MATH 208 Section 003
Test 3
April 24, 2015
Name:
You must show me enough work that I can reproduce your results.
1. (8 points) Parameterize the plane containing the points P = (1, −1, 2), Q = (3, 4, 1), and R = (5, 2, −2).
2. Let F~ = (3x2 y + ex )~i + (x3 − 4y)~j.
(a) (6 points) Use the curl test to show that F~ is path-independent. Explain why you can use the curl
test here.
(b) (4 points) Find a potential function f for F~ .
(c) (4 points) Use your answer to part (b) to compute the line integral of F~ from (0, 1) to (1, 2) along
the curve parameterized by ~r(t) = t~i + (2t2 − t + 1)~j. [Note: if you did not answer part (b), you
can still answer this part using another method.]
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3. (12 points) Let F~ = (y − 23 y 3 )~i + (x3 + xy 2 + 2y)~j and let C be the curve consisting of the x-axis from
√
√
√
√
(0, 0) to ( 2, 0), the circle x2 + y 2 = 2 from ( 2, 0) to (0, 2), and the y-axis from ( 2, 0) to (0, 0),
oriented counter-clockwise (see the picture to the right). Calculate the circulation integral
I
F~ · d~r.
C
4. (12 points) Let F~ = (3y − 12)~i + x2~j and let C be the portion of the graph of y = 4 − x2 from (−2, 0)
to (2, 0). Calculate the line integral
Z
F~ · d~r.
C
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5. (12 points) Let F~ = 2xy~i + ~j + z~k and let S be the part of the graph of z = x + 3 sin(y) + 5 above the
rectangle 0 ≤ x ≤ 1, 0 ≤ y ≤ π, oriented upward. Compute the flux of F~ through S.
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6. let F~ = 2xyz~i + (3y − ez )~j + (2z − yz 2 )~k.
(a) (4 points) Calculate the divergence of F~ , div F~ .
(b) (8 points) Let S be the closed cylinder (i.e., including the top and bottom) of radius 3 centered on
the z-axis, with top in the plane z = 2 and base in the plane z = −2, oriented outward. Compute
the flux of F~ through S.
Question:
1
2
3
4
5
6
Total
Points:
8
14
12
12
12
12
70
Score:
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