Math 208 (Lutz)
Exam IV Practice Problems
4/23/14
1. Compute the flux of F! = x!i + y!j + z!k over the quarter cylinder S given by x2 + y 2 = 1,
0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1, oriented away from the z-axis.
2. Find a potential function for the vector field F! = F1!i + F2!j, where
F1 =
2 cos2 x
2
− 2 cos x sin x ln(2x + y) + 4e2y cos(4x + y) + 3x2 y,
2x + y
F2 =
cos2 x
2
2
+ 4ye2y sin(4x + y) + e2y cos(4x + y) + x3 .
2x + y
3. Evaluate the circulation of F! = 3y!i + xy!j around the unit circle (in 2-space), oriented
counterclockwise.
4. Calculate the flux of F! = x2!i + y 2!j + z!k through the cone z =
with x2 + y 2 ≤ 1, x ≥ 0, y ≥ 0.
!
x2 + y 2 , oriented upward,
! = xy!i + z!j + 3y!k around the square of side 6, centered at the
5. Evaluate the circulation of G
origin, lying in the yz-plane, and oriented counterclockwise viewed from the positive x-axis.
6. Calculate the line integral of F! = y(x + 1)−1!i + ln(x + 1)!j along C, where C is the curve
from the origin along the x-axis to the point (3, 0) and
√ then√counterclockwise around the
circumference of the circle x2 + y 2 = 9 to the point (3/ 2, 3/ 2).
7. Compute the flux of F! = y!i − x!j + z!k through the sphere of radius 1 centered at the origin,
oriented outward.
8. Compute the flux of F! = !x, y" through the surface S, which is oriented toward the origin,
and is parameterized by x = 2s, y = s + t, z = 1 + s − t, for 0 ≤ s ≤ 1, 0 ≤ t ≤ 1.