(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I'm asked to get Maxwell's equations using the Euler-lagrange equation:

[tex]\partial\left(\frac{\partial L}{\partial\left\partial_{\mu}A_{\nu}\right)}\right)-\frac{\partial L}{\partial A_{\nu}}=0[/tex]

with the EM Langrangian density:

[tex]L=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-j_{\mu}A^{\mu}[/tex]

where the electromagnetic field tensor is:

[tex]F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}[/tex]

3. The attempt at a solution

I'm able to multiply out the density with the full form of the tensor F to get:

[tex]\frac{1}{4}F_{\mu\nu}F^{\mu\nu}=\frac{1}{2}\partial_{\mu}A_{\nu}\left(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}\right)=\frac{1}{2}\left(\partial_{\mu}A_{\nu}F^{\mu\nu}\right)[/tex]

My problem is that I know that the derivative w.r.t the scalar potential A for the 1st term in the density is zero as it only contains derivatives. i.e

[tex]\frac{\partial }{\partial A_{\mu}}\left(\partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}-\partial_{\mu}A_{\nu}\partial^{\nu}A^{\mu}\right)=0[/tex]

But I'm unable to show it explicity

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# Homework Help: Euler-Lagrange 2nd derivative

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