= Material transport of magnetic helicity density
{title2=$D_t(\mathbf A\cdot\mathbf B/\rho)$}
For smooth <ideal magnetohydrodynamics>, write $\mathbf B=\nabla\times\mathbf A$ and $\partial_t\mathbf A=\mathbf u\times\mathbf B-\nabla\varphi$. The <continuity equation> and induction equation give $D_t(\mathbf B/\rho)=(\mathbf B/\rho)\cdot\nabla\mathbf u$. The vector-potential equation becomes $D_t\mathbf A=\nabla(\mathbf u\cdot\mathbf A-\varphi)-(\nabla\mathbf u)^T\mathbf A$. Contracting these two equations cancels their velocity-gradient terms and proves
$$
D_t\left(\frac{\mathbf A\cdot\mathbf B}{\rho}\right)=\frac{\mathbf B}{\rho}\cdot\nabla(\mathbf u\cdot\mathbf A-\varphi).
$$
Consequently the <magnetic helicity> in a material volume satisfies $dH_m/dt=\int_{\partial V}(\mathbf u\cdot\mathbf A-\varphi)\mathbf B\cdot\mathbf n\,dS$. 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.
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