Past exam of the geometry course of the University of Oxford / 2008 / A / 2 / iii / c

For the helicoid,Hence $E=1+v^2$, $F=0$, $G=1$. Since $r_{uu}=(-v\cos u,-v\sin u,0)$ and $r_{vv}=0$, both $L$ and $N$ vanish. The mean-curvature formula therefore givesFor comparison, $M=1/\sqrt{1+v^2}$, so $K=-1/(1+v^2{)}^2$; zero mean curvature does not mean zero Gaussian curvature.