The electromagnetic continuity equation, expressed as $\nabla \cdot \mathbf{J} + \frac{\partial \rho}{\partial t} = 0$, mathematically states that the divergence of the current density ($ \mathbf{J}$) out of a differential volume is equal to the negative rate of change of the charge density ($\rho$) within that volume. This is a direct statement of the conservation of electric charge.