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Homework Help: Dual vector is the covariant derivative of a scalar?

  1. Apr 21, 2012 #1


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    1. The problem statement, all variables and given/known data

    In Wald's text on General Relativity he makes an assertion that I'm not sure why it is allowed mathematically. Here's the basic setup:

    Let [itex]\omega_{b}[/itex] be a dual vector, [itex]\nabla_{b}[/itex] and [itex]\tilde{\nabla}_{b}[/itex] be two covariant derivatives and [itex]f\in\mathscr{F}[/itex]. Then we may let [itex]\omega_{b}=\nabla_{b}f=\tilde{\nabla}_{b}f[/itex]

    This is in chapter 3 on curvature between equations 3.1.7 and 3.1.8...

    2. Relevant equations

    If it is relevant he is using this assertion to show:


    [itex]\nabla_{a}\nabla_{b}f=\tilde{\nabla}_{a}\tilde{ \nabla}_{b}f-C^{c}_{ab}\nabla_{c}f[/itex]

    3. The attempt at a solution

    I know how to plug in his assertion I just don't get why the heck it is allowed? I've tried reading up on dual vector spaces but I haven't found what I'm looking for... any help would be appreciated.
  2. jcsd
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