Let $S$ be the surface defined by $z = e^{1-x^2-y^2}$, $z\geq 1$, oriented by the upward normal, and let $\mathbf{F}(x,y,z) = (x,y,2-2z)$. Find the total flux of $\mathbf{F}$ across $S$.
So I find the boundary of the surface which is a closed circle C, and use stokes theorem to switch from a surface integral to a different kind of integral. But I don't know what to do next. Thank you.