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 submanifold, and the fact that this restriction is compatible with the exterior derivative. This is one possible approach to certain over-determined systems, for example. A Pfaffian system is specified by 1-forms alone, but the theory includes other types of example of differential system.

Given a collection of differential 1-forms ${\displaystyle \textstyle \alpha _{i},i=1,2,\dots ,k}$ on an ${\displaystyle \textstyle n}$-dimensional manifold ${\displaystyle M}$, an integral manifold is a submanifold whose tangent space at every point ${\displaystyle \textstyle p\in M}$ is annihilated by each ${\displaystyle \textstyle \alpha _{i}}$.

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