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 $\mathbf{B}\cdot \mathbf{A}$.

Parameterize a material surface by fixed particle labels $a,b$, with tangents $\mathbf{a}={\partial }_a\mathbf{x}$, $\mathbf{b}={\partial }_b\mathbf{x}$. The material derivative gives $D\mathbf{a}/Dt=(\nabla \mathbf{u})\mathbf{a}$ and similarly for $\mathbf{b}$. Differentiate $d\mathbf{S}=(\mathbf{a}\times \mathbf{b})dadb$ and use the cross product identityThis proves the displayed oriented-area evolution law, with $(\nabla \mathbf{u}{)}_{ij}={\partial }_j{u}_i$. It allows both area change and rotation; even incompressible flow need not preserve the magnitude of a material area element.