Frobenius theorem on Pfaffian systems

From Encyclopedia of Mathematics
Revision as of 17:24, 7 February 2011 by (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

A theorem on the conditions for a system of Pfaffian equations (cf. Pfaffian equation) to be completely integrable, or (in geometrical terms) on conditions under which a given field of -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.


[1] G. Frobenius, "Ueber das Pfaffsche Problem" J. Reine Angew. Math. , 82 (1877) pp. 230–315
[2] P. Hartman, "Ordinary differential equations" , Birkhäuser (1982)


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:
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article