The General Theory for One First-Order Equation
In contrast to ordinary differential equations, there is no unified theory of partial differential equations. Some equations have their own theories, while others have no theory at all. The reason for this complexity is a more complicated geometry. In the case of an ordinary differential equation a locally integrable vector field (that is, one having integral curves) is defined on a manifold. For a partial differential equation a subspace of the tangent space of dimension greater than 1 is defined at each point of the manifold. As is known, even a field of two-dimensional planes in three-dimensional space is in general not integrable.
KeywordsTangent Plane Contact Structure Regular Point Quasilinear Equation Integral Manifold
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