The divergence theorem, more commonly known especially in older literature as Gauss's theorem (e.g., Arfken 1985) and also known as the Gauss-Ostrogradsky theorem, is
a theorem in vector calculus that can be stated as follows. Let 
be a region in space with boundary 
. Then the volume integral
of the divergence 
of 
over 
and the surface integral
of 
over the boundary 
of 
are related by
 |
(1)
|
The divergence theorem is a mathematical statement of the physical fact that, in the absence of the creation or destruction of matter, the density within a region of space can change only by having it flow into or away from the region through its boundary.
A special case of the divergence theorem follows by specializing to the plane. Letting 
be a region in the plane with boundary 
, equation (1) then collapses to
 |
(2)
|
If the vector field 
satisfies certain constraints, simplified forms can be used.
For example, if 
where 
is a constant vector 
, then
 |
(3)
|
But
 |
(4)
|
so
 |  | ![int_V[(del v)·c+vdel ·c]dV](/images/equations/DivergenceTheorem/Inline17.svg) |
(5)
|
 |  |  |
(6)
|
and
 |
(7)
|
But 
,
and 
must vary with 
so that 
cannot always equal zero. Therefore,
 |
(8)
|
Similarly, if 
,
where 
is a constant vector 
, then
 |
(9)
|
