Cartan–Kähler theorem

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In mathematics, the Cartan–Kähler theorem is a major result on the integrability conditions for differential systems, in the case of analytic functions, for differential ideals I.


It is named for Élie Cartan and Erich Kähler.


It is not true that merely having dI contained in I is sufficient for integrability. There is a problem caused by singular solutions. The theorem computes certain constants that must satisfy an inequality in order that there be a solution.

Proof and assumptions[edit]

The Cauchy-Kovalevskaya theorem is required, so the analyticity is necessary.


  • Jean Dieudonné, Eléments d'analyse, vol. 4, (1977) Chapt. XVIII.13
  • R. Bryant, S. S. Chern, R. Gardner, H. Goldschmidt, P. Griffiths, Exterior Differential Systems, Springer Verlag, New York, 1991.

External links[edit]