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Tips/Methods for differentiating indexed equations

  1. Dec 10, 2011 #1
    Hi,
    I'm not quite sure how to differentiate indexed equations in a quick way, and I'm not sure that the way I use is correct. Does anyone have any tips/methods/resources that I could use to do these kind of operations.

    By indexed equations I mean equations like
    [itex]\frac{\partial F^i}{\partial v_j}[/itex]
    and similar.

    Below I have an example of a problem that requires this

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

    Using the Lagrangian
    [itex]L=-\frac{1}{4 \mu_0}F^{ab}F_{ab}[/itex]

    derive Maxwell's equations

    2. Relevant equations

    [itex]F^{ab}=\partial^a A^b - \partial^b A^a[/itex]
    A is electromagnetic 4-potential
    Plus the Euler-Lagrange equations (E-L)

    3. The attempt at a solution
    From the E-L equations

    [itex]\partial_i \left(\frac{\partial L}{\partial (\partial_i A_j)}\right) = \frac{\partial L}{\partial A_j}[/itex]

    subbing in the definition of [itex]F^{ab}[/itex] into L gives

    [itex]L = -\frac{1}{2\mu_0}(\partial_a A_b \partial^a A^b - \partial^b A^a \partial_a A_b)[/itex]

    So how do I differentiate this expression with respect to [itex]\partial_i A_j[/itex] as required by the E-L equation?

    My method has been to re-express [itex]\partial^a A^b[/itex] as [itex]\eta^{a \alpha} \eta^{b \beta}\partial_{\alpha} A_{\beta}[/itex] where eta is the minkowski metric tensor. After doing this I continue with the derivative assuming that
    [itex]\frac{\partial}{\partial_i A_j}\left(\partial_a A_b\right) = \delta^{i}_{a} \delta^{j}_{b}[/itex]

    (Stopping here)
    Anyone have any better ways?

    Thanks
     
  2. jcsd
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