Math 320
Calculus III
Sixth Test
______________________
Name
This 50-minute 6-page test covers sections 16.4-9 of "Calculus" by Stewart.
1. Fill in the blank with the letter of the answer which most closely matches.
grad f
a. indicates the "flow out of " points
curl F
b. indicates the "circulation around" points
divF
c. indicates the rate and direction increase at points
2. True or False? Place T for true or F for false in the blanks below.
 =
(2 points each)
(2 points each)



i + j + k is the "diff" operator.
x
y
z
To apply Stoke’s theorem the surface must be oreintable.
The Möbius strip is not an oreintable surface.
x ·F = 0 (when F has continuous second order partials).
xxF = 0 (when F has continuous second order partials).
3
3
2
2
3. Evaluate 
 y dx – x dy along the curcle x + y = 1 (with positive orientation).
(8 points)
4. Determine whether or not F = y cos xy i + x cos xy j + sin z k is conservative. If so, find a
potential function f such that f = F.
(6 points)
z
z
z
5. Let F = e sin y i + e cos x j + e k .
a. Find the curl of F
(6 points)
b. Find the divergence of F
(6 points)
6. Identify the surface with the vector equation: r(u,v) = u i + u cos v j + u sin v k
(6 points)
2
7. Evaluate the surface integral 

S z dS where S is the surface x = y + 2z , 0 < y < 1, 0 < z < 1.
(10 points)
8. Find the flux of F = xy sin z i + cos xz j + y cos z k across the surface of the region below
the paraboloid x2 + y2 = z and above the cone x2 + y2 = z2.
(10 points)
9. Use Stokes’ theorem to evaluate 
 F dr where
C
F = yz i + 2xz j + exy k
and C is the intersection of the plane z = 5 and the cylinder x2 + y2 = 25 oriented
counterclockwise as viewed from above.
(10 points)
10. Use the divergence theorem to find 

S F dS where
F = 4x3z i + 4y3z j + 3z4 k
and S is the sphere of radius R centered on the origin.
(10 points)