Math 317, Fall 2012, Section 101
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Midterm Exam II
November 7, 2012
No books. No notes. No calculators. No electronic devices of any kind.
Name
Student Number
Problem 1. (5 points)
Evaluate the line integral
Z
xy 4 ds ,
C
where C is the right half of the circle x2 + y 2 = 16 in the plane.
Math 317, Midterm Exam II
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1
2
3
4
5
Problem 2. (5 points)
Use the fundamental theorem to evaluate the line integral
Z
F~ · d~r ,
C
where C is the parametrized curve
~r(t) = ht2 , tπ, 2t + 1i ,
0 ≤ t ≤ 1,
and F~ is the vecctor field
F~ (x, y, z) = hy cos(xy), x cos(xy) + z sin(yz), y sin(yz)i .
Find the numerical value of the integral.
total/25
Math 317, Midterm Exam II
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Problem 3. (5 points)
Use Green’s theorem to evaluate the line integral
I
hey , 2xey i · hdx, dyi ,
C
Where C is the rectangle with sides x = ±1, y = ±2.
Math 317, Midterm Exam II
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Problem 4. (6 points)
Assume that the vector field F~ has continuous partial derivatives everywhere in the
domain D, where D is R2 without the origin. True or false?
(a) If curl(F~ ) = 0 throughout D, then F~ is conservative,
(b) if curl(F~ ) = 0 throughout the first quadrant, then F~ is conservative in the
first quadrant,
R
R
(c) If curl(F~ ) = 2 throughout D, then the line integral C1 F~ · d~r > C1 F~ · d~r,
where C1 and C2 are semicircles from (2, −1) to (2, 1), and C1 goes to the left,
and C2 to the right of the point (2, 0).
(d) If curl(F~ ) < 0, throughout D, and F~ is the velocity field of a fluid flow, then
the fluid will push any chain of boats constrained to move along a simple
closed curve which does not enclose the origin, in a clockwise direction.
H
(e) if curl(F~ ) = 3 throughout D, then the integral F~ · d~r, where C is a circle of
radius 2 around the origin (oriented counterclockwise) is equal to 12π.
(f) if curl(F~ ) = 3 throughout D, then any paddle wheel inserted into the fluid
in such a way that it avoids the origin, will rotate counterclockwise, with an
angular speed of 6 radians per unit time.
Math 317, Midterm Exam II
Problem 5. (4 points)
Compute the curl and the divergence of the vector field
F~ (x, y, z) = xey~ı + sin(z) cos(x)~ + (x2 + y 2 + z 2 )~k
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Math 317, Midterm Exam II
Overflow space I.
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Math 317, Midterm Exam II
Overflow space II.
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