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4. (30) Let S be the drinking cup which is 4 units tall, whose sides are the cylinder x² + y² = 9, with bottom at z = 0, and which has no top (or how would you drink?). (See picture, previous page.) Let F(x, y, z) = < −y, x,x+z>. Compute the flux of curl(F) through S. Parameterize the two pieces of S (side and bottom, see previous page for picture), compute the 2 fluxes, add, and compare to problem 3 Be sure that your normal vectors point "out". Note that your r vectors should have 2 parameters, but not the same 2 parameters. Finally, one of these integrals evaluates to 0. Just saying. curl(F) = "Sides" (cylinder shell) "Bottom" (disk)
3. (20) Let S be the drinking cup which is 4 units tall, whose sides are the cylinder x² + y² = 9 (i.e., the radius is 3, centered around z-axis), with bottom at z = 0, and which has no top (or how would you drink?). (Be clear with this picture! Bottom and sides, cylinder, no top.) boundary S Surface r(t) = dř (t) = F (†) = break [ ³ · d² = C no top Let F(x, y, z) =< -y, x,x+z>. We are going to verify Stokes' for this object, in this and the next problem. Do stuff in the order asked. Compute the circulation of F around the boundary, C, of S. Parameterize C. This is just the circle (not disk) at the top (rim) of the cup. Be sure your circle is at the top, not the bottom. Then compute what's asked for. "sides bottom (include the limits)