= Solution
Take $x^\mu=(ct,x,y,z)$ and <metric signature> $(+,-,-,-)$. The <four-current>, <four-potential> and <four-gradient> are
$$
J^\mu=(c\rho,\mathbf j),\qquad
A^\mu=(\phi/c,\mathbf A),\qquad
\partial_\mu=(c^{-1}\partial_t,\nabla),\qquad
\partial^\mu=(c^{-1}\partial_t,-\nabla).
$$
The first two transform as contravariant <four-vectors> under an inertial-frame <Lorentz transformation>, within a consistently transformed electromagnetic gauge. The lowered-index <gradient> is a <covector>. Their contraction is a <Lorentz scalar>:
$$
\partial_\mu J^\mu=\partial_t\rho+\nabla\cdot\mathbf j=0.
$$
Indeed, $J'^\mu=\Lambda^\mu{}_{\nu}J^\nu$ and $\partial'_\mu=(\Lambda^{-1})^\alpha{}_{\mu}\partial_\alpha$, so the two matrices cancel in the contraction. \b[Local <charge conservation> in one <inertial frame> implies local <charge conservation> in every <inertial frame>.] Integrating over a region gives the charge change equal to minus its outward current <flux>; global conserved charge requires no <flux> through the system boundary.
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