Difference between revisions of "Frobenius theorem on Pfaffian systems"
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| − | A theorem on the conditions for a system of | + | A theorem on the conditions for a system of [[Pfaffian equation]]s to be completely integrable, or (in geometrical terms) on conditions under which a given field of $n$-dimensional tangent subspaces on a differentiable manifold is the tangent field of some [[Foliation|foliation]]. For several equivalent formulations of the Frobenius theorem, see the articles [[Involutive distribution]]; [[Cauchy problem]]; for a version with minimum differentiability requirements see [[#References|[2]]]. The name of the theorem is connected with the account of it in [[#References|[1]]], but does not accord with the information given there about its history. |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> G. Frobenius, "Ueber das Pfaffsche Problem" ''J. Reine Angew. Math.'' , '''82''' (1877) pp. 230–315</TD></TR> | ||
| + | <TR><TD valign="top">[2]</TD> <TD valign="top"> P. Hartman, "Ordinary differential equations" , Birkhäuser (1982) {{MR|0658490}} {{ZBL|0476.34002}} </TD></TR> | ||
| + | </table> | ||
====Comments==== | ====Comments==== | ||
| − | G. Frobenius actually also treats the [[ | + | G. Frobenius actually also treats the [[normal form]] of a [[differential form]] of degree 1. |
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| + | {{TEX|done}} | ||
Latest revision as of 19:10, 24 September 2017
A theorem on the conditions for a system of Pfaffian equations to be completely integrable, or (in geometrical terms) on conditions under which a given field of $n$-dimensional tangent subspaces on a differentiable manifold is the tangent field of some foliation. For several equivalent formulations of the Frobenius theorem, see the articles Involutive distribution; Cauchy problem; for a version with minimum differentiability requirements see [2]. The name of the theorem is connected with the account of it in [1], but does not accord with the information given there about its history.
References
| [1] | G. Frobenius, "Ueber das Pfaffsche Problem" J. Reine Angew. Math. , 82 (1877) pp. 230–315 |
| [2] | P. Hartman, "Ordinary differential equations" , Birkhäuser (1982) MR0658490 Zbl 0476.34002 |
Comments
G. Frobenius actually also treats the normal form of a differential form of degree 1.
How to Cite This Entry:
Frobenius theorem on Pfaffian systems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Frobenius_theorem_on_Pfaffian_systems&oldid=18039
Frobenius theorem on Pfaffian systems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Frobenius_theorem_on_Pfaffian_systems&oldid=18039
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article