= Material surface element evolution
{title2=$D(d\mathbf S)/Dt=[(\nabla\cdot\mathbf u)\mathsf I-(\nabla\mathbf u)^{\mathsf T}]d\mathbf S$}
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 $d\mathbf S=(\mathbf a\times\mathbf b)\,da\,db$ and use the <cross product> identity
$$
(A\mathbf a)\times\mathbf b+\mathbf a\times(A\mathbf b)
=[(\operatorname{tr}A)\mathsf I-A^{\mathsf T}](\mathbf a\times\mathbf b).
$$
This proves the displayed oriented-area evolution law, with $(\nabla\mathbf u)_{ij}=\partial_ju_i$. It allows both area change and rotation; even <incompressible flow> need not preserve the magnitude of a material area element.
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