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Defition of lie derivatives

  1. Aug 22, 2010 #1
    suppose there is a vector field V on a manifold M

    V generates a flow on M

    suppose \gamma(t) is an integral curce in this flow

    now there is another vector field W on M

    why not define the lie derivative of W with respect to V as the limit of the divide

    (W(\gamma+\delta \gamma)-W(\gamma))/\delta \gamma

    here the difference is taken by components

    i think this is very natural from our experience in the calculus course in undergraduate.

    why we need to drag?
  2. jcsd
  3. Aug 23, 2010 #2


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    The difference W(\gamma+\delta \gamma)-W(\gamma)) does not make sense since W(\gamma+\delta \gamma) and W(\gamma)) are elements of different vector spaces.
  4. Aug 24, 2010 #3


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    The expression [itex]\gamma+\delta\gamma[/itex] (which I assume means [itex]\gamma(t)+\delta\gamma(t)[/itex]) doesn't make sense either, since a manifold doesn't necessarily have a vector space structure. For most manifolds, both the + and the multiplication in [itex]\delta\gamma(t)[/itex] are undefined.
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