Due date: Monday 11. March 2024, 10 AM.
Exercise 1
Consider a closed interval \(I=[a,b]\) and a smooth curve \(\gamma\colon I \longrightarrow\mathbb{R}^m.\) Denote by \(\widetilde{\gamma}:= \gamma\circ\phi^{-1}\colon J\longrightarrow\mathbb{R}^m\) the smooth curve corresponding to a parametrization \(\phi^{-1}\colon J\longrightarrow I.\) Prove that the length of a smooth curve is independent of parametrization, i.e. that \(\ell(\gamma)=\ell(\widetilde{\gamma}).\)
Exercise 2
Let \(\gamma\colon(0,\pi)\longrightarrow \mathbb{R}^2\) be the curve defined by \[\gamma(t)=\left(\operatorname{sin}(t),\operatorname{cos}(t)+\operatorname{log}\left(\operatorname{tan}\left(\frac{t}{2}\right)\right)\right).\] The image of \(\gamma\) is know as the tractrix.
Prove that \(\gamma\) is smooth.
Show that \(\dot{\gamma}(t)\) is non-vanishing for all \(t\in (0,\pi)\setminus\left\{\frac{\pi}{2}\right\}.\)
Prove that the length of the segment of the tangent line from the point of tangency to the \(y\)-axis is \(1.\)
Compute the arc length parametrization of \(\gamma\) starting from \(t=\frac{\pi}{2}.\)
Compute the curvature of \(\gamma,\) where it is defined.
Exercise 3
Let \(\gamma : I \to \mathbb{R}^2\) be a smooth unit speed curve and \(t_0 \in I\) a time point where the signed curvature \(\kappa\) of \(\gamma\) is non zero, \(\kappa(t_0)\neq 0.\)
Show that there is a unique unit speed curve \(\delta : J \to \mathbb{R}^2\) defined on some suitable intervall \(J,\) so that the image of \(\delta\) is a circle and so that \[\gamma(t_0)=\delta(0), \qquad \gamma^{\prime}(t_0)=\delta^{\prime}(0), \qquad \gamma^{\prime\prime}(t_0)=\delta^{\prime\prime}(0).\]
Interpret the curvature \(k : I \to \mathbb{R}\) in terms of \(\delta.\)