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Integral Curves and Flows

  • John M. Lee
Part of the Graduate Texts in Mathematics book series (GTM, volume 218)

Abstract

The primary geometric objects associated with smooth vector fields are their integral curves, which are smooth curves whose velocity at each point is equal to the value of the vector field there. The collection of all integral curves of a given vector field on a manifold determines a family of diffeomorphisms of (open subsets of) the manifold, called a flow. The main theorem of this chapter, the fundamental theorem on flows, asserts that every smooth vector field determines a unique maximal integral curve starting at each point, and the collection of all such integral curves determines a unique maximal flow. After proving the fundamental theorem, we show how “flowing out” from initial submanifolds along vector fields can be used to create useful parametrizations of larger submanifolds. We then introduce the Lie derivative, which is a coordinate-independent way of computing the rate of change of one vector field along the flow of another. In the last section, we apply flows to the study of first-order partial differential equations.

Keywords

Vector Field Cauchy Problem Smooth Manifold Integral Curve Infinitesimal Generator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

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    Boyce, William E., DiPrima, Richard C.: Elementary Differential Equations and Boundary Value Problems, 9th edn. Wiley, New York (2009) Google Scholar
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    Evans, Lawrence C.: Partial Differential Equations. Am. Math. Soc., Providence (1998) Google Scholar
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    Folland, Gerald B.: Introduction to Partial Differential Equations, 2nd edn. Princeton University Press, Princeton (1995) Google Scholar
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    John, Fritz: Partial Differential Equations. Applied Mathematical Sciences, vol. 1, 4th edn. Springer, New York (1991) Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • John M. Lee
    • 1
  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

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