Exterior differential system

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 dα 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.

An exterior differential system on a manifold M is a differential ideal

One can express any partial differential equation system as an exterior differential system with independence condition. Say that we have kth order partial differential equation systems for maps ${\displaystyle f:\mathbb {R} ^{m}\rightarrow \mathbb {R} ^{n}}$, given by

The solution of this partial differential equation system is the submanifold ${\displaystyle \Sigma }$ of the jet space consisting of integral manifolds of the pullback of the contact system to ${\displaystyle \Sigma }$.

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