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Lie Derivatives

  • Stephen Bruce Sontz
Part of the Universitext book series (UTX)

Abstract

Lie derivatives are discussed in this brief chapter, which I have included in part because it is so traditional. There are two more concrete reasons for this diversion. The first is that Lie derivatives offer some sort of introduction to the idea behind the Frobenius theorem. The second is that they give us an inadequate way of transporting vectors along curves. Why inadequate? This is a technicality, difficult to describe for now. But what we really need to transport vectors in a “parallel” manner is a connection. This is what a connection will do for us. I also discuss why integral curves are not always so important in physics. But be warned that my point of view here is rather heretical.

Bibliography

  1. 4.
    Y. Choquet-Bruhat, C. DeWitt-Morette, and M. Dillard-Bleick, Analysis, Manifolds and Physics, North-Holland Pub. Co., revised edition, 1982.Google Scholar
  2. 32.
    S. Lang, Fundamentals of Differential Geometry, Graduate Texts in Mathematics, Vol. 191, Springer, 1999.Google Scholar
  3. 33.
    J. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, Vol. 218, Springer, 2003.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Stephen Bruce Sontz
    • 1
  1. 1.Centro de Investigación en Matemáticas, A.C.GuanajuatoMexico

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