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10) A) Consider the vector field F = (r, y

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10) A) Consider the vector field F = (r, y. 0). (a) Compute V x F and V . F. (b) Let C be the closed curve in the plane = = y + 3 lying over the curve r = 2cosd in the zy-plane. Compute ( F . T ds. (c) Compute the flux of F out of the cylinder a + y' =4 for -2% s = $ 2x. (d) Compute the flux of F out of the wavy cylinder r = 2 + sin z for -21 $ 2 5 2x. 10.B. Let F(r, y,) = G(r, y,!) = be two vector fields, where r = ri + yj + zk. (a) Let S be the sphere of radius 3 centered at the origin. Compute: (i) The flux of F out of S. (ii) The flux of G out of S. (b) Describe a surface 7 such that the flux of F through T equals the flux of G through T. Give an equation for T. 10.C. A wire along the z-axis carrying an electric current / generates a magnetic field (-yi+ cj B(r. V. = ) = 6 \+ 1/2 where c is the speed of light. (a) Describe the domain of definition of B (in R"). (b) Show that V x B is zero away from the z-axis. (c) Compute the circulation of B around the unit circle C, in the zy-plane centered at the origin, oriented counterclockwise when viewed from above. (d) Describe a closed loop in R" around which the circulation of B is zero. (e) Compute the circulation of B around the ellipse Cy in the plane z = 2 given by equation z' + 9y" =9, oriented counterclockwise when viewed from above. 10.D. You are holding a crystal ball of radius 1, four feet above the ground in an energy vector field F = (y' -x x' - y , 75=) Find the a and y coordinates of the ball's center at which the flux of F out of the ball will be the most positive. Express your answer in the form (z, y, 4). Find another location where the flux will be negative. 10.E. Let S be the "black hole" surface in IR" parameterized by r(r,0) = (rcos#, rsine, Inr) (0 < rs 1,050s 2x) where r and & are the usual parameters in cylindrical coordinates. Let F = (0,0, vaz + y?) (a) Compute the vectors r, and r, tangent to S at S(1/4, w/4) (in R3 ). (b) Compute the flux of F upwards through S, for 0 s r s 1.