Vector surface area element
= Vector surface area element
{title2=$\mathbf n\,dS=\pm(\mathbf X_u\times\mathbf X_v)\,du\,dv$}
A vector area element combines the scalar <surface area element> with a chosen <unit normal vector>. Reversing the order of the two parameter directions reverses its sign. <Flux> through an oriented surface is the integral of the <dot product> of a <vector field> with this vector element. For a graph $z=f(x,y)$, the upward choice is $(-f_x,-f_y,1)\,dx\,dy$.