 Solution

Take $A^{\mu }=(\phi /c,\mathbf{A})$, ${\partial }_{\mu }=(c^{-1}{\partial }_t,\nabla )$ and $J^{\mu }=(c\rho ,\mathbf{j})$. ThenExpanding the cyclic derivative of $F^{ab}={\partial }^a{A}^b-{\partial }^b{A}^a$ produces six second derivatives, which cancel in pairs because partial derivatives commute. HenceThis Bianchi identity contains $\nabla \cdot \mathbf{B}=0$ and $\nabla \times \mathbf{E}=-{\partial }_t\mathbf{B}$. The inhomogeneous Maxwell equations areThe temporal component is $\nabla \cdot \mathbf{E}=\rho /{ϵ}_0$, since ${ϵ}_0{\mu }_0{c}^2=1$; the spatial components give $\nabla \times \mathbf{B}-c^{-2}{\partial }_t\mathbf{E}={\mu }_0\mathbf{j}$. The source suppresses the coupling factor, as in units with ${\mu }_0=1$ or a rescaled current.

For the physical Lorentz force density on matter, the consistent tensor formula isIts temporal component is power transferred to matter per unit volume divided by $c$. The printed expression $J^{\nu }{F}_{\nu \mu }$ has the opposite sign for the stated definition of $F$ in this convention. A stationary positive charge in a positive $x$ electric field is a concrete check: the physical upper spatial force is $+\rho E_x$, whereas the printed expression, raised to an upper index, gives $-\rho E_x$. It can represent the opposite momentum transfer to the electromagnetic field, but must not be identified with the force on the charge without changing a convention.