In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Cartan–Kähler theorem is a major result on the integrability conditions for differential systems

Integrability conditions for differential systems

In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. The idea is to take advantage of the way a differential form restricts to a...

, in the case of analytic function

Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...

s, for differential ideal

Differential ideal

In the theory of differential forms, a differential ideal I is an algebraic ideal in the ring of smooth differential forms on a smooth manifold, in other words a graded ideal in the sense of ring theory, that is further closed under exterior differentiation d...

s I. It is named for Élie Cartan

Élie Cartan

Élie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...

 and Erich Kähler

Erich Kähler

was a German mathematician with wide-ranging geometrical interests.Kähler was born in Leipzig, and studied there. He received his Ph.D. in 1928 from the University of Leipzig. He held professorial positions in Königsberg, Leipzig, Berlin and Hamburg...

.

It is not true that merely having dI contained in I is sufficient for integrability. There is a problem caused by singular solution

Singular solution

A singular solution ys of an ordinary differential equation is a solution that is singular or one for which the initial value problem fails to have a unique solution at some point on the solution. The set on which a solution is singular may be as small as a single point or as large as the full...

s. The theorem computes certain constants that must satisfy an inequality in order that there be a solution.

The Cauchy-Kovalevskaya theorem

Cauchy-Kovalevskaya theorem

In mathematics, the Cauchy–Kowalevski theorem is the main local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems...

is required, so the analyticity is necessary.