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Difference between revisions of "Frobenius theorem on Pfaffian systems"

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A theorem on the conditions for a system of Pfaffian equations (cf. [[Pfaffian equation|Pfaffian equation]]) to be completely integrable, or (in geometrical terms) on conditions under which a given field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041800/f0418001.png" />-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|Involutive distribution]]; [[Cauchy problem|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.
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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"> 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>
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<table>
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<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>
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<TR><TD valign="top">[2]</TD> <TD valign="top"> P. Hartman, "Ordinary differential equations" , Birkhäuser (1982) {{MR|0658490}} {{ZBL|0476.34002}} </TD></TR>
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</table>
  
  
  
 
====Comments====
 
====Comments====
G. Frobenius actually also treats the [[Normal form|normal form]] of a [[Differential form|differential form]] of degree 1.
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G. Frobenius actually also treats the [[normal form]] of a [[differential form]] of degree 1.
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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=24447
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article