Math 250B Practice Questions for Test 4

Math 250B
Practice Questions for Test 4
1. Find the divergence and curl of F = x 2 yz i + 3xyz 3 j + (x 2 − z 2 ) k.
Answers:
∇ · F = 2xyz + 3xz 3 − 2z
∇ × F = −9xyz 2 i + (x 2 y − 2x) j + (3yz 3 − x 2 z) k
2. Determine if F = (e x cos y + yz) i + (xz − e x sin y) j + (xy − 2z) k is conservative and if
yes, find a potential function.
Answer: The field is conservative and f = e x cos y + xyz − z 2 + C is a potential function
for any constant C.
3. Evaluate the line integral ∫ z ds for the curve C with parametrization
C
Answer: 10π 2
x = 3 cos t, y = 3 sin t, z = 4t, 0 ≤ t ≤ π.
4. Evaluate the line integral ∫ C 2x ds where C consists of the parabola y = x2 from (0, 0 to
(1, 1) followed by the vertical line segment from (1, 1) to (1, 2).
Answer: 2 + (5 √ 5 − 1)/6
5. Evaluate the line integral ∫ C xeyz ds where C is the line segment from (0, 0, 0) to (1, 2, 3).
Answer: √ 14(e 6 − 1)/12
6. Evaluate ∫ C F · dr where F = xy i + x2 j and the path C is described by
Answer: 5/7
r(t) = t 2 i + t 3 j, 0 ≤ t ≤ 1.
7. Evaluate ∫ C F · dr where F = 4x3 i + 2yz 3 j + 3y 2 z 2 k and the path C is described by
r(t) = √ t 3 + 1 i + (t 2 − 1) j + (t + 1) k, 0 ≤ t ≤ 2.
Answer: Use the fundamental theorem of line integrals to get ∫ F · dr = 322.
C
8. Evaluate ∮
(2x + y 2 ) dx + (x 2 + 2y) dy
C
where C is the positively oriented closed curve formed by y = 0, x = 2, and y = x 3 /4.
Answer: 72/35

Math 250B Practice Questions for Test 4 Page 2 of 2
9. Evaluate the outward flux
∮
Φ =
C
F · n ds
where F = x 3 i+y 3 j and C is the positively oriented closed curve formed by x 2 +y 2 = 4.
Answer: 24π
10. Find the surface area of the surface described by
r(u, v) = a sin u cos v i + a sin u sin v j + a cos u k, 0 ≤ u ≤ π, 0 ≤ v ≤ 2π.
Answer: The surface is a sphere of radius a with surface area 4πa 2 .
11. Evaluate the surface integral ∫∫ S z dS where S is the portion of the cone z2 = x 2 + y 2
from z = 0 to z = 1.
Answer: 2√ 2π
3
12. Evaluate the surface integral ∫∫ yz dS where S is the part of the plane x + y + z = 1 in
S
the first-octant.
Answer: √ 3
24
13. Evaluate the surface integral ∫∫ S (x2 z+y 2 z) dS where S is the hemisphere z = √ 4 − x 2 − y 2 .
Answer: 16π
14. Find the flux of F = y i − x j + 4 k upward through S, where S is the part of the
paraboloid z = 1 − x 2 − y 2 in the first octant.
Answer: Φ = π
15. Find the outward flux of F = e x sin y i + e x cos y j + yz 2 k through the box S bounded
by x = 0, x = 1, y = 0, y = 1, z = 0, z = 2.
Answer: Φ = 2
16. Find the outward flux of F = x 3 i+ x 2 e z j+ 3zy 2 k through the closed surface S bounded
by the cylinder x 2 + y 2 = 4 and the planes z = 1 and z = 3.
Answer: Φ = 48π
17. Find the outward flux of F = 3x i + 2y j + z k through the sphere S of equation x 2 +
y 2 + z 2 = 4.
Answer: Φ = 64π
18. Find the flux of
2x i + 2y j
F = + k
x 2 + y 2
downward through S, where S is described by
Answer: Φ = 7π
r(u, v) = u cos v i + u sin v j + 2u k, 0 ≤ u ≤ 1, 0 ≤ v ≤ 2π.