词条 | Cartan–Kähler theorem |
释义 |
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 . HistoryIt is named for Élie Cartan and Erich Kähler. MeaningIt is not true that merely having contained in 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. Statement of the theoremLet be a real analytic EDS. Assume that is a connected, -dimensional, real analytic, regular integral manifold of with (i.e., the tangent spaces are "extendable" to higher dimensional integral elements). Moreover, assume there is a real analytic submanifold of codimension containing and such that has dimension for all . Then there exists a (locally) unique connected, -dimensional, real analytic integral manifold of that satisfies . Proof and assumptionsThe Cauchy-Kovalevskaya theorem is used in the proof, so the analyticity is necessary. References
External links
2 : Partial differential equations|Theorems in analysis |
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