Wolfram Alpha:

```Divergence Theorem (aka Gauss's Theorem and different from Gauss's Law)
-----------------------------------------------------------------------
_
The divergence of a vector field, F, is defined as:
_ _
∇.F = ∂Fx/∂x +  ∂Fy/∂y + ∂Fz/∂z --- a SCALAR

The divergence can be regarded as the spreading of a field away from center.

Consider a region bounded by a 3D surface (i.e. a sphere).  Then, the Divergence
Theorem can be stated as:
_        _ _
∫∇.F dV = ∫(F.n) dσ

where dσ is a small part of the surface area.

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. For example, imagine a lake with constant depth fed by a source.  The
water has to 'overflow' through its boundary for the lake to stay at the same
depth and keep M/V = ρ constant.

Example:  A sphere
_
∫∇.F dV = F4πR2

Let LHS = M (mass) so that,
_            _
=> F = (M/4πR2)(R/R)

We have considered a point mass but it doesn't matter how big the source is as
long as it doesn't exceed the size of the sphere. A real gravitational field
has negative divergence (convergence)```