Talk:Darboux's theorem

From Wikipedia, the free encyclopedia
WikiProject Mathematics (Rated Start-class, Mid-importance)
WikiProject Mathematics
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
Start Class
Mid Importance
 Field:  Geometry

Reference needed

I'd like to see a reference for this theorem. Planetmath does not count. Ideally, you should find (1) a modern treatment of this theorem (I have one somewhere, but need to dig it up), and (2) the original Darboux paper addressing this theorem. Silly rabbit 18:31, 3 May 2007 (UTC)

Ok, I've more or less handled it myself. Silly rabbit 19:25, 3 May 2007 (UTC)

Some comments

I'm making a few changes to give a more accurate reflection of the Darboux theorem:

  • First, it isn't a theorem in symplectic topology any more that the chain rule is a theorem in differential geometry. Rather it's unambiguously the foundation of symplectic topology. (I think symplectic topologists would feel a bit underappreciated and misunderstood otherwise.)
  • Secondly, the statement should be for rank p forms. I guess this is a forgivable omission, since Darboux doesn't get around to stating the general version until much later in the paper.
  • Third, I think it's important to point out some of the applications of this theorem to other areas of mathematics. (Ok, Hamiltonian systems is kind of a dead giveaway.) But it's also, if I recall correctly, a foundational result in the theory of Lie groups. Silly rabbit 21:36, 4 May 2007 (UTC)
  • Question: what is dx_(p+1)? if there are only coordinates x_(1,...,p)? —Preceding unsigned comment added by 132.248.196.4 (talk) 21:38, 20 September 2008 (UTC)
The situation you describe will never occur. The rank p is always ≤n/2 and the coordinates are (). So unless n=2p (which is the symplectic case), is certainly going to be well-defined. siℓℓy rabbit (talk) 22:44, 20 September 2008 (UTC)

Clarify

The article claims

Suppose that θ is a differential 1-form on an n dimensional manifold, such that dθ has constant rank p

but the exterior derivative of a k-form is a (k+1)-form, so this statement can not be consistent. What is the correct rank for θ? --Leo C Stein (talk) 01:43, 28 January 2011 (UTC)

From the equations, it's clear that θ is indeed a 1-form, and dθ is a 2-form. The usage of "rank" is unclear, however -- sometimes a p-form is called rank p. That is not the usage here. What is meant by rank here? Leo C Stein (talk) 17:11, 30 January 2011 (UTC)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Talk:Darboux%27s_theorem&oldid=410990364"
This content was retrieved from Wikipedia : http://en.wikipedia.org/wiki/Talk:Darboux's_theorem
This page is based on the copyrighted Wikipedia article "Talk:Darboux's theorem"; it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License (CC-BY-SA). You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA