Conserved current
= Conserved current
= Current conservation
{synonym}
A conserved current has components $j^\mu=(\rho,\mathbf j)$ satisfying the continuity equation $\partial_t\rho+\nabla\cdot\mathbf j=0$. It relates local accumulation of a quantity to its transport.
For a fixed volume $V$, the divergence theorem gives
$$
\frac{d}{dt}\int_V\rho\,d^3x=-\int_{\partial V}\mathbf j\cdot d\mathbf S.
$$
Thus the integrated charge density, not an arbitrary spatial integral of every current component, is constant when boundary flux vanishes. Electric charge provides one example; a continuous field symmetry can provide a <Noether current> for another. A measured local density can change while the global charge remains conserved, because current redistributes it across the sample.