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I have a modified Lagrangian for an electromagnetic field:

$$L=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu} - \frac{\xi}{2} (\partial_\mu A^\mu)^2.$$

All the symbols and variables have their usual meanings. I need to perform integration by parts on it to get to a version of the lagrangian which looks like this:

$$L=-\frac{1}{2}\partial_{\mu}A_{\nu} \partial^{\mu}A^{\nu}+\frac{1-\xi}{2} (\partial_\mu A^\mu)^2.$$

The problem is I have absolutely no clue how to perform integration by parts on the first lagranian written here. I know what integration by parts is and how to apply it to a simple equation like $xln(x)$ but when it comes to tensors and all I do not know where to start. I would still like to try and do this by myself, but I was hoping for some hints on where to start and what variables I need to integrate etc.

Qmechanic
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Chris G
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    Hint: start from$$\begin{align}\frac12F_{\mu\nu}F^{\mu\nu}&=\partial_\mu A_\nu(\partial^\mu A^\nu-\partial^\nu A^\mu)\&\sim\partial_\mu A_\nu\partial^\mu A^\nu+A_\nu\partial^\nu\partial_\mu A^\mu\&\sim\partial_\mu A_\nu\partial^\mu A^\nu-\partial^\nu A_\nu\partial_\mu A^\mu,\end{align}$$where $\sim$ indicates equivalence up to a total derivative. – J.G. Oct 31 '23 at 16:35

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