Differential system

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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