UCLA: Math 32B Problem set 9 Spring, 2020 This week on the problem set you will get practice applying and understanding Green’s theorem and Stokes’ theorem. Homework: The homework will be due on Friday 5 June. It will consist of questions 3, 4, 5 below. *Numbers in parentheses indicate the question has been taken from the textbook: J. Rogawski, C. Adams, Calculus, Multivariable, 3rd Ed., W. H. Freeman & Company, and refer to the section and question number in the textbook. 1. (Section 18.1) 3, 7, 8, 9, 12, 19, 20, 21, 23, 24 25, 29, 36∗, 41, 45. (Use the following translations 4th 7→ 3rd editions: 7 7→ 5, 8 7→ 6, 9 7→ 7, 12 7→ 10, 19 7→ 15, 20 7→ 16, 21 7→ 17, 23 7→ 19, 24 7→ 20, 25 7→ 21, 29 7→ 25, 36 7→ 32, 41 7→ 37, 45 7→ 41 otherwise the questions are the same). 2. (Section 18.2) 5, 8, 9, 18, 19. (Use the following translations 4th 7→ 3rd editions: 18 7→ 16, 19 7→ 17, otherwise the questions are the same). 3. Let F(x, y, z) = 〈x, x + y3, x2 + y2 − z〉 and let S be the surface z = x2 − y2 where x2 + y2 ≤ 1 with upward orienation and boundary C (with the usual boundary orientation). Find ∫ C F · dr. 4. Let F = 〈x, y,−2z + ex4+y2〉 and let S be the part of the hyperboloid x2 + y2 = 1 + z2 where z2 ≤ 3 oriented so that at points with positive z values the z coordinate of the normal vector is negative (i.e. with outward pointing normal). What is ∫∫ S F · dS? Hint: Find a simpler surface with the same boundary. 5. Consider the 3 dimensional polyhedron pictured below with vertices (0, 0, 2) (0, 0,−1) (0, 1, 0) (1, 0, 0) (0, 1, 1) (1, 0, 1) with outward pointing orientation. Find the flux of F = 〈2×2 − 3xy2, xz2ez + y3, sin(x2 + y2)〉 through S. *The questions marked with an asterisk are more difficult or are of a form that would not appear on an exam. Nonetheless they are worth thinking about as they often test understanding at a deeper conceptual level.
