Material surface element
= Material surface element
{title2=$D\mathbf A/Dt=(\nabla\cdot\mathbf u)\mathbf A-(\nabla\mathbf u)^T\mathbf A$}
If $\mathbf A=\mathbf t_a\times\mathbf t_b\,da\,db$ and each tangent satisfies $D\mathbf t/Dt=(\nabla\mathbf u)\mathbf t$, differentiation of the cross product gives $D\mathbf A/Dt=(\nabla\cdot\mathbf u)\mathbf A-(\nabla\mathbf u)^T\mathbf A$. Combining this with ideal induction proves pointwise conservation of $\mathbf B\cdot\mathbf A$.