Material surface element evolution ($D(d\mathbf{S})/Dt=[(\nabla \cdot \mathbf{u})\mathsf{I}-(\nabla \mathbf{u}{)}^{\mathsf{T}}]d\mathbf{S}$)

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.