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How to differentiate a term with respect to metric

  1. Apr 20, 2015 #1
    1. The problem statement, all variables and given/known data
    for example: ##\frac{\partial(F^{ab}F_{ab})}{ \partial g^{ab}} ## where F_{ab} is electromagnetic tensor.
    or ##\frac{\partial N_{a}}{\partial g^{ab}}## where ##N_{a}(x^{b}) ## is a vector field.

    2. Relevant equations


    3. The attempt at a solution
    i saw people write ##F^{ab}F_{ab}## as ## g^{ca}g^{db}F_{cd} F_{ab}## and say we write the term in a way that exposes the dependence on the metric. but exactly what it means ?
     
  2. jcsd
  3. Apr 21, 2015 #2

    Orodruin

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    You need to write your function in terms of the variables you have, in your case ##g^{ab}## and ##F_{ab}##. The contravariant ##F^{ab}## is not independent of those and so will have non zero partial derivatives wrt both. Writing it in terms of your variables lets you differentiate easier.

    As a similar example: Take ##f(x,y) = xy##. The partial derivative ##\partial_yf## is simply ##\partial_y (xy) = x##. This is essentially the same. Do not be confused by the fact that you use the same letter to denote the covariant and contravariant tensors.
     
  4. Apr 21, 2015 #3
    for example if ##N_{\mu}(x^\nu)## be unit time like dynamical four velocity, what's happening ? i don't know explicit form of ##N_{\mu}##.
     
  5. Apr 21, 2015 #4
    ok we can write for first example:##\frac {\partial (g^{ca}g^{db}F_{cd}F_{ab})}{\partial g^{ef}}## = ##\frac{\partial g^{ca}}{\partial g^{ef}}g^{db}F_{cd}F_{ab}+\frac{\partial g^{db}}{\partial g^{ef}}g^{ca}F_{cd}F_{ab}+\frac{\partial F_{cd}}{\partial g^{ef}}g^{ca}g^{db}F_{ab}+\frac{\partial F_{ab}}{\partial g^{ef}}g^{ca}g^{db}F_{cd}## . is it correct ?

    ##\frac{\partial g^{ca}}{\partial g^{ef}}=\delta^c_{e} \delta^a_{f}## is it correct ?

    what's happening for ## \frac{\partial F_{cd}}{\partial g^{ef}}## ?
     
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