Material magnetic-flux transport theorem ($d{\Phi }_B/dt={\int }_{S(t)}[{\partial }_t\mathbf{B}-\nabla \times (\mathbf{u}\times \mathbf{B})+\mathbf{u}\nabla \cdot \mathbf{B}]\cdot d\mathbf{S}$)

Combine the material surface element evolution law with the product rule for $\mathbf{B}\cdot d\mathbf{S}$. The result isExpanding 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.