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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 ...