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Derivative of contravariant metric tensor with respect to covariant metric tensor

  1. Nov 17, 2011 #1
    1. The problem statement, all variables and given/known data
    Show that [tex]\frac{{\partial}g^c{^d}}{{\partial}g_a{_b}}=-\frac{1}{2}(g^a{^c}g^b{^d}+g^b{^c}g^a{^d})[/tex]

    2. Relevant equations


    3. The attempt at a solution
    It seems like it should be simple, but I just do not see how to come up with the above solution. This is what I am coming up with: [tex]\frac{\partial[\delta^c{_a}\delta^d{_b}(g_a{_b})^-{^1}]}{{\partial}g_a{_b}}[/tex][tex]-\delta^c{_a}\delta^d{_b}(g_a{_b})^-{^2}=-(g^c{^d})^2[/tex]
    Maybe I am not applying the Kronecker deltas correctly.
     
    Last edited: Nov 17, 2011
  2. jcsd
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