Talk:Darboux's theorem
WikiProject Mathematics  (Rated Startclass, Midimportance)  


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 welldefined. siℓℓy rabbit (talk) 22:44, 20 September 2008 (UTC)
Clarify
The article claims
Suppose that θ is a differential 1form on an n dimensional manifold, such that dθ has constant rank p
but the exterior derivative of a kform 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 1form, and dθ is a 2form. The usage of "rank" is unclear, however  sometimes a pform 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)
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