The covariant Maxwell equations

Use $\eta=\operatorname{diag}(1,-1,-1,-1)$, $x^0=ct$, $F^{0i}=-E_i/c$, $F^{ij}=-\epsilon^{ijk}B_k$, and $J^\mu=(c\rho,\mathbf j)$. Then the sourced equation is $\partial_\mu F^{\mu\nu}=\mu_0J^\nu$. For the temporal choice of the free index, only the spatial derivatives survive and give $\nabla\cdot\mathbf E/c=\mu_0c\rho$. Since $\mu_0\epsilon_0c^2=1$, this is Gauss's electric law. One power of c comes from the field component and another from the current's time component; both are required.

For a spatial free index, the temporal derivative contributes minus the time derivative of E divided by c squared, while spatial derivatives produce the curl of B. Rearranging gives the Ampère–Maxwell law. The displacement-current term follows from the same tensor equation as electric Gauss law; it is not a separate correction chosen only for capacitors. In SI the common vacuum coefficients ensure the equations have one invariant propagation speed. Macroscopic material equations additionally require specified polarization, magnetization, and constitutive assumptions.

Write the second tensor statement using lower field indices: the cyclic sum of three derivatives vanishes. With three spatial indices it becomes zero magnetic divergence. With one temporal and two spatial indices it becomes Faraday's induction law, including its minus sign. This notation avoids introducing an additional orientation convention for the four-dimensional dual tensor. It uses the same lower components already established, especially $F_{0i}=+E_i/c$, so no new field convention is hiding in the compact expression.

If $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$ for a sufficiently smooth local potential, the cyclic identity follows because mixed derivatives commute. Thus the homogeneous equations are automatically satisfied by the potential representation, while the sourced equations determine its dynamics. This argument is local; nontrivial topology can require different potential patches. Magnetic monopoles, if introduced into a different theory, would modify this homogeneous structure. We assume ordinary Maxwell electrodynamics without magnetic charge throughout the present module.

Take another divergence of the sourced equation. The derivative pair is symmetric, while the contracted field indices are antisymmetric, so $\partial_\nu\partial_\mu F^{\mu\nu}=0$. It follows that $\partial_\nu J^\nu=0$. A proposed external current that violates local continuity cannot be a source for any solution of these Maxwell equations. Smoothness can be relaxed to distributions when derivatives are treated consistently, allowing point sources and idealized surfaces, but arbitrary inconsistency cannot be repaired by changing gauge.

There is also a useful initial-value interpretation. Electric Gauss law and zero magnetic divergence constrain an initial spatial slice, while the curl equations evolve the fields. Taking divergences of the evolution equations shows that compatible constraints remain satisfied when continuity holds. This does not remove the need for boundary conditions or supply an instantaneous action-at-a-distance solver. The tensor form guarantees consistent transformation between observers; constructing a particular physical solution still requires sources, initial data, boundaries, and an appropriate causal prescription.