Material surface ($S(t)$)

A material surface is transported with its fluid elements, including its boundary when open. Its oriented material surface element evolves by tangent transport. Magnetic flux freezing conserves the flux through such an open surface.

If $\mathbf{A}={\mathbf{t}}_a\times {\mathbf{t}}_{b}dadb$ 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 $\mathbf{B}\cdot \mathbf{A}$.