An exterior differential system consists of a smooth manifold and a differential ideal
- .
An integral manifold of an exterior differential system consists of a submanifold having the property that the pullback to of all differential forms contained in vanishes identically.
One can express any partial differential equation system as an exterior differential system with independence condition. Suppose that we have a kth order partial differential equation system for maps , given by
- .
The graph of the -jet of any solution of this partial differential equation system is a submanifold of the jet space, and is an integral manifold of the contact system on the -jet bundle.
This idea allows one to analyze the properties of partial differential equations with methods of differential geometry. For instance, we can apply the Cartan–Kähler_theorem to a system of partial differential equations by writing down the associated exterior differential system. We can frequently apply Cartan's equivalence method to exterior differential systems to study their symmetries and their diffeomorphism invariants.