Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 25H / a / i / Solution

The coordinate tangent vectors arewhere $f,f^{\prime },g^{\prime }$ are evaluated at $v$. Their cross product isand thereforeThe positivity follows because $f(v)>0$ and the profile is a regular curve. Hence $d\varphi$ has rank two everywhere, so $\varphi$ is an immersion. The angle interval $(0,2\pi )$ and the absence of profile self-intersections make $\varphi$ injective with a continuous local inverse. It is therefore an embedded surface parametrization, and its image is a regular surface.