= Solution
Use <Faraday's law> $\partial_t\mathbf B=-\nabla\times\mathbf E$ and the <solenoidal magnetic-field constraint> $\nabla\cdot\mathbf B=0$. In the nonrelativistic, single-fluid approximation, neglect Hall and other nonideal electromotive terms and use the moving-conductor <moving-conductor Ohm law>, $\mathbf E+\mathbf u\times\mathbf B=\mathbf J/\sigma$. Infinite <electrical conductivity> with finite current gives $\mathbf E=-\mathbf u\times\mathbf B$, hence
$$
\boxed{\partial_t\mathbf B=\nabla\times(\mathbf u\times\mathbf B).}
$$
This is the <ideal magnetohydrodynamic induction equation>. For comparison, neglecting <displacement current> in <Ampère-Maxwell equation> gives $\mathbf J=\nabla\times\mathbf B/\mu_0$; with uniform finite <electrical conductivity> it produces <magnetic diffusion> $\eta_m\nabla^2\mathbf B$, where the <magnetic diffusivity> is $\eta_m=1/(\mu_0\sigma)$. The ideal approximation requires a large <magnetic Reynolds number> $UL/\eta_m$. Dropping displacement current is useful for this finite-conductivity comparison, but Faraday's law and the ideal Ohm relation already suffice for the ideal induction equation.
Expanding the curl and using the <solenoidal magnetic-field constraint> gives the <material derivative> form
$$
\frac{D\mathbf B}{Dt}=(\mathbf B\cdot\nabla)\mathbf u-\mathbf B\nabla\cdot\mathbf u.
$$
Combine this with <mass conservation>, $D\rho/Dt=-\rho\nabla\cdot\mathbf u$, to obtain
$$
\boxed{\frac D{Dt}\left(\frac{\mathbf B}{\rho}\right)=\left(\frac{\mathbf B}{\rho}\cdot\nabla\right)\mathbf u.}
$$
Now parametrize a <material curve> by a fixed label $a$: $\mathbf X(a,t)$ obeys $\partial_t\mathbf X=\mathbf u(\mathbf X,t)$. Differentiating with respect to $a$ shows that its tangent $\boldsymbol\ell=\partial_a\mathbf X$ evolves by $D\boldsymbol\ell/Dt=(\boldsymbol\ell\cdot\nabla)\mathbf u$. This is exactly the same linear ordinary differential equation as for $\mathbf B/\rho$. Initially parallel tangents remain parallel by uniqueness, with a label-dependent proportionality factor constant along each particle trajectory. Thus \b[<magnetic field lines> are transported as <material curves>], wherever the field and fluid flow are smooth and the field is nonzero. This is the field-line form of <magnetic flux freezing>.
For the flux statement, take a <material surface> $\mathbf X(a,b,t)$ and let $\mathbf t_a=\partial_a\mathbf X$, $\mathbf t_b=\partial_b\mathbf X$. Both tangents obey $D t_i/Dt=(\partial_j u_i)t_j$. Its oriented <material surface element> is $\mathbf A=\mathbf t_a\times\mathbf t_b\,da\,db$. Differentiating the cross product, rather than assuming its transport rule, gives
$$
\frac{DA_i}{Dt}=\varepsilon_{ijk}\big[(\partial_\ell u_j)t_{a\ell}t_{bk}+t_{aj}(\partial_\ell u_k)t_{b\ell}\big]da\,db
=(\partial_j u_j)A_i-(\partial_i u_j)A_j.
$$
Equivalently, $D\mathbf A/Dt=(\nabla\cdot\mathbf u)\mathbf A-(\nabla\mathbf u)^T\mathbf A$, with $(\nabla\mathbf u)_{ij}=\partial_j u_i$. Contracting this derived rule with the induction equation gives a pointwise cancellation:
$$
\frac D{Dt}(\mathbf B\cdot\mathbf A)=\big[(\nabla\mathbf u)\mathbf B-\mathbf B\nabla\cdot\mathbf u\big]\cdot\mathbf A+\mathbf B\cdot\big[(\nabla\cdot\mathbf u)\mathbf A-(\nabla\mathbf u)^T\mathbf A\big]=0.
$$
Integrating over the fixed material labels therefore proves \b[conservation of flux through an open <material surface>]:
$$
\boxed{\frac d{dt}\int_{S(t)}\mathbf B\cdot d\mathbf S=0.}
$$
The surface need not be closed; its boundary is carried with the fluid. The result follows from material transport, rather than from the zero flux through a closed surface.
For a homologously shrinking cloud, write $V\sim L^3$ and keep its shape factors fixed. Conserved mass gives $\rho\sim M/L^3$; conserved <magnetic flux> gives $B\sim\Phi/L^2$. Thus the gravitational and <magnetic energies> scale as
$$
\boxed{W_g=-C_g\frac{GM^2}{L},\qquad E_B=C_B\frac{\Phi^2}{\mu_0L},}
$$
where $C_g,C_B>0$ are dimensionless geometry factors. Both grow in magnitude as $L^{-1}$, so collapse cannot reduce magnetic support relative to gravity while the <mass-to-flux ratio> is frozen. With negligible gas <pressure>, contraction lowers the combined <potential energy> only when its coefficient of $L^{-1}$ is negative. Consequently \b[a necessary <critical mass-to-flux ratio> condition] is
$$
\boxed{\frac M\Phi>\frac{\lambda}{\sqrt{\mu_0G}},\qquad \lambda=\sqrt{C_B/C_g}.}
$$
Here $\Phi$ denotes the magnitude of the conserved threading flux. The numerical coefficient depends on geometry and boundary conditions; the scaling argument does not determine it or make the condition sufficient in the presence of other support.
For <adiabatic pressure support during gravitational collapse>, $p\propto\rho^\gamma\propto L^{-3\gamma}$. The pressure-support scale is $pV\propto L^{3-3\gamma}$, so relative to either gravity or <magnetic energy>,
$$
\boxed{\frac{pV}{|W_g|}\ \propto\ L^{4-3\gamma}.}
$$
<Pressure> becomes more important as $L$ decreases if $\gamma>4/3$, equally important in scaling if $\gamma=4/3$, and less important if $\gamma<4/3$. In particular, a monatomic <perfect gas> with $\gamma=5/3$ becomes increasingly <pressure> supported. For <isothermal pressure support during gravitational collapse>, the <isothermal equation of state> gives $p\propto\rho\propto L^{-3}$, so $pV$ is constant and $pV/|W_g|\propto L$: \b[isothermal <pressure> becomes less important during collapse]. The same comparisons hold against magnetic support because its energy has the same $L^{-1}$ scaling as gravity.
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