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

[Matthias Kawski]

On Fri, 30 Apr 2004, Smith, Alexander J. wrote:

> .... This statement does not need any of the fancy set up for
> the abstract Stokes' theorem, i.e., forms on a manifold. All you
> need is an inner product in R^n...which you can define again in
> the obvious way.

I believe it is a very long way from the usual first encounter
with line integrals in calculus to any reasonably deep under-
standing. But just because we may not want to tell our calculus
studenst the whole truth, does not mean that we should not do
the very best to understand it. It was many years after Ph.D.
that I got to a level where I am reasonably happy with it.

I believe the following are a few key observations:

 * In line and surface integrals do not integrate (tangent)
   vector fields, but differential forms (that is, cotangent
   vector fields).

 * Line and surface integrals do NOT involve any inner products.
   (Even Stewart gets this right -- but it is very painful there.
   I prefer to use dot-products when convenient, but keep in mind
   that the integral theorems are solidly on the diff form side.)

 * However, if a metric is present (e.g. same units along the
   axes), then the metric can be used to convert tangent into
   cotangent vector fields (column into row vectors) and vice
   versa. (In linear algebra v --> v^T or v --> v^T*G=<v^TG,.>)

 * Some of the most powerful applications of the integral thms
   of vector calculus are not in physical 3-space, but e.g. in
   thermodymanics. The best known is the conversion of heat into
   mechanical energy via e.g. the Carnot-cyle. Green's theorem
   integrates the field pdV around the contour etc. Putting
   arrows and angles into this picture is a disasetr from all
   that I know.
   We probably don't want to mention these applic's in the first
   time around, but we should be reasonably correct and forward-
   looking that the theorems we do in the first round actually
   can be generalized, and are not plain DEAD ENDS.

 * I personally found it very useful in my classes even in the
   1st round to quite casually distinguish contra and covariant
   vector fields -- it is so easy, just bring in units:
   The differential of a function, say, temperature is measured
   in degrees per inch. The velocity of the bug in inches per
   second. Now change from inches to centimeters, and one field
   is multiplied by 2.54, the other divided by 2.54. Clearly
   something is going on. Students accept this very readily,
   and it helps much as an organizing principle: fields that
   arise as derivatives (gradient fields) should be irrotaional
   (no spinning), whereas in velocity fields (e.g fluid flow)
   the first look is whether they are divergence free (neither
   expanding nor contracting).
   At least according to AbrahamMarsden the gradient is the
   "transpose" of the differential ... this can cause trouble
   w/ units... don't go too far in the first round. What matters
   is that the default plan for gradient fields is to evaluate
   circulation/work/line integrals. The default plan for fluid
   velocity fields is to look for flux across a curve / surface.

   This situation gets quite a bit harder when looking at fields
   that mix both properties -- most typically the Liouville
   "vector field" representation of complex analytic functions
   (analyticity <--> Cauchy-Schwarz-equations < --> curl and
   divergence free). 3D is even more fun -- and this is where
   E and M fields are. The problem with them is that they have
   TOO MUCH symmetry, i.e. they lack some of the distinctive
   features that are give-awayas for what is to be done with
   other fields.
   This is why I hold off with fully 3-D E-M fields until late.

   This is closely related to the most fundamental fact that
   d^2=0 (curl-test, divergence-test). It is very natural to
   take a curl of a gradient, and to take a div of a curl,
   but unless in dim 2 where all is same, it is strange to
   look at the div of a agrad or at the curl fo a curl.
   They do occur (e.g. the Laplacian is very important),
   but they always involve a Hodge-star operator, which
   converts a k-form into an n-k form. I prefer to develop
   most intuition in settings where this operator does not
   come in. (Easiest write-up[ is in old version of Marsden/
   Tromba in the appendix< or advanced in Flanders).
   Just on the side recall that Hodge star makes sense only
   in Riemannina settings, i.e. when a metric is given, or when
   the coordnate axes are labels w/ the same (or comparable
   units).

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

   I understand that Tevian disagrees -- but when I see my fluid
   mechanics coallgues play w/ Navier Stokes etc. i see all the
   time models for 3D flows that have a translational symmetry,
   and very often are analyzed via cross-sections. Thus the flow
   across a surface in 3D becomes the flow of a 2D field across
   a curve (w/ trivial product in perpendicular direction).
   In class we talk about flood irrigation -- 1/2 a foot deep
   flow across a few acres. The vertical components and variation
   of the fluid flow are negligible. In EM, consider the fields
   along a wire carrying a current -- they are 3D, but all
   analysis takes place in a plane perpendicular to the wire
   (times a trivial height).

   The absolutely crucial difference is that when considering
   2D- fields and BOTH line integrals in depths, then one can
   use visual images to devlop intuition. In 3D hardly any
   nontrivial fiedl can bve pictured by a nonexpert (w/o prior
   info on what one expects to see). I do 6 weeks of 2D and
   then a few days of 3D.

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


Matthias
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Matthias Kawski                http://math.asu.edu/~kawski
Dept. of Mathematics and Statistics         kawski at asu.edu
Arizona State University            office: (480) 965 3376
Tempe, Arizona 85287-1804           home:   (480) 893 0107
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