Vector surface element of a graph
= Vector surface element of a graph
{title2=$d\mathbf S=(-g_x,-g_y,1)\,dx\,dy$}
The graph $z=g(x,y)$ has upward-oriented vector area element $d\mathbf S=(-g_x,-g_y,1)\,dx\,dy$ and scalar area element $dS=\sqrt{1+g_x^2+g_y^2}\,dx\,dy$. These follow from the <cross product> of the parametrization derivatives $(1,0,g_x)$ and $(0,1,g_y)$. The vector element is used in flux <surface integrals> and <Stokes theorem>; reversing orientation changes its sign.