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

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.

The first fundamental form of a surface of revolution isThe geodesic Lagrangian is consequentlyThe rotation angle $u$ is a cyclic coordinate, so its conjugate momentum is conserved by the Euler-Lagrange equation:Thus $f(v(s){)}^2{u}^{\prime }(s)$ is constant.

The positively oriented unit tangent to the parallel of a surface of revolution through $\gamma (s)$ is $e_u={\varphi }_u/f(v)$. Since $\gamma$ has unit speed and $\theta$ is its angle with that parallel,Multiplication by $f(v(s))$ yieldswhich is Clairaut's relation.