From: Sascha.Unzicker@lrz.uni-muenchen.de
Newsgroups: sci.math
Subject: Re: De Rham Theorem : applications to physics
Date: 26 May 1998 14:41:57 GMT
mo31g111 writes:
>Hi everybody,
> I'm doing a memoir on De Rham Theorem. I would like to study some of
>its applications to physics. If somebody knows such applications it
>would be very nice of him to communicate it.
> By advance thanks you very much.
Hi, this old posting of mine may be interesting:
Cohomology tells you something about the connectedness of a manifold.
You have to know the difference between a *closed* and *exact*
differential form.
A form w is closed, if its exterior derivative vanishes: d w=0.
A form w is exact, if its the derivative of another form: d v =w.
A exact form is always closed, since dd v=0 is valid for all forms
(this contains Div Curl=0, Curl Grad=0, etc).
But the contrary is not true, and cohomology measures the 'failure'
of the statement closed-> exact.
An example :
Take the following 1-form w (vector field) in R2\{0}:
(-y, x)/(x^2+y^2)
w is closed, d w=0 (Curl w=0 in ordinary vector analysis), but
its not exact, because there is no global potential v with d v=w!
According to a theorem of de Rham, this analytical fact is
related to the (topological!) property that R2\{0} is double
connected.
You can now ask: how meany independent 1-forms (like the above one)
(or p-forms)exist, that are closed, but not exact? This number is the
so-called 1st (or pth) betti number, a topological invariant
that you can calculate from simplicial complexes or CW-complexes.
The alternating sum of betti numbers is the Euler characteristic.
The cohomolgy groups form with the exterior product a cohomology ring.
Literature:
1)Samuel Goldberg, Curvature and Homology (Dover reprint 1962)
2)Bishop & Goldberg, Tensor analysis on manifolds (Dover reprint)
3)M.Nakahara, Geometry, Topology and Physics (Inst. of Phys. publ, 1995)
chap. 6.
Alexander
http://www.lrz-muenchen.de/~u7f01bf/WWW/dg.html