Editing Darboux's theorem (section)

==Statement==
Suppose that <math>\theta </math> is a differential 1-form on an ''<math>n </math>''-dimensional manifold, such that <math>\mathrm{d} \theta </math> has constant [[Exterior algebra#Rank of a k-vector|rank]] ''<math>p </math>''. Then

* if <math> \theta \wedge \left(\mathrm{d}\theta\right)^p = 0 </math> everywhere, then there is a local system of coordinates <math> (x_1,\ldots,x_{n-p},y_1,\ldots, y_p) </math> in which <math display="block"> \theta=x_1\,\mathrm{d}y_1+\ldots + x_p\,\mathrm{d}y_p; </math>
* if <math> \theta \wedge \left( \mathrm{d} \theta \right)^p \ne 0 </math> everywhere, then there is a local system of coordinates <math> (x_1,\ldots,x_{n-p},y_1,\ldots, y_p) </math> in which<math display="block"> \theta=x_1\,\mathrm{d}y_1+\ldots + x_p\,\mathrm{d}y_p + \mathrm{d}x_{p+1}.</math>

Darboux's original proof used [[Mathematical induction|induction]] on ''<math>p </math>'' and it can be equivalently presented in terms of [[Distribution (differential geometry)|distributions]]<ref>{{Cite book |last=Sternberg |first=Shlomo |url=https://archive.org/details/lecturesondiffer0000ster |title=Lectures on Differential Geometry |publisher=[[Prentice Hall]] |year=1964 |isbn=9780828403160 |pages=140-141 |author-link=Shlomo Sternberg}}</ref> or of [[Differential ideal|differential ideals]].<ref name=":0">{{Cite journal |last=Bryant |first=Robert L. |author-link=Robert Bryant (mathematician) |last2=Chern |first2=S. S. |author-link2=Shiing-Shen Chern |last3=Gardner |first3=Robert B. |author-link3=Robert Brown Gardner |last4=Goldschmidt |first4=Hubert L. |last5=Griffiths |first5=P. A. |author-link5=Phillip Griffiths |date=1991 |title=Exterior Differential Systems |url=https://doi.org/10.1007/978-1-4613-9714-4 |journal=Mathematical Sciences Research Institute Publications |language=en |doi=10.1007/978-1-4613-9714-4 |issn=0940-4740|url-access=subscription }}</ref>

=== Frobenius' theorem ===
Darboux's theorem for ''<math>p=0 </math>'' ensures that any 1-form ''<math>\theta \neq 0 </math>'' such that ''<math>\theta \wedge d\theta = 0 </math>'' can be written as ''<math>\theta = dx_1 </math>'' in some coordinate system <math> (x_1,\ldots,x_n) </math>.

This recovers one of the formulation of [[Frobenius theorem (differential topology)|Frobenius theorem]] in terms of differential forms: if <math> \mathcal{I} \subset \Omega^*(M) </math> is the differential ideal generated by <math> \theta </math>, then ''<math>\theta \wedge d\theta = 0 </math>'' implies the existence of a coordinate system <math> (x_1,\ldots,x_n) </math> where <math> \mathcal{I} \subset \Omega^*(M) </math> is actually generated by <math> d x_1 </math>.<ref name=":0" />