61. a. What do you think will be the electric flux through the cylindrical surface that is placed as shown in the constant electric field in Figure 19.16? Why? b. What if the cylinder is placed upright, as shown in Figure 19.17? Explain. Figure 19.16 Figure 19.17 >Answer > Solution 62. Let S be part of a cylinder centered on the y-axis. Explain why the three vectors fields F, G, and H have the same flux through S. Do not compute the flux. F = xi + 2yzk G = xi + ysinx] + 2yzk H = xi + cos(x2 + 2 )j + 2yzk66. Let S be the cube with side length 2, faces parallel to the coordinate planes, and centered at the origin. a. Calculate the total flux of the constant vector field v= -i + 2j + k out of S by computing the flux through each face separately. b. Calculate the flux out of S for any constant vector field v = ai + bj + ck. c. Explain why the answers to parts (a) and (b) make sense. 67. Let S be the tetrahedron with vertices at the origin and at (1, 0, 0), (0, 1, 0) and (0, 0, 1). a. Calculate the total flux of the constant vector field v= -i + 2j + k out of S by computing the flux through each face separately. Answer b. Calculate the flux out of S in part (a) for any constant vector field v. Answer c. Explain why the answers to parts (a) and (b) make sense. 68. Let P(x, y, z) be the pressure at the point (x, y, z) in a fluid. Let F(x, y, z) = P(x, y, z)k. Let S be the surface of a body submerged in the fluid. If S is oriented inward, show that Ss F . dA is the buoyant force on the body, that is, the force upward on the body due to the pressure of the fluid surrounding it. [Hint: F . dA = P(x, y, 2) k . dA = (P(x, y, 2)dA) . k]. 69. A region of 3-space has a temperature which varies from point to point. Let T(x, y, z) be the temperature at a point (x, y, z). Newton's law of cooling says that grad Tis proportional to the heat flow vector field, F, where F points in the direction in which heat is flowing and has magnitude equal to the rate of flow of heat. a. Suppose F = k grad T for some constant k. What is the sign of k? Solution b. Explain why this form of Newton's law of cooling makes sense. Solution c. Let W be a region of space bounded by the surface S. Explain why Rate of heat loss from W k / (grad I ) . d.A . Solution 70. The z-axis carries a constant electric charge density of 1 units of charge per unit length, with 1 > 0. The resulting electric field is E. a. Sketch the electric field, E, in the xy-plane, given E(x, y, z) = 2x xi+ yj ac2 + 2 2