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SOLUTIONS TO 18.02 PRACTICE MIDTERM #3B
BJORN POONEN
1) For each of (a)(c) below: If the statement is true, write TRUE. If the statement is
false, write FALSE. (Please do not use the abbreviations T and F.) No explanations are
required in this problem.
(a) Let
F
be a continuously diﬀerentiable vector ﬁeld on
R
2
, and let
C
be a simple closed
oriented curve. If div
F
= 0 at every point inside
C
, then
H
C
F
·
d
r
= 0.
Solution.
FALSE. For example, if
F
is the rotational ﬂow given by
F
(
x, y
) =
h
y, x
i
, and
C
is the counterclockwise unit circle centered at the origin, then div
F
= 0 + 0 = 0 at every
point, but
H
C
F
·
d
r
is positive since
F
is parallel to the unit tangent vector at each point of
C
.
(If we replace div
F
by curl
F
, then the statement is true by Green’s theorem.)
±
(b) Any region obtained by removing a ray (halfline) from
R
2
is simply connected.
Solution.
TRUE, because if a simple closed curve is contained in the region (i.e., does not
intersect that ray), then its interior is contained in the region too: it’s impossible for a simple
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 Fall '08
 Auroux
 Multivariable Calculus

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