Divergence Theorem and Variations
Theorem.
If
V
is a solid with surface
∂V
ii
F
·
ˆ
n
dS
=
iii
V
∇ ·
F
dV
f
ˆ
n
dS
=
V
∇
f dV
ˆ
n
×
F
dS
=
V
∇ ×
F
dV
where
ˆ
n
is the outward unit normal of
.
Memory Aid.
All three formulae can be combined into
ˆ
n
∗
˜
F dS
=
V
∇ ∗
˜
F dV
where
∗
can be either
·
,
×
or nothing. When
∗
=
·
or
∗
=
×
, then
˜
F
=
F
. When
∗
is
nothing,
˜
F
=
f
.
Proof:
The Frst formula is the divergence theorem and was proven in class.
To prove the second formula, assuming the Frst, apply the Frst with
F
=
f
a
, where
a
is any constant vector.
f
a
·
ˆ
n
dS
=
V
∇ ·
(
f
a
)
dV
=
V
b
(
∇
f
)
·
a
+
f
∇ ·
a
B
dV
=
V
(
∇
f
)
·
a
dV
To get the second line, we used vector identity # 8. To get the third line, we just used
that
a
is a constant, so that it is annihilated by all derivatives. Since
a
is a constant, we
can factor it out of both integrals, so
a
·
f
ˆ
n
dS
=
a
·
V
∇
f dV
=
⇒
a
·
±ii
f
ˆ
n
dS
−
V
∇
f dV
²
= 0
March 3, 2013
Divergence Theorem and Variations
1
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 Fall '13
 JoelFeldman
 Calculus, Derivative, Vector Space, Vector Motors, Vector field, Surface integral

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