Editing Darboux's theorem (section)

{{short description|Foundational result in symplectic geometry}}
{{About|Darboux's theorem in differential geometry|Darboux's theorem related to the intermediate value theorem|Darboux's theorem (analysis)}}

In [[differential geometry]], a field in [[mathematics]], '''Darboux's theorem''' is a [[theorem]] providing a normal form for special classes of [[differential forms|differential 1-forms]], partially generalizing the [[Frobenius integration theorem]]. It is named after [[Jean Gaston Darboux]]<ref>{{cite journal |last=Darboux |first=Gaston |author-link=Jean Gaston Darboux |year=1882 |title=Sur le problème de Pfaff |trans-title=On the Pfaff's problem |url=http://gallica.bnf.fr/ark:/12148/bpt6k68005v |journal=Bull. Sci. Math. |language=fr |volume=6 |pages=14–36, 49–68 |jfm=05.0196.01}}</ref> who established it as the solution of the [[Johann Friedrich Pfaff|Pfaff]] problem.<ref>{{cite journal |last=Pfaff |first=Johann Friedrich |author-link=Johann Friedrich Pfaff |year=1814–1815 |title=Methodus generalis, aequationes differentiarum partialium nec non aequationes differentiales vulgates, ultrasque primi ordinis, inter quotcunque variables, complete integrandi |trans-title=A general method to completely integrate partial differential equations, as well as ordinary differential equations, of order higher than one, with any number of variables |url=https://archive.org/details/abhandlungenderp14akad/page/76/mode/1up?view=theater |journal=Abhandlungen der Königlichen Akademie der Wissenschaften in Berlin |language=la |pages=76–136}}</ref>

It is a foundational result in several fields, the chief among them being [[symplectic geometry]]. Indeed, one of its many consequences is that any two [[symplectic manifold]]s of the same dimension are locally [[symplectomorphism|symplectomorphic]] to one another. That is, every <math>2n </math>-dimensional symplectic manifold can be made to look locally like the [[linear symplectic space]] <math>\mathbb{C}^n </math> with its canonical symplectic form.

There is also an analogous consequence of the theorem applied to [[contact geometry]].