For example having $F(x,y,z)=xz\vec i+\vec k$ through the paraboloid on cylindrical coordinates $z=4-r^2,\,0\le r\le2,\,0\le\theta\le2\pi,$ oriented according the inner norm.
I think by Gauss we have $\displaystyle\int_{0}^{2}{\int_{0}^{2\pi }{\int_{0}^{4-{{r}^{2}}}{(z+1)r\,dz}\,d\theta }\,dr},$ but the info about the norm drives me crazy.