Talk:Darboux's theorem

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Mathematics rating:	Start Class	Mid Priority	Field: Geometry
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Geometry guy 20:48, 1 June 2007 (UTC)

[edit] 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)

[edit] 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)