Localized steady-current moment identities ($\int j_i=0,\int x_i{j}_k=-\int x_k{j}_i$)

A localized steady electric current density satisfies $\nabla \cdot \mathbf{j}=0$. Integrate the divergences of $x_i\mathbf{j}$ and $x_i{x}_k\mathbf{j}$ over a region enclosing its support; all boundary fluxes vanish. The first identity removes the monopole term from the magnetic vector potential expansion. The antisymmetric first moment is equivalent to the magnetic dipole moment $\frac{1}{2}\int \mathbf{x}\times \mathbf{j}dV$.