Undergrad Coleman Lecture: Varying E-M Lagrangian - Problem 3.1 Explained

[Pnin]

This is from Coleman Lectures on Relativity, p.63. I understand that he uses integration by parts, but just can't see how he gets to the second equation. (In problem 3.1 he suggest to take a particular entry in 3.1 to make that more obvious, but that does not help me.)

 

[vanhees71]

You can simplify the task a bit by writing
$$\delta \mathcal{L}=-\frac{1}{4} \delta (F_{\mu \nu} F^{\mu \nu}) = -\frac{1}{2} \delta F_{\mu \nu} F^{\mu \nu}=-\delta (\partial_{\mu} A_{\nu}) F^{\mu \nu}.$$
Then you have, indeed via partial integration)
$$\delta I = -\int \mathrm{d}^4 x \partial_{\mu} \delta A_{\nu} F^{\mu \nu} = + \int \mathrm{d}^4 x \delta A_{\nu} \partial_{\mu} F^{\mu \nu} \stackrel{!}{=}0,$$
and from this you get the free Maxwell equations
$$\partial_{\mu} F^{\mu \nu}=0, \quad F_{\mu \nu}=\partial_{\mu} A_{\nu} -\partial_{\nu} A_{\mu}$$
for the potential.