Cartan–Kähler theorem

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

It is named for Élie Cartan and Erich Kähler.

Meaning

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

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

References

• 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

• Alekseevskii, D.V. (2001), "Pfaffian problem", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• E. Cartan, "On the integration of systems of total differential equations," transl. by D. H. Delphenich
• E. Kähler, "Introduction to the theory of systems of differential equations," transl. by D. H. Delphenich