# How One Elegant 4D Equation Unifies Gauss's Law and the Ampère-Maxwell Law The unification of Maxwell's equations into a single, elegant fourdimensional relativistic tensor equation: $$\partial\_\mu F^{\mu \nu}=K^\nu$$. This equation unifies the two inhomogeneous Maxwell's equations-Gauss's Law ( $$\nabla \cdot F =\rho / \epsilon\_0$$ ) and the Ampère-Maxwell Law ( $$\nabla \times B =\mu\_0 J+\mu\_0 \epsilon\_0 \frac{\partial F}{\partial t}$$ )-which are collectively called "inhomogeneous" because they are directly sourced by charge density ( $$\rho$$ ) and current density ( J ), embedded in the four-current density $$K^\nu$$. Specifically, the $$\nu=0$$ (time) component of the tensor equation yields Gauss's Law, while the $$\nu=j$$ (spatial) components yield the Ampère-Maxwell Law, which relates the curl of the magnetic field to both current and the time-changing electric field (displacement current). This relativistic formulation, which embeds the electric (E) and magnetic (B) fields into the electromagnetic field tensor $$F^{\mu \nu}$$ and uses the four-dimensional spacetime coordinate $$x^\mu$$, reveals that electromagnetism is inherently consistent with Special Relativity. {% embed url="" %} {% embed url="" %}