11. (question 4?) A beautiful ux integral Consider the potential function (x. y, z) = 6(9), G is any twice differentiable function and p = 1hr: + y2 + 22: therefore. G depends only on the distance from the origin. a. Show that the gradient vector eld associated with d: is given by: F = th = G '60)? , where r = {15, y,z) and p = Irl. b. Let S be the sphere of radius a centered at the origin and let D be the region enclosed by S. Show that the flux ofF across 3 is @SF - ndS = 4na26'(a). c. Show that v - F = v - we = 259'" + 6'00. (1. Use part (c) to show that the ux across 5 {as given in parttbl] is also obtained by the volume integral HID V - Fdl'. (Hint: use spherical coordinates and integrate by parts). 12. Suppose that F = (Fr, 1"}.in {subscripts here indicating vector components, not partial derivatives) is a continuously differentiable vector eld. Let Ch be the solid cylinder: Ch = {(1. J'. 2) I I: + J12 S 1. E E z S g} and let D be the 2-dimensional disc of radius 1 centred at the origin in the xy plane. D ={(x,y.z)|xz+yzsl, 2:0} Show from rst principles (i.e., without using the divergence theorem) that l 3F um- mag naysayers "Huh 35,. 063 an