Aha - you made an edit!
I should hope so!
The maths is hardly deep!
$$ \eta_{ab} \mapsto g_{ab}, \qquad \partial_a \mapsto \nabla_a $$
So Maxwell's equations can be written
$$ \nabla^a F_{ab} =0 , \qquad \nabla_{[a} F_{bc]} =0 $$
which are clearly invariant under coordinate transformations..
I was afraid that that is what you will write. You realize that "the clearly invariant under coordinate transformations" is true for the Lorentz transforms and general linear transforms but not true for Shubert's cockamamie transforms, right? This is why I asked you to start by calculating the partial derivative wrt t.
Alternatively, use the exterior calculus version of Maxwell's equations and use the fact that pull-backs commute with exterior derivatives
Under what conditions is the above true? Do the pull-backs commute with the exterior derivatives for nonlinear transforms? Are A and $$\Phi$$ four-vectors wrt non-linear transforms? You realize that the tensorial formalism is nothing but a different notation for the classical formalism using partial derivatives, so what is not true for the classical formalism cannot become magically true if one uses four-vectors or tensors.
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