Proof that Maxwell equations are Lorentz invariant In Peskin and Schroeder page 37, it is written that


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*Using vector and tensor fields, we can write a variety of Lorentz-invariant equations.

*Criteria for Lorentz invariance: In general, any equation in which each term has the same set of uncontracted Lorentz indices will naturally be invariant under Lorentz transformations.


I would like to explicitly show that the above criteria is valid for Maxwell's equations $\partial^{\mu} F_{\mu \nu} = 0$ or $\partial^{2}A_{\nu}-\partial_{\nu}\partial^{\mu}A_{\mu}=0$.


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*Solution 1: Maxwell's equations follow from the Lagrangian $$\mathcal{L}_{MAXWELL}=-\frac{1}{4}(F_{\mu \nu})^{2} = -\frac{1}{4}(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})^{2}$$ which is a Lorentz scalar, so this means that the equation of motion is Lorentz-invariant as well. That's one way to convince yourself that the above Maxwell's equations are, in fact, Lorentz invariant. Is this correct?

*Solution 2: I would like to actively transform the electromagnetic field strength tensor $F_{\mu \nu}$ and show that the Maxwell's equations $\partial^{\mu} F_{\mu \nu} = 0$ or $\partial^{2}A_{\nu}-\partial_{\nu}\partial^{\mu}A_{\mu}=0$ remain Lorentz invariant.
I can see that $\partial^{2}$ and $\partial^{\mu}A_{\mu}$ will not Lorentz transform as they are Lorentz scalars.
Under an active Lorentz transformation, $V^{\mu}(x) \rightarrow \Lambda^{\mu}_{\nu}V^{\nu}(\Lambda^{-1}x)$. So, will $A_{\nu}$ and $\partial_{\nu}$ Lorentz transform in the same way?
 A: Maxwell's equations can be written in the form
$$
\partial_{\mu}F^{\mu\nu} = \frac{4\pi}{c} j^{\nu},\qquad
\partial_{\lambda}F_{\mu\nu}+ \textrm{cyclic}(\lambda,\mu,\nu)=0
$$
with $F_{\mu\nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}$. Let us look at the first set: the right hand side is a vector (and this can be proven looking at the equation for the conservation of the charge), therefore the left hand side must be a vector too. Being $\partial_{\mu}$ the covariant components of a dual form, $F^{\mu\nu}$ must be the components of a rank $(0,2)$ tensor. As such, under Lorentz transformations, they will transform as
$$
{F^{\mu\nu}}'(x') = \Lambda^\mu_{\phantom{\mu}\gamma}(x)\,\Lambda^\nu_{\phantom{\nu}\rho}(x)F^{\gamma\rho}(x) 
$$
(where one can equivalently take $\Lambda^{-1}$ according to how such map is defined). Expanding the above in terms of the electric and magnetic field (as derived from the vector potential $A_{\mu}$ such that $F_{\mu\nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}$) one can obtain the explicit transformation laws for the two fields. Once so, you can directly plug them into the Maxwell's equations by brute force and check that they are left invariant.
Another way, as you pointed out yourself, is to notice that the action is left invariant under the effect of some particular transformations; hence, the equations of motion must be as well. 
