For these increasing warps, the <square-root velocity function> is $Q(h)(t)=\sqrt{h'(t)}$. The <chain rule> gives
$$
Q(h_i\circ\gamma)(t)
=Q(h_i)(\gamma(t))\sqrt{\gamma'(t)}.
$$
Therefore
$$
\begin{aligned}
\lVert Q(h_i\circ\gamma)-Q(h_j\circ\gamma)\rVert_2^2
&=\int_0^1
|Q(h_i)(\gamma(t))-Q(h_j)(\gamma(t))|^2\gamma'(t)\,dt\\
&=\int_0^1|Q(h_i)(u)-Q(h_j)(u)|^2\,du,
\end{aligned}
$$
where the last equality again uses the <change of variables formula>. Taking square roots proves invariance under common right composition by $\gamma$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 225 / 3 / a / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-225/3/a/ii/solution