Surface area element
= Surface area element
{title2=$dS=|\mathbf X_u\times\mathbf X_v|\,du\,dv$}
For a regular <parametrization> $\mathbf X(u,v)$ of a surface in three-dimensional <Euclidean space>, the scalar surface area element is the area of the infinitesimal parameter parallelogram after mapping. Its value is the displayed <cross product> magnitude. The <vector surface area element> additionally records the chosen orientation.