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BJORN POONEN
November 22, 2009, 1:05pm to 1:55pm (50 minutes)
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) If
C
is a piecewise smooth curve in
R
3
from (
a
1
, a
2
, a
3
) to (
b
1
, b
2
, b
3
), then
R
C
y dx
=
b
1
b
2

a
1
a
2
.
Solution.
FALSE. In fact, the value of
R
C
y dx
depends on more than just the endpoints of
C
, because
y dx
is not an exact diﬀerential. To see this, notice that
curl
h
y,
0
,
0
i
=
±
±
±
±
±
±
i
j
k
∂
∂x
∂
∂y
∂
∂z
y
0
0
±
±
±
±
±
±
=

k
6
=
0
.
±
(b) If
f
(
x, y, z
) is a function with continuous second partial derivatives on
R
3
, then
div(
∇
f
) = 0 at every point.
Solution.
FALSE. By deﬁnition,
div(
∇
f
) =
²
∂
∂x
,
∂
∂y
,
∂
∂z
³
· h
f
x
, f
y
, f
z
i
=
f
xx
+
f
yy
+
f
zz
,
but this need not be 0. For example, if
f
(
x, y, z
) :=
x
2
, then we get div(
∇
f
) = 2. (In fact,
div(
∇
f
) =
∇ · ∇
f
=:
∇
2
f
and
∇
2
is called the
Laplace operator
.)
±
2) Let
F
= (cos
x
+ 2
y
2
+ 5
yz
)
i
+ (4
xy
+ 5
xz
)
j
+ (5
xy
+ 3
z
2
)
k
on
R
3
. Show that
F
is
conservative, and ﬁnd a potential function for
F
. For full credit, use a systematic method
(not just guessing), and show your work.
Solution.
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This note was uploaded on 03/08/2010 for the course MATH 18.02 taught by Professor Auroux during the Fall '08 term at MIT.
 Fall '08
 Auroux
 Multivariable Calculus

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