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## Homework Statement

Hi, I have a problem calculating the variation of the action using tensor algebra because two derivative indices are causing some problem.

## Homework Equations

Generally you have the action [itex]S = \int L(A^{\mu}, A^{\mu}_{\;,\nu}, x^{\mu})d^4x [/itex]

where:

[itex] A ^{\mu}= A^{\mu}(x^{\nu}) [/itex]

[itex] A ^{\mu}_{\;,\nu} = \frac{\partial A^{\mu}}{\partial x^{\nu}} [/itex]

[itex] x^{\nu} = (x^0, x^1, x^2 ,x^3) [/itex]

[itex] d^{4}x = dx^0 dx^1dx^2dx^3 [/itex]

Would I be correct in stating that the variation of the action is [itex] \delta S = \int ( \frac{\partial L}{\partial A^{\mu} } \delta A^{\mu} + \frac{\partial L}{\partial A ^{\mu}_{\;,\nu} } \delta A ^{\mu}_{\;,\nu} ) d^{4} x [/itex] ?

## The Attempt at a Solution

Say that our function L looks like this:

[itex] L = A_{\mu, \nu}-A_{\nu, \mu} [/itex]

where [itex] A_{\mu} = \eta _{\mu \nu} A^{\nu} [/itex] and [itex] \eta_{\mu \nu} [/itex] is the Minkowski metric tensor.

How do I make sense of this in context of the variation [itex]\delta S[/itex] above? More specifically what should the derivative index of the variation [itex] \delta A^{\mu}[/itex] be? Because L is the [itex]\nu[/itex] derivative of [itex]A_{\mu}[/itex] minus [itex]A_{\nu}[/itex] derivated with respect to the [itex]\mu[/itex] derivative.

Specifically what I want to accomplish is to rewrite the right hand side of [itex]\delta S[/itex] as the sum of two parts that looks something like this

[itex]\frac{\partial L}{\partial A_{\mu , \nu} } \delta A_{\mu, \nu}=\frac{\partial }{\partial x^{\nu}} (\frac{\partial L}{\partial A_{\mu,\nu}} \delta A_{\mu})-\delta A_{\mu} \frac{\partial }{\partial x^{\nu}}( \frac{\partial L}{\partial A_{\mu,\nu}}) [/itex] but for me to be able to do that I need a common derivative index in L which I don't have. I have two separate derivative indexes and I have not idea what to do with them. Thank you for your help.

I know the equations are physically nonsensical since I have removed all the clutter beside the actual problem, so this is mainly a mathematical question. I could not decide whether it should be in the Physics section or Math section. If a moderator think it should be moved somewhere else please feel free to do it there or tell me and I'll do it later.

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