Math 208-002
Exam #3
December 5, 2013
UNL Student ID Number:
Name:
Turn off your cell phones.
Page:
1
2
3
4
5
6
Total
Points:
22
22
14
14
14
14
100
Score:
Instructions:
1. No books or notes may be used on the exam.
2. Credit or partial credit will be given only when the appropriate explanation and/or work is
shown.
3. Read and follow directions carefully.
4. This exam has 8 questions, for a total of 100 points. There are 6 pages (in addition to this
cover sheet).
5. You will have 50 minutes to complete the exam.
6. ALL CELL PHONES MUST BE TURNED OFF DURING THE EXAM. Phones
that ring, vibrate, or otherwise cause a distraction will be confiscated and may result in the
student receiving no credit for the exam.
DO NOT OPEN YOUR EXAM UNTIL INSTRUCTED TO DO SO!!
(10 points) 1. Give a parameterization in vector form of the plane containing the points (1, 2, 3), (1, 3, 5),
and (2, 1, 4).
R
(12 points) 2. Let F~ = 2x~i + 6y 2~j + 12z 3~k. Calculate C F~ · d~r, where C is the path from (4, 0, 0) along the
quarter circle in the xy-plane to (0, 4, 0), followed by the path to (0, 0, 0) along the y-axis, and
then along the z-axis to (0, 0, 5).
R
(10 points) 3. Calculate C ((x2 − y)~i + (y 2 + x)~j) · d~r, where C is the path along the graph of the parabola
y = x2 + 1 from (0, 1) to (1, 2).
R
(12 points) 4. Use Green’s Theorem to calculate C ((2x2 + 3y)~i + (2x + 3y 2 )~j) · d~r, where C is the path
around the triangle with vertices (2, 0), (0, 3), (−2, 0), oriented counterclockwise.
Page 2
(14 points) 5. Let S be the open cylindrical can with a bottom, but no top, given by the union of the cylinder
x2 +y 2 = 9, 0 ≤ z ≤ 2, and the disk x2 +y 2 ≤ 9, z = 0. If S is oriented outward and downward,
find
Z
~
curl(−y~i + x~j + z~k) · dA.
S
(Hint: The boundary of S is the circle x2 + y 2 = 9 in the plane z = 2.)
Page 3
(14 points) 6. Use the Divergence Theorem to find the flux of F~ through the closed cylinder of radius 2,
centered on the z-axis, with 3 ≤ z ≤ 7, where F~ = (x + 3eyz )~i + (ln(x2 z 2 + 1) + y)~j + z~k.
Page 4
(14 points) 7. Calculate directly the flux of the vector field F~ = x~i + y~j through the part of the surface
z = 25 − (x2 + y 2 ) above the disk of radius 5 centered at the origin in the xy-plane, oriented
upward.
Page 5
R
(14 points) 8. Use Stokes’ Theorem to find C (−z~j + y~k) · d~r, where C is a circle of radius 2 in the plane
x + y + z = 3, centered at (1, 1, 1), and oriented clockwise when viewed from the origin.
(Hint: A unit normal vector for the plane x + y + z = 3 is ~n =
Page 6
√1 (~i
3
+ ~j + ~k).)