Question 5 Let C be the intersection of two surfaces, x2 + y - 2x = 0 and z = \\4 - x2 - y2 . Find the work along C with the Tries clockwise orientation if seen from the positive direction of the z axes, done by the force field G = (2y - 1, x, z) remaining: 3 Solution: Marked out of 10.00 Flag The curve C is the intersection of two surfaces, the upper hemisphere x2 + y + z = 4, z 2 0, and the cylinder question x2 + y2 - 2x =0, i.e. (x - 1)2 + y = 1. To parametrizatione first parameterize the circle (x - 1)2 + y = 1. in the xy-plane, which is a projection of C, into xy-plane. By the requirement, this circle needs to be clockwise oriented. We obtain: x(t) = 1 + cos(t), y(t) = - sint, where t is in:(choose one of the following) a) [0, ] b) [-it/2, 7/2] c) [0, 27t] d) [-it/2, it] Note that the orientation on the circle is clockwise if seen from the positive direction of the z-axis. Since 2x = x2 + y ,we get z(1) = \\4 - 2(1 + cost) = V2(1 - cost) = 2 |sin =|. However, sin 5 2 * fort in : (choose one of the following) a) [0, 7] b) [-7/2, it/2] c) [0, 2x] d) [-it/ 2, it]. Finally, the parameterization of C is r(1) = (1 + cost, - sint, 2 sin ->, for t in: (choose one of the following) a) [0, z] b) [-it/2, 7/2] c) [0, 2x] d) [-7/2, it]. Thus, we obtain W = / G . dr = ... = Check Question 6 For each of the following vector fields on R decide if it is conservative. If it is conservative provide a potential. If it is Tries not conservative enter 'DNE' remaining: 2 F(x, y) = (3 . x2 . cos(y + 7), -x3 . sin(y + 7)) f (x, y) = Marked out of 10.00 G(x, y ) = (x2 . 13, x4. y? ) g(x, y ) = Flag h (x, y) = question H(x, y) = (sin(y + 3 . x), cos(7 . y + x)) Check