I’ve been really busy with other projects — and work — of late, but I think we can get started again. We left off right after defining hypersurface integrals, which puts us in position to prove the divergence theorem.
So, let be a hypersurface with dimension in an -manifold , and let be a vector field on some region containing , so we can define the hypersurface integral
And if corresponds to a -form , we can write this as
where is the oriented volume form of and is a -form that “points perpendicular to” in . We take the given inner product and integrate it as a function against the volume form of itself.
A little juggling lets us rewrite:
where we take our -form and flip it around to the “perpendicular” form . Integrating this over involves projecting against , which is basically what the above formula connotes.
Now, let’s say that the surface is the boundary of some -dimensional submanifold of , and that it’s outward-oriented. That is, we can write . Then our hypersurface integral looks like
Next we’ll jump over to the other end and take the divergence and integrate it over the region . In terms of the -form , this looks like
But Stokes’ theorem tells us that
which tells us in our vector field notation that
This is the divergence — or Gauss’, or Gauss–Ostrogradsky — theorem, and it’s yet another special case of Stokes’ theorem.