= Material magnetic-flux transport theorem
{title2=$d\Phi_B/dt=\int_{S(t)}[\partial_t\mathbf B-\nabla\times(\mathbf u\times\mathbf B)+\mathbf u\nabla\cdot\mathbf B]\cdot d\mathbf S$}
Combine the <material surface element evolution> law with the <product rule> for $\mathbf B\cdot d\mathbf S$. The result is
$$
\frac d{dt}\int_{S(t)}\mathbf B\cdot d\mathbf S
=\int_{S(t)}\left[\frac{D\mathbf B}{Dt}+\mathbf B\nabla\cdot\mathbf u-(\mathbf B\cdot\nabla)\mathbf u\right]\cdot d\mathbf S.
$$
Expanding the <curl> gives the equivalent title formula. The <ideal magnetohydrodynamic induction equation> and <Gauss's law for magnetism> make the integrand zero and prove <magnetic flux freezing>, without requiring <incompressible flow>.
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