TA: My Huynh
MATH 1920 Sections 209/214
Discussion 16 - Chapters 17.1, 17.2, 17.3
October 4, 2016
1. Calculate div(F) and curl(F) for F = (ey , sin(x), cos(x)).
2. Find a potential function for the vector field F by inspection or show that one does
not exist.
a) F = (yexy , xexy )
b) F = (2xyz, x2 z, x2 yz)
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c) F = (2xzex , 0, ex )
3. Let C = C1 + C2 where C1 is the quarter circle x2 + y 2 = 4, z = 0 from (0, 2, 0) to
(2, 0, 0), and C2 is the line segment from 2, 0, 0 to (3, 3, 3). Compute the work done
along C by the force F = (−y + z, z − x, x + y + z).
R
4. Let I = C f (x, y, z)ds. Assume that f (x, y, z) ≥ m for some number m and all
points (x, y, z) on C. Which of the following conclusions is correct? Explain.
(a) I ≥ m
(b) I ≥ mL where L is the length of C.
5. The following statement is false:
If F is a gradient vector field, then the line integral of F along every curve is zero.
Which single word must be added to make it true?
6. Which of the following statements are true for all vector fields, and which are true
for only conservative vector fields? Why?
a) The line integral along a path from P to Q does not depend on which path is
chosen.
b) The line integral over an oriented curve C does not depend on how C is parametrized.
c) The line integral around a closed curve is zero.
d) The line integral changes sign if the orientation is reversed.
e) The line integral is equal to the difference of a potential function at two endpoints.
f) The line integral is equal to the integral of the tangential component along the
curve.
g) The cross partials of the components are equal.
7. Let F be a vector field on an open, connected domain D with continuous second
partial derivatives. Which of the following statements are always true, and which
are true under additional hypotheses on D?
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a) If F has a pontential function, then F is conservative.
b) If F is conservative, then the cross partials of F are equal.
c) If the cross partials of F are equal, then F is conservative.
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8. Is the vector field F = (4x ln(y), 2x y−1 ), for y > 0, conservative? If yes, find the
potential function.
9. Let C be the semicircle given by x2 + y 2 = 4 and y ≥ 0 oriented from left to right.
Calculate each of the followings integrals.
Z
a)
1ds
C
Z
b)
F · dr
C
Z
F · dr where F is the gradient vector field of the function f (x, y) = ln(x2 ).
c)
C
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Partial answers:
1. div(F) = 0, curl(F) = (0, sin(x), cos(x) − ey ).
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2. a) exy , b) DNE, c) zex .
3. 27/2
4. b)
5. closed
6. a) conservative b) all c) conservative d) all e) conservative f) all g) conservative.
7. a) always true, b) always true, c) need D to be simply-connected.
8. yes, f (x, y) = 2x2 ln(y) − ln(y).
9. a) 0, b) 0, c) 0.
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