= Solution
Put $D_t=\partial_t+\mathbf u\cdot\nabla$. The <continuity equation> and the expanded <ideal magnetohydrodynamic induction equation> are
$$
D_t\rho=-\rho\nabla\cdot\mathbf u,\qquad D_t\mathbf B=(\mathbf B\cdot\nabla)\mathbf u-\mathbf B\nabla\cdot\mathbf u,
$$
so $D_t(B_i/\rho)=(B_j/\rho)\partial_ju_i$. Because $\mathbf B=\nabla\times\mathbf A$, the induction equation permits the <magnetic vector potential> equation $\partial_t\mathbf A=\mathbf u\times\mathbf B-\nabla\varphi$, with an arbitrary scalar electric potential $\varphi$ carrying the gauge choice. Using $(\mathbf u\times\nabla\times\mathbf A)_i=u_j\partial_iA_j-u_j\partial_jA_i$ yields
$$
D_tA_i=\partial_i(\mathbf u\cdot\mathbf A-\varphi)-A_j\partial_i u_j.
$$
Contracting with $B_i/\rho$ and differentiating that factor gives
$$
\begin{aligned}
D_t\left(\frac{A_iB_i}{\rho}\right)
&=\frac{B_i}{\rho}\partial_i(\mathbf u\cdot\mathbf A-\varphi)-\frac{B_iA_j}{\rho}\partial_i u_j+\frac{A_iB_j}{\rho}\partial_j u_i\\
&=\frac{\mathbf B}{\rho}\cdot\nabla(\mathbf u\cdot\mathbf A-\varphi).
\end{aligned}
$$
The last two terms cancel by exchanging their dummy indices. Thus the requested <material derivative> is
$$
\boxed{D_t(\mathbf A\cdot\mathbf B/\rho)=\frac{\mathbf B}{\rho}\cdot\nabla(\mathbf u\cdot\mathbf A-\varphi).}
$$
This is <material transport of magnetic helicity density>.
For a material volume, <mass conservation> gives $d\int_{V(t)}\rho h\,dV/dt=\int_{V(t)}\rho D_th\,dV$. Apply this with $h=\mathbf A\cdot\mathbf B/\rho$, and use $\nabla\cdot\mathbf B=0$ and the <divergence theorem>:
$$
\frac{dH_m}{dt}=\int_{V(t)}\mathbf B\cdot\nabla(\mathbf u\cdot\mathbf A-\varphi)\,dV=\int_{\partial V(t)}(\mathbf u\cdot\mathbf A-\varphi)\mathbf B\cdot\mathbf n\,dS.
$$
If the <magnetic field lines> do not penetrate the boundary, $\mathbf B\cdot\mathbf n=0$, so $\boxed{dH_m/dt=0}$.
For uniqueness under a regular, single-valued <gauge transformation> $\mathbf A\mapsto\mathbf A+\nabla\Lambda$,
$$
\Delta H_m=\int_V\nabla\Lambda\cdot\mathbf B\,dV=\int_{\partial V}\Lambda\mathbf B\cdot\mathbf n\,dS=0.
$$
Consequently this <magnetic helicity> is both conserved and gauge independent. As usual, vector potentials are understood to be globally regular potentials for the same field, rather than unrelated harmonic-circulation choices on a multiply connected restricted domain.
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