 Suppose that we have a "k"th order partial differential equation system for maps formula_7, given by

The graph of the formula_9-jet formula_10 of any solution of this partial differential equation system is a submanifold formula_5 of the jet space, and is an integral manifold of the contact system formula_12on the formula_9-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.