Section 13.9 wrapup:
Let’s use the Divergence Theorem to prove
(div
v
)(
P
) = lim
r
→
0
(
∫∫
S
(
r
)
v
⋅
d
S) /
Vol(
B
(
r
))
where
B
(
r
) is a ball with center
P
and radius
r
, and
S
(
r
) =
∂
B
(
r
):
By the Divergence Theorem, the flux of
v
out of
B
(
r
) is
given by
∫∫ ∫
B
(
r
)
(div
v
)
dV
.
Thus
(1) the average divergence of
v
in
B
(
r
)
equals
(2) the flux of
v
out of
B
(
r
) divided by the volume of
B
(
r
).
If we let
r
go to 0, (1) converges to the divergence of
v
at
P
, and the right hand side converges to
lim
r
→
0
(flux of
v
out of
B
(
r
))/vol(
B
(
r
)).
In other words, divergence can be interpreted as flux per
unit volume, just as the components of the curl vector can
be interpreted as circulation per unit area.
True/False questions for chapter 13:
1. “If
F
is a vector field, then div
F
is a vector field.”
..?..
False; div
F
is a scalar field.
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2. “If
F
is a vector field, then curl
F
is a vector field.”
..?..
True.
See Definition 13.5.1.
3. “If
f
has continuous partial derivatives of all orders on
R
3
, then div(curl
∇
f
) = 0.”
..?..
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 Fall '11
 Staff
 Vector Calculus, Vector field, div

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