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# Math 53 Homework 13 Due Friday 4/29/16 in section (The problems in parentheses are for extra practice and optional. Only turn in the underlined...

Hey. I'd like to get help for Problem 1. I have drawn the diagram but am unable to identify U and T - I only observe 2 surfaces, S and U(or T). How shall I visualize the problem and proceed to solve the sub-problems?

Math 53 Homework 13 Due Friday 4/29/16 in section (The problems in parentheses are for extra practice and optional. Only turn in the underlined problems.) Monday 4/18: The divergence theorem (continued) Read: section 16.9. Work: 16.9: 17 , 19 , 27 , (29). Problems 1 and 2 below. Wednesday 4/20: Stokes’ theorem Read: section 16.8. Work: 16.8: (1), (3), (5), (7), (9), 13 1 , 15 . Problems 3 and 4 below. Friday 4/22: Review for midterm 2 Monday 4/25: MIDTERM 2 Wednesday 4/27: Stokes’ theorem (continued) Read: sections 16.8 and 16.10. Problem 1. Consider the space region bounded below by the right-angled cone z = r x 2 + y 2 , and above by the sphere x 2 + y 2 + z 2 = 2. These two surfaces intersect in a horizontal circle; let T be the horizontal disk having this circle as boundary, S the spherical cap forming the upper surface, and U the cone forming the lower surface. Orient S , T , U “upwards”, so the normal vector has a positive ˆ k-component. a) For each of the three surfaces, determine geometrically (without calculation) whether the ﬂux of the vector ±eld v F = x ˆ ı + y ˆ  is positive or negative. b) Calculate the ﬂux of v F across each surface (with the upwards orientation). (Do not use the divergence theorem). c) Use the divergence theorem to ±nd the ﬂux of v F = x ˆ ı + y ˆ  out of the solid cone bounded by T and U . Same question with the ice-cream cone bounded by S and U . d) Show that the answers you found in part (c) are consistent with those you found in part (b). (Be careful with orientations!) 1 6th/7th ed: do the 8th ed problem: v F ( x, y, z ) = - y ˆ ı + x ˆ - 2 ˆ k, S is the cone z 2 = x 2 + y 2 , 0 z 4, oriented downwards. 1
Problem 2. a) Let f ( x, y, z ) = 1 = ( x 2 + y 2 + z 2 ) - 1 / 2 . Calculate v F = f , and describe geometrically the vector feld v F . b) Evaluate the ﬂux oF v F over the sphere oF radius a centered at the origin. c) Show that div v F = 0. Does the answer obtained in (b) contradict the divergence theorem? Explain. d) Let S be a surFace in the frst octant, whose boundary lies in the three coordinate planes (see picture). Show that ii S v F · ˆn dS is independent oF the choice oF S , and calculate its value. (Hint: apply the divergence theorem to a suitable portion oF the frst octant). ˆn Problem 3. a) Calculate the curl oF v F = 2 xz ˆ ı + y 2 ˆ k. b) Using Stokes’ theorem, show that c C v F · dv r = 0 For any simple closed curve C drawn on the unit sphere x 2 + y 2 + z 2 = 1. Problem 4. Consider the tetrahedron with vertices at P 0 = (0 , 0 , 0), P 1 = (1 , 0 , 1), P 2 = (1 , 0 , 1), and P 3 = (1 , 1 , 0). a) Say which orientation (order oF vertices) oF the boundary curve oF each Face is compatible with the choice oF the normal vector pointing out oF the tetrahedron. b) Compute the work done by the vector feld v F = yz ˆ y 2 ˆ k around the boundary curve oF the Face P 0 P 1 P 3 directly using line integrals (using the orientation From part (a)). c) Use Stokes’ theorem to compute the work done around each oF the Four Faces (including the one you computed directly in part (b)). Use the orientations From part (a). (Note: the symmetry z → − z exchanges two oF the Faces oF the tetrahedron, and can be used to avoid one calculation – iF you choose to use symmetry, you need to explain why it is legitimate.) d) The sum oF the Four values you Found in part (c) should be zero. Explain this in two di±erent ways: (i) geometrically, by considering the various line integrals that are being added to- gether; (ii) by using the divergence theorem to compute the ﬂux oF curl v F out oF the tetra- hedron. 2

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