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ch14_07 - P1 OSO/OVY GTBL001-14-nal P2 OSO/OVY QC OSO/OVY...

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14-75 SECTION 14.7 . . The Divergence Theorem 1189 56. S yz dS , where S is the portion of x 2 + y 2 = 1 with x 0 and z between z = 1 and z = 4 y 57. S ( y 2 + z 2 ) dS , where S is the portion of the paraboloid x = 9 y 2 z 2 in front of the yz -plane 58. S ( y 2 + z 2 ) dS , where S is the hemisphere x = 4 y 2 z 2 59. S x 2 dS , where S is the portion of the paraboloid y = x 2 + z 2 to the left of the plane y = 1 60. S ( x 2 + z 2 ) dS , where S is the hemisphere y = 4 x 2 z 2 61. S 4 dS , where S is the portion of y = 1 x 2 with y 0 and between z = 0 and z = 2 62. S ( x 2 + z 2 ) dS , where S is the portion of y = 4 x 2 be- tween z = 1 and z = 4 63. Explain the following result geometrically. The flux integral of F ( x , y , z ) = x , y , z across the cone z = x 2 + y 2 is 0. 64. In geometric terms, determine whether the flux in- tegral of F ( x , y , z ) = x , y , z across the hemisphere z = 1 x 2 y 2 is 0. 65. For the cone z = c x 2 + y 2 (where c > 0), show that in spher- ical coordinates tan φ = 1 c . Then show that parametric equa- tions are x = u cos v c 2 + 1 , y = u sin v c 2 + 1 and z = cu c 2 + 1 . 66. Find the surface area of the portion of z = c x 2 + y 2 below z = 1, using the parametric equations in exercise 65. 67. Find the flux of x , y , z across the portion of z = c x 2 + y 2 below z = 1. Explain in physical terms why this answer makes sense. 68. Find the flux of x , y , z across the entire cone z 2 = c 2 ( x 2 + y 2 ). 69. Find the flux of x , y , 0 across the portion of z = c x 2 + y 2 below z = 1. Explain in physical terms why this answer makes sense. 70. Find the limit as c approaches 0 of the flux in exercise 69. Explain in physical terms why this answer makes sense. EXPLORATORY EXERCISES 1. If x = 3 sin u cos v, y = 3 cos u and z = 3 sin u sin v , show that x 2 + y 2 + z 2 = 9. Explain why this equation doesn’t guarantee that the parametric surface defined is the entire sphere, but it does guarantee that all points on the surface are also on the sphere. In this case, the parametric surface is the entire sphere. To verify this in graphical terms, sketch a picture showing geometric interpretations of the “spher- ical coordinates” u and v . To see what problems can oc- cur, sketch the surface defined by x = 3 sin u 2 u 2 + 1 cos v, y = 3 cos u 2 u 2 + 1 and z = 3 sin u 2 u 2 + 1 sin v . Explain why you do not get the entire sphere. To see a more subtle example of the same problem, sketch the surface x = cos u cosh v, y = sinh v, z = sin u cosh v . Use identities to show that x 2 y 2 + z 2 = 1 and identify the surface. Then sketch the surface x = cos u cosh v, y = cos u sinh v, z = sin u and use identities to show that x 2 y 2 + z 2 = 1. Explain why the sec- ond surface is not the entire hyperboloid. Explain in words and pictures exactly what the second surface is. 14.7 THE DIVERGENCE THEOREM Recall that at the end of section 14.5, we had rewritten Green’s Theorem in terms of the divergence of a two-dimensional vector field. We had found there (see equation 5.3) that C F · n ds = R ∇ · F ( x , y ) dA .
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