Material transport of magnetic helicity density ($D_t(\mathbf{A}\cdot \mathbf{B}/\rho )$)

For smooth ideal magnetohydrodynamics, write $\mathbf{B}=\nabla \times \mathbf{A}$ and ${\partial }_t\mathbf{A}=\mathbf{u}\times \mathbf{B}-\nabla \phi$. The continuity equation and induction equation give $D_t(\mathbf{B}/\rho )=(\mathbf{B}/\rho )\cdot \nabla \mathbf{u}$. The vector-potential equation becomes $D_t\mathbf{A}=\nabla (\mathbf{u}\cdot \mathbf{A}-\phi )-(\nabla \mathbf{u}{)}^T\mathbf{A}$. Contracting these two equations cancels their velocity-gradient terms and provesConsequently the magnetic helicity in a material volume satisfies $d{H}_m/dt={\int }_{\partial V}(\mathbf{u}\cdot \mathbf{A}-\phi )\mathbf{B}\cdot \mathbf{n}dS$. It is conserved when the magnetic field is tangent to the boundary. A single-valued gauge transformation changes $H_m$ by ${\int }_{\partial V}\Lambda \mathbf{B}\cdot \mathbf{n}dS$, which also vanishes under that condition.