Proving Lorentz invariance of Maxwell equations I've read somewhere that one does not need to prove Lorentz invariance of the Maxwell
equations 
$F_{\mu\nu,\sigma}+F_{\nu\sigma,\mu}+F_{\sigma\mu,\nu}=0$
because it is "manifestly Lorentz invariant" or "because they are tensor equations"? What is meant by that? I've read that this could mean that space and time are treated "on equal footing". How can this replace a mathematical proof?
 A: You aren't "replacing a mathematical proof". What the statements you are referring to mean is that in tensor notation, the proof is immediate, so that nothing needs to be written down. This is because if you have a tensor equation as above, in order to prove Lorentz invariance, do a Lorentz transformation and go to another set of coordinates $x^{\mu'}$. Then using the usual transformation laws we get that ${\partial_{\mu}} = \Lambda^{\mu'}_{\mu}\partial_{\mu'}$ and $F_{\mu\nu} = \Lambda^{\mu'}_{\mu}\Lambda^{\nu'}_{\nu}F_{\mu'\nu'} $, we can write the Maxwell equation in terms of the new coordinates to become
$\Lambda^{\mu'}_{\mu}\Lambda^{\nu'}_{\nu}\Lambda^{\sigma'}_{\sigma}(F_{\mu'\nu',\sigma'}+F_{\nu'\sigma',\mu'}+F_{\sigma'\mu',\nu'})=0.$
However, this can only hold if the thing inside the brackets is zero itself. Namely Maxwell's equation in the primed coordinate system also holds.
More succintly, what a "tensor equation" means is that there was nothing special about the coordinate system in which the equations were derived. You could have equally well chosen another system and derived the same equations. Thus invariance under coordinate change is immediate.
