Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2004 / Paper 2 / 7 / b / Solution

The defining function $x^3+xy-1$ is independent of $z$, so a normal vector is $(3x^2+y,x,0)$. It never vanishes here because $x>0$. Every normal is horizontal, whereas $\mathbf{G}=(0,0,x)$ is vertical and tangent to the surface. Their dot product is zero at every point, givingThis is independent of orientation. Although the surface extends without a bound on $x$, the integrand vanishes identically, so no convergence problem arises for this flux.