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:
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:
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.