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

[Tevian Dray]

>>>>> Alexander Smith writes:

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

Yes, so long as you interpret "special case" as in the 2d/3d discussion below.

>>>>> Matthias Kawski writes:

    MK> Re the divergence form of Green's theorem in the plane -- it
    MK> is a straight-forward analogy to the 3 D divergence theorem
    MK> (much closer than the circulation formulaiton of Green's thm
    MK> is to the special case of Stokes for surfaces in 3D).

I think this depends on what you mean by "straightforward".  Stokes' Theorem
really does follow from Green's Theorem "analytically"; the divergence
theorems are related to each other "by analogy".  But Matthias goes on to
outline a more rigorous view:

    MK> Thus the flow across a surface in 3D becomes the flow of a
    MK> 2D field across a curve (w/ trivial product in perpendicular
    MK> direction). In class we talk about flood irrigation -- 1/2 a
    MK> foot deep flow across a few acres. The vertical components
    MK> and variation of the fluid flow are negligible. In EM, consider
    MK> the fields along a wire carrying a current -- they are 3D, but
    MK> all analysis takes place in a plane perpendicular to the wire
    MK> (times a trivial height).

In fact, I like this approach -- which is also what Hughes Hallett et al do.
But this amounts to saying that divergence only makes sense in 3d, although
of course in some cases the third dimension doesn't matter.

    MK> Re distinguishing all the different kinds of fields, the best
    MK> that I know is Gabriel Weinreich's little book on "Geometric
    MK> Vectors" (or similau). U of Chicago Press. about $20. small
    MK> paperback. As I recall he is a physicist who tried to teach
    MK> us math folks who simple it is to organize all these fields.

This book goes a bit overboard, distinguishing some types of fields which are
really the same.  But lots of fun nonetheless.  I reviewed it for the American
Journal of Physics (AJP 67, 553-554 (1999).

Tevian