I Maxwell's equations with differential forms

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Hello!

I was not quite sure about posting in this category, but I think my question fits here.

I am wondering about Maxwell equations in vacuum written with differential forms, namely:
\begin{equation} \label{pippo}
dF = 0 \qquad d \star F = 0
\end{equation}
I know ##F## is a 2-form, and It can be written by using the 1-form ## F = d A##. I am not now interested in the derivation (even if the first is straightforward), but I want to see the equivalence with the Maxwell equation written in a covariant way:
\begin{equation} \label{pluto}
\varepsilon^{\mu \nu \rho \sigma} \partial_\nu F_{\rho \sigma} = 0 \qquad \partial^\mu F_{\mu \nu} = 0
\end{equation}

If I explicit everything, knowing the components of ##F_{\mu \nu}##, I can recover both \ref{pluto} equations starting from \ref{pippo}. I'm interested in a more "direct" derivation, but I cannot find any reference in textbooks.
Thank you in advance,

Francesco

NB: I am not a native english speaker, sorry for that.
 

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