|
||
|
||
|
The divergence theorem is the form of the fundamental theorem of calculus
that applies when we integrate the divergence of a vector v over a region
R of space. As in the case of Green's or Stokes' theorem, applying the one dimensional
theorem expels one of the three variables of integration to the boundaries,
and the result is a surface integral over the boundary of R, which is directed
normally away from R.
The one dimensional fundamental theorem in effect converts thev
in the integrand to an n
v
on the boundary, where n is the outward directed unit vector
normal to it.
Another way to say the same thing is: the flux integral of v over a bounding
surface is the integral of its divergence over the interior.
Divergence Theorem: For R a reasonably nice region in whichv
exists, we have
where the normal is taken to face out of R everywhere on its boundary, R.