3 and y = l, oriented with normals pointing toward the y-axis. Let F(a:, y,z) = (0,y,0). (3.) Evaluate [g. s (b) This GeoGebra applet shows 1:". and 5. Your friend doesn't understand graphically why the answer to #1(a) has the sign that it does; write an explanation for your friend. (0) Suppose the axes are measured in meters and that F(:r,y,z) represents the velocity, in m/s, of water at point (:17, y, z) in a river. If S is a mesh surface placed in the river, how much water ows through 5 "m 3 seconds? Include units in your answer. 2. (Problem Set 32, #2) In each part, you're given a vector eld I3 and a surface S; evaluate /f F. - dig. s Hint: In one part, you can do this without parameterizing 8. (a) 13"(33, y, z) : (4y, 3:, 2), and S is the part of the plane :1: + 219+ 32 : 6 which lies inside the cylinder 2:2 + 22 = 9, oriented with downward pointing normals. (b) 13(32, 3;, z) = {myz,y, z), and S is the part of the cylinder y2 + Z2 = 9 between a: = 0 and m = 3, oriented so that the unit normal vector at (1, 0, 3) is k. 3. (Problem Set 30, #3) We 've seen that, if r5 2's 0; vector eld on R2, we can measure the rotation of I3 using the 2-dtmenstonat curt. Next time, we '1! tatk about how to measure the rotation of vector elds on R3, and it will be important to be realty comfortable ua'th the 2-dtmeustonat curt. Suppose (its, 3;) = (cry + e", 2:2 yz) represents the velocity of water in a very shallow stream. If you put a paddlewheel in the stream at (2, 1), will it spin clockwise, spin counterclockwise, or not spin at all? Be sure to justify your