MATH 208 Section 003
Practice Test 3
Name:
You must show me enough work that I can reproduce your results.
1. Consider the vector field F~ sketched below.
(a) (6 points) Give a formula for F~ . (Many answers are possible.)
(b) (4 points) Draw an oriented curve C on the grid below such that the circulation of F~ around C is
positive. Clearly indicate the orientation of the curve.
2
1
−2
−1
1
−1
−2
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2
2
2
2. (12 points) Let F~ = 2xex y~i + (2ex − 4y)~j and let C be the boundary of the rectangle 0 ≤ x ≤ 4,
0 ≤ y ≤ 2, oriented counterclockwise. Calculate the circulation integral
I
F~ · d~r.
C
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3. Let F~ =
√ 1 ~i
2 x−y
−
~j.
√1
2 x−y
(a) (4 points) Calculate the (two-dimensional or scalar) curl of F~ .
(b) (2 points) What is the domain of F~ ?
(c) (6 points) Explain whether you can use the curl test to conclude that F~ is path-independent. Justify
your answer carefully.
4. (12 points) Determine the work done in moving an object from (0, 0) to (π, 0) along the graph of
y = sin(x) by the force F~ = y~i − ~j.
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5. (12 points) Let F~ = y~i − x~j + z~k. Let S be the part of the graph of the paraboloid z = x2 + y 2 − 12
that lies below the xy-plane, oriented toward the origin. Compute the flux of F~ through S.
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6. let F~ = (ey + 3xy 2 )~i − 2y 3 z~j + (4z − x4 )~k.
(a) (4 points) Calculate div F~ .
(b) (8 points) Let S be the boundary of the cube 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1, oriented outward.
Calculate the flux of F~ through S.
Question:
1
2
3
4
5
6
Total
Points:
10
12
12
12
12
12
70
Score:
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