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Covariant Derivative derivation.

  1. Aug 20, 2012 #1
    1. The problem statement, all variables and given/known data
    Using the Leibniz rule and:
    [tex] \nabla_{c}X^{a}=\partial_{c}X^a+\Gamma_{bc}^{a}X^b [/tex]
    [tex] \nabla_{a}\Phi=\partial\Phi [/tex]

    Show that [itex] \nabla_c X_a = \partial_c X_a - \Gamma^{b}_{ac}X_{b} [/itex].
    The question is from Ray's Introducing Einsteins relativity,

    My attempt:
    [tex] \nabla_c(X^aX_a)=\nabla_c(X^a)X_a+X^a\nabla_c(X_a) [/tex]
    [tex] = (\partial_{c}X^a+\Gamma_{bc}^{a}X^b)X_a+X^a\nabla_c(X_a) [/tex]

    From here I'm not sure how to introduce the scaler field phi, or how doing so would help. Cheers for any help!
     
  2. jcsd
  3. Aug 20, 2012 #2
    You already introduced it!
     
  4. Aug 20, 2012 #3
    Of course! The left hand side is exactly the scaler field I need. All comes out nicely after that! Thank you sir,
     
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