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Multivariable Calculus Final Exam - Prof. P. Wong, December 14, 2006, Exams of Mathematics

A final exam for a multivariable calculus course taught by prof. P. Wong, held on december 14, 2006. The exam covers various topics such as matrix representation of functions, change of variables in double integrals, directional derivatives, green's theorem, parametrized surfaces, and vector calculus theorems.

Typology: Exams

2012/2013

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MATH206A MULTIVARIABLE CALCULUS - PROF. P.
WONG
FINAL EXAM - DECEMBER 14, 2006
NAME:
Instruction: Read each question carefully. Explain ALL your work
and give reasons to support your answers.
Advice: DON’T spend too much time on a single problem.
Problems Maximum Score Your Score
1. 15
2. 20
3. 15
4. 20
5. 15
6. 15
7. 20
Total 120
1
pf3
pf4
pf5
pf8

Partial preview of the text

Download Multivariable Calculus Final Exam - Prof. P. Wong, December 14, 2006 and more Exams Mathematics in PDF only on Docsity!

MATH206A MULTIVARIABLE CALCULUS - PROF. P.

WONG

FINAL EXAM - DECEMBER 14, 2006

NAME:

Instruction: Read each question carefully. Explain ALL your work and give reasons to support your answers. Advice: DON’T spend too much time on a single problem.

Problems Maximum Score Your Score

  1. 15
  2. 20
  3. 15
  4. 20
  5. 15
  6. 15
  7. 20 Total 120

1

2 FINAL EXAM - DECEMBER 14, 2006

  1. Let f : R^2 → R^2 be given by

f(u, v) = (3u, −v).

(5 pts) (i) Find a matrix B so that f(u, v) = B

[

u v

]

(5 pts.) (ii) If D is the unit square [0, 1] × [0, 1], describe the image D∗^ = f(D).

(5 pts.) (iii) Compute (^) ∫ ∫

D∗

x + y dA

by transforming the double integral into a double integral over D.

4 FINAL EXAM - DECEMBER 14, 2006

  1. Consider the following function f : R^2 → R given by f(x, y) = x^5 + 2xy + y^3.

(8 pts) (i) Find a direction (give a unit vector) in which f increases most rapidly at the point (2, −2).

(7 pts) (ii) Find an equation of the line tangent to the level curve f(x, y) = 16 at the point (2, −2).

MATH206A MULTIVARIABLE CALCULUS - PROF. P. WONG 5

  1. (8 pts) (i) Let C be the path formed by the square with vertices (0, 0), (1, 0), (0, 1), and (1, 1), oriented counterclockwise. Use Green’s theorem to evaluate the line integral ∮ C

y^2 dx + x^2 dy.

(12 pts) (ii) Let F (x, y) = (2x sin y, x^2 cos y). Determine whether the vector field F is path independent. If F is path independent, find a function f so that ∇f = F.

MATH206A MULTIVARIABLE CALCULUS - PROF. P. WONG 7

  1. Let F (x, y, z) = (3x − xy, xy − yz, z(y − x) + z 22 ). (5 pts) (i) Find the divergence div F of F.

(10 pts) (ii) Use Gauss’ (or Divergence) theorem to evaluate the surface integral (^) ∫∫

S^2

© F · n dσ

where S^2 is the 2-dimensional unit sphere. [The volume of a ball of radius R is 43 πR^3 .]

8 FINAL EXAM - DECEMBER 14, 2006

  1. Let F (x, y, z) = (z, x, y). Consider the parametrized surface M given by

f(s, t) = (s cos t, s sin t, t) where 0 ≤ s ≤ 1 , 0 ≤ t ≤ π 2.

(5 pts.)(i) Find curl F.

(5 pts.)(ii) Rewrite (not evaluate) the surface integral ∫ ∫ M

curl F · n dσ

as a double integral over the region R = {(s, t)| 0 ≤ s ≤ 1 , 0 ≤ t ≤ π 2 }.

(10 pts.)(iii) Use Stokes’ theorem to evaluate the path integral ∮ ∂M

F · dx [Hint: Use part (ii).].