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". In other words, for any form α in "I", the exterior derivative "d"α is also in "I".
In the theory of differential algebra, a differential ideal "I" in a differential ring "R" is an ideal which is mapped to itself by each differential operator.
Exterior differential systems and partial differential equations.
An exterior differential system consists of a smooth manifold formula_1 and a differential ideal
An integral manifold of an exterior differential system formula_3 consists of a submanifold formula_4 having the property that the pullback to formula_5 of all differential forms contained in formula_6 vanishes identically.
One can express any partial differential equation system as an exterior differential system with independence condition. 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. We can frequently apply Cartan's equivalence method to exterior differential systems to study their symmetries and their diffeomorphism invariants.
Perfect differential ideals.
A differential ideal formula_14 is perfect if it has the property that if it contains an element formula_15 they contain any element formula_16 such that formula_17 for some formula_18.