i
1. If true, justify and if false, give a counterexample, or explain why.
(a) Let
f
(
x, y, z
) =
y

x
. Then the line integral of
∇
f
around the
unit circle
x
2
+
y
2
= 1 in the
xy
plane is
π
, the area of the circle.
Solution.
This is false. The line integral of any gradient around
a closed curve is zero.
(b) Let
F
be a smooth vector field in space and suppose that the
circulation of
F
around the circle of radius 1 centered at (0
,
0
,
0)
and lying in the
xy
plane, is zero. Then (
∇ ×
F
)(0
,
0
,
0) = 0.
Solution.
This is false. There are two reasons this is wrong.
First, one has to have zero circulation for circles of arbitrarily
small radius. Second, one has to have the circles in planes with
arbitrary normal vectors. An explicit counterexample is
F
=
y
k
,
which has
∇ ×
F
=
i
. The circulation around any circle in the
xy
plane is zero, but the curl at (0
,
0
,
0) is not zero.
(c) The center of mass of the region between the spheres
x
2
+
y
2
+
z
2
= 4 and
x
2
+
y
2
+
z
2
= 9 having mass density
δ
(
x, y, z
) = sin[
π
(7

x
2

y
2
+ 5
z
)]
lies somewhere on the
z
axis between
z
=

3 and
z
= 3.
Solution.
This is true. The mass density is symmetric about
the
z
axis (this is because the function
δ
depends on
x
and
y
only in the combination
r
2
=
x
2
+
y
2
) and so the center of mass
lies on this axis of symmetry. On the other hand, the center
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 Spring '08
 Ramakrishnan
 Calculus, Linear Algebra, Algebra, Unit Circle, 2 J

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