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February 6th, 2009
Math 20e  Assignment #5 Solutions
Problem 1
(page 310 #2)
.
Find the divergence of the vector ﬁeld
V
(
x,y,z
) =
zy
i
+
xz
j
+
xy
k
.
Solution.
Let
V
(
x,y,z
) =
h
yz,xz,xy
i
=
h
V
1
(
x,y,z
)
,V
2
(
x,y,z
)
,V
3
(
x,y,z
)
i
. Then we com
pute the divergence by taking
div
V
=
∇ ·
V
=
∂V
1
∂x
+
∂V
2
∂y
+
∂V
3
∂z
=
∂
∂x
(
yz
) +
∂
∂y
(
xz
) +
∂
∂z
(
xy
)
= 0 + 0 + 0
= 0
.
Problem 2
(page 311 #7)
.
Sketch a few ﬂow lines for
F
(
x,y
) =
y
i
. Calculate
∇ ·
F
and
explain why your answer is consistent with your sketch.
Solution.
∇ ·
F
=
∂F
1
∂x
+
∂F
2
∂y
=
∂
∂x
(
y
) +
∂
∂y
(0)
= 0
.
1
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F
=
∇ ·
F
= 0 is consistent with the graph because zero divergence means the vector
ﬁeld should have no expansions or contractions, and the ﬂow lines do not spread out or get
closer together.
Problem 3
(page 311 #8)
.
Sketch a few ﬂow lines for
F
(
x,y
) =

3
x
i

y
j
. Calculate
∇·
F
and explain why your answer is consistent with your sketch.
Solution.
∇ ·
F
=
∂F
1
∂x
+
∂F
2
∂y
=
∂
∂x
(

3
x
) +
∂
∂y
(

y
)
=

4
.
div
F
=
∇ ·
F
=

4 is consistent with the graph because negative divergence means the
vector ﬁeld should contract, and the ﬂow lines do not spread out or get closer together.
Problem 4
(page 312 #26)
.
Show that
F
= (
x
2
+
y
2
)
i

2
xy
j
is
not
a gradient ﬁeld.
Solution.
If
F
was a gradient ﬁeld, then we could write
F
=
∇
f
for some scalar function
f
. But then by Theorem 1 on page 303 we know that the curl of a gradient is always zero,
which means
0 =
∇ × ∇
f
=
∇ ×
F.
2
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This note was uploaded on 01/30/2010 for the course MATH 20E 20E taught by Professor Enright during the Fall '09 term at UCSD.
 Fall '09
 Enright

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