Differential Ideal - Exterior Differential Systems and Partial Differential Equations

Exterior Differential Systems and Partial Differential Equations

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

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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, given by

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The solution of this partial differential equation system is the submanifold of the jet space consisting of integral manifolds of the pullback of the contact system to .

This idea allows one to analyze the properties of partial differential equations with methods of differential geometry. For instance, we can apply Cartan's method on partial differential equation systems by writing down the exterior differential system associated with it.

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