Past exam of the geometry of surfaces course of the University of Oxford / 2023 / 3 / c / Solution

Set $C=\cos u$, $S=\sin u$, and $D=1+S^2=2-C^2$. At $v=0$, the first partial derivatives areTheir dot products give $E=1$, $F=1+C$, $G=3+2C$. HenceSince $EG-F^2=3+2C-(1+C{)}^2=D>0$, the parametric surface is regular everywhere. Its cross product and unit normal at $v=0$ areThe second partial derivatives at $v=0$ areTheir dot products with $N$ give $e=f=-1/\sqrt{D}$ and $g_2=-(1+C)/\sqrt{D}$. ThusPart (b) transports these first fundamental form and second fundamental form coefficients to every $v$. Finally,The last expression is the surface area element from $\sqrt{EG-F^2}$. In particular the Gaussian curvature changes sign where $\cos u$ does; it vanishes at $u=\pi /2$ modulo $\pi$.