10. (10 points) Let $\mathbf{F}(x,y,z)=x\mathbf{i}+y\mathbf{j}-2z\mathbf{k}$. Suppose that the closed surface $S$ is defined by the paraboloid $y=x^2+z^2$ with $0 \le y \le 1$. Find the flux of $\mathbf{F}$ outward across $S$.