Math 23
Sections 110113
B. Dodson
Week 10 Homework:
[due April 15]
16.1, 16.2
vector fields and line integrals
16.3
Fundamental Theorem for line integrals
16.4
Green’s Formula
Problem 16.1.26. Find and sketch the
gradient field of
f
(
x, y
) =
1
2
(
x
+
y
)
2
.
Solution. The gradient field
→
∇
f
=
< f
x
, f
y
>,
so
f
x
=
1
2
·
2(
x
+
y
) =
x
+
y,
and
f
y
=
1
2
·
2(
x
+
y
) =
x
+
y
gives
→
∇
f
=
< x
+
y, x
+
y > .
For the sketch, we pick a few values
→
∇
f
(1
,
2) =
<
3
,
3
>,
→
∇
f
(1
,
3) =
<
4
,
4
>,
→
∇
f
(1
,

2) =
<

1
,

1
>
and draw displacement vectors
<
3
,
3
>, <
4
,
4
>, <

1
,

1
>
starting at
(1
,
2)
,
(1
,
3)
,
(1
,

2) respectively.
Problem 16.2.5:
Evaluate the line integral
Z
C
xy dx
+ (
x

y
)
dy,
where
C
is the curve consisting of line segments
from (0
,
0) to (2
,
0) and from (2
,
0) to (3
,
2)
.
Solution:
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2
.
By definition,
Z
C
=
Z
C
1
+
Z
C
2
.
For
C
1
we pick the simplest parameterization
x
=
t, y
= 0 for 0
≤
t
≤
2
.
Then substituting the
parameterization,
xy
= 0
,
so
Z
C
1
xy dx
= 0;
and,
y
= 0 gives
dy
= 0
·
dt,
so
Z
C
1
(
x

y
)
dy
= 0
.
For
C
2
we look for a parameterization with 0
≤
t
≤
1
,
expecting that to simplify the integral. We take (2
,
0)
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 Spring '06
 YUKICH
 Fundamental Theorem Of Calculus, Integrals, Line integral, Line segment

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