Math 200, Spring 2010
Handout
#30
Section 173 Divergence Theorem
The Divergence Theorem (Gauss’s Theorem)
Suppose
W
is a
simple solid region
and
S
is the boundary surface of
W
, given with positive (outward)
orientation.
If
F
is a vector field whose components have continuous partial derivatives on an open region
containing
W
, then
( )
div
S
W
d
dV
=
∫∫
∫∫∫
F
S
F
i
.
This says that, under the given conditions, the flux of
F
across the boundary surface of
W
is equal to the triple
integral of the divergence of
F
over
W
.
The Divergence Theorem applies to simple solid regions or regions that are a finite union of simple solid regions.
Examples of simple solid regions are regions bounded by ellipsoids or rectangular boxes.
Note: If
F
is a velocity field of a fluid in motion, then the total flux of the velocity field across
the boundary is equal to the triple integral of the divergence of the velocity field over the solid.
The positively oriented boundary surface of the solid region
W
often written as
W
∂
, and
( )
div
F
as
∇
F
i
,
so the Divergence Theorem can be expressed as
W
W
dV
d
∂
=
∇
∫∫∫
∫∫
F
F
S
i
i
Observe the similarity to Green’s and Stokes’ Theorems:
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 Spring '10
 JAMESDCAMPBELL
 Math, Vector Calculus, Vector field, Stokes' theorem

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