# 辅导案例-32B

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 = 〈2x2 − 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.  Email:51zuoyejun

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