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## Homework Statement

$$ L = -\frac{1}{2} (\partial_{\mu} A_v) (\partial^{\mu} A^v) + \frac{1}{2} (\partial_{\mu} A^v)^2$$

calculate $$\frac{\partial L}{\partial(\partial_{\mu} A_v)}$$

## Homework Equations

$$ A^{\mu} = \eta^{\mu v} A_v, \ and \ \partial^{\mu} = \eta^{\mu v} \partial_{v}$$

## The Attempt at a Solution

rewrite equation as $$ L = - \frac{1}{2} \eta^{\mu a} \eta^{v b} (\partial_{\mu} A_v) (\partial_{a} A_b) + \frac{1}{2} \eta^{v b} \eta^{v b} (\partial_{\mu} A_b) (\partial_{\mu} A_b )$$

now taking $$\frac{\partial L}{\partial(\partial_{\mu} A_v)}$$

we get $$ - \frac{1}{2} ( \eta^{\mu a} \eta^{v b} \partial_{a} A_b + \eta^{\mu a} \eta^{v b} (\partial_{\mu} A_v) \delta^{\mu v}_{a b} ) + \frac{1}{2} \eta^{v b} \eta^{v b}( \delta_{b}^{v} \partial_{\mu} A_b + \partial_{\mu} A_b \delta_{b}^{v}) $$

This is what I've got so far. I'm pretty certain I've made a number of mistakes here. On the left hand side I've got the mu's and v's repeating 3 times which probably isn't a good sign and on the right hand side the indices of the delta don't seem to be consistent.

The answer is

$$ \frac{\partial L}{\partial(\partial_{\mu} A_v)} = - \partial^{\mu} A^v + (\partial_{\rho} A^{\rho} ) \eta^{\mu v} $$

Any help would be appreciated.