[BUG] PDT as a special case of the (n dim) Divergence Theorem

[Smith, Alexander J.]

1. The circulation-curl form of Green's theorem for a "flat pancake
region" is a special case of Stokes' Theorem for a surface in space.

2. The divergence-flux form of Green's theorem is equivalent to the
circulation-curl form. You apply the circulation-curl form of a planar
vector field to the field that you get by rotating each vector by 90
degrees, and you suddenly end up with the divergence-flux form. 

3.  So we are left wondering if the divergence theorem for 2d vector
fields is in some way a special case of the 3d divergence theorem.
Tevian suggests "no" and I concur.

BUT...is it not true that if you have a vector field F in R^n and you do
a multiple integral over an "n-cube" of the divergence of F...defined in
the obvious way, you get the same thing that you get by calculating the
flux of F out of the 2n sides of the n-cube? This statement does not
need any of the fancy set up for the abstract Stokes' theorem, i.e.,
differential forms on a manifold. All you need is an inner product in
R^n...which you can define again in the obvious way.

This n-dimensional divergence theorem feels true in view of the
"physical" proof of the divergence theorem (flow meters arranged around
the boundary, and flow meters arranged on the inside of the cube). 

Thus from this point of view, the div-flux form of Green's theorem for a
vector field in the plane and the 3d divergence theorem are special
cases of this low-tech n-dimensional statement, which an undergraduate
could believe. (Is it true?)

Alex