The General Theory for One First-Order Equation

  • Vladimir I. Arnold
Part of the Universitext book series (UTX)


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.


Tangent Plane Contact Structure Regular Point Quasilinear Equation Integral Manifold 
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  1. 1.
    Arnold, V.I.: Ordinary Differential Equations. Springer, Berlin, Chap. 2, §11 (1992)Google Scholar
  2. 2.
    Arnold, V.I.: Geometrical Methods in the Theory of Ordinary Differential Equations, 2nd edition. Springer, New York, Chap. 2 (1988)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Vladimir I. Arnold
    • 1
    • 2
  1. 1.Steklov Mathematical InstituteMoscowRussia
  2. 2.CEREMADEUniversité de Paris-DauphineParis Cedex 16France

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