This is a second, very, very important statement of the Fundamental Theorem:

In this expression should be read in your mind as ``over the open volume bounded by the closed surface '', and is an arbitrary vector quantity, typically a vector field like or or a vector current density such as . Note well that the right hand side you should be reading as ``the flux of the vector function out through the closed surface S''.

You might also see this written as:

where is read as ``the surface bounding the volume ''. This is slightly more compact notation, but a student can easily be confused by what appears to be a partial differential in the surface limits.

A simple consequence of the divergence theorem is:

Proof: Assume

then

so that

Since is constant and arbitrary, we can factor it out from the integral:

Since this has to be true for *any* nonzero
, we can
essentially divide out the constant and conclude that:

Q.E.D.

You should prove on your own (using exactly the same sort of reasoning) that:

There thus is one such theorem for (acting on any scalar ), (acting on any vector function ) or acting on any vector function . We can use all of these forms in integration by parts.