Starting from the action
$$S=-\frac{1}{4}\int{F_{\mu\nu}F^{\mu\nu}d^4x}$$ (until here this is physics but my question is about the math) that is:
I got the following equation of motion solving for that: $$\partial_{\mu}F^{\mu\nu}=0$$ but we usually know that starting from the action above, we should get $d^\star F=0$ which I am not getting.
My question are $d^\star F$ and $\partial_{\mu}F^{\mu\nu}$ equivalent?