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.