APPM 2350
HOMEWORK 12
FALL 2010
Due Friday, November 19, 2010 at 4 pm under your TA’s door.
1. Let
°
F
(
x, y
)=
−
y
x
2
+
y
2
ˆ
i
+
x
x
2
+
y
2
ˆ
j.
(a) Is
°
F
(
x, y
) continuous for all
x
and
y
? If not, then where is it discontinuous?
(b) Using the convention
°
F
(
x, y
)=
M
(
x, y
)
ˆ
i
+
N
(
x, y
)
ˆ
j
, show that
M
y
=
N
x
.
(c) If
C
is any closed contour not around the origin, ±nd
°
C
°
F
·
d°r ,
and justify your answer.
(d) Let
C
°
be
x
2
+
y
2
=
±
2
,
±>
0, where you move in a counterclockwise fashion around the circle.
Now ±nd
°
C
°
°
F
·
d°r
. Note, your answer should be independent of
±
. How can you reconcile your
result with the fact that
M
y
=
N
x
?
2. Now keep
°
F
the same as the previous problem, but let
C
be some arbitrary curve around the origin
in the xyplane. Imagine something like the ±gure shown immediately below.
!
"
#
We are now going to ±gure out how to ±nd
°
C
°
F
·
d°r
. To do this, we ±rst look at a similar but slightly
di²erent curve as shown in the picture below.
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 Fall '07
 ADAMNORRIS
 Line segment, l2 ∪ Ce

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