# Tips/Methods for differentiating indexed equations

1. Dec 10, 2011

### thepopasmurf

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
$\frac{\partial F^i}{\partial v_j}$
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
$L=-\frac{1}{4 \mu_0}F^{ab}F_{ab}$

derive Maxwell's equations

2. Relevant equations

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

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

$\partial_i \left(\frac{\partial L}{\partial (\partial_i A_j)}\right) = \frac{\partial L}{\partial A_j}$

subbing in the definition of $F^{ab}$ into L gives

$L = -\frac{1}{2\mu_0}(\partial_a A_b \partial^a A^b - \partial^b A^a \partial_a A_b)$

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

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

(Stopping here)
Anyone have any better ways?

Thanks