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.

History[edit]

It is named for Élie Cartan and Erich Kähler.

Meaning[edit]

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.

References[edit]

  • 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]