SECTION
18.3
Divergence Theorem
(ET Section 17.3)
1245
(b)
By Stokes’ Theorem,
Z
C
r
F
·
d
s
=
ZZ
S
r
curl
(
F
)
·
d
S
By part(a) we have
m
(
r
)
≤
1
π
r
2
Z
C
r
F
·
d
s
≤
M
(
r
)
(2)
We take the limit over the circles of radius
r
centered at
Q
,as
r
→
0. As
r
→
0, the regions
S
r
are approaching the
center
Q
. The continuity of the curl implies that
lim
r
→
0
m
(
r
)
=
lim
r
→
0
M
(
r
)
=
curl
(
F
)(
Q
)
·
e
n
(
Q
)
=
curl
(
F
)(
Q
)
·
e
Therefore,
lim
r
→
0
m
(
r
)
≤
lim
r
→
0
1
r
2
Z
C
r
F
·
d
s
≤
lim
r
→
0
M
(
r
)
curl
(
F
)(
Q
)
·
e
≤
lim
r
→
0
1
r
2
Z
C
r
F
·
d
s
≤
curl
(
F
)(
Q
)
·
e
Hence,
lim
r
→
0
1
r
2
Z
C
r
F
·
d
s
=
curl
(
F
)(
Q
)
·
e
18.3 Divergence Theorem
(ET Section 17.3)
Preliminary Questions
1.
What is the Fux of
F
= h
1
,
0
,
0
i
through a closed surface?
SOLUTION
The divergence of
F
= h
1
,
0
,
0
i
is div
(
F
)
=
∂
P
∂
x
+
∂
Q
∂
y
+
∂
R
∂
z
=
0, therefore the Divergence Theorem
implies that the Fux of
F
through a closed surface
S
is
S
F
·
d
S
=
ZZZ
W
div
(
F
)
dV
=
W
0
=
0
2.
Justify the following statement: The Fux of
F
=
±
x
3
,
y
3
,
z
3
®
through every closed surface is positive.
The divergence of
F
=
D
x
3
,
y
3
,
z
3
E
is
div
(
F
)
=
3
x
2
+
3
y
2
+
3
z
2
Therefore, by the Divergence Theorem, the Fux of
F
through a closed surface
S
is
S
F
·
d
S
=
W
(
3
x
2
+
3
y
2
+
3
z
2
)
Since the integrand is positive for all
(
x
,
y
,
z
)
6=
(
0
,
0
,
0
)
, the triple integral, hence also the Fux, is positive.
3.
Which of the following expressions are meaningful (where
F
is a vector ±eld and
ϕ
is a function)? Of those that are
meaningful, which are automatically zero?
(a)
div
(
∇
)
(b)
curl
(
∇
)
(c)
∇
curl
(
)
(d)
div
(
curl
(
F
))
(e)
curl
(
div
(
F
))
(f)
∇
(
div
(
F
))
(a)
The divergence is de±ned on vector ±elds. The gradient is a vector ±eld, hence div
(
∇
)
is de±ned. It is not automat-
ically zero since for
=
x
2
+
y
2
+
z
2
we have
div
(
∇
)
=
div
h
2
x
,
2
y
,
2
z
i =
2
+
2
+
2
=
6
0
(b)
The curl acts on vector valued functions, and
∇
is such a function. Therefore, curl
(
∇
)
is de±ned. Since the
gradient ±eld
∇
is conservative, the cross partials of
∇
are equal, or equivalently, curl
(
∇
)
is the zero vector.
(c)
The curl is de±ned on vector ±elds rather than on scalar functions. Therefore, curl
(
)
is unde±ned. Obviously,
∇
curl
(
)
is also unde±ned.