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\begin{document}
\begin{center}
\large{\textbf{Math 4250/6250 Homework \#1}}
\end{center}
\bigskip
This homework assignment covers Section 1.1 in Shifrin.
Shifrin, Exercises 1.1: 2, 4, 5, 8, 9, 10.
Challenge problem for 6250 students: \#14.
\end{document}
\begin{problems}
\item (Do Carmo, 1-3, \#2) A circular disk of radius 1 in the $xy$ plane rolls along the $x$ axis without slipping. The curve described by a point on the rim of the disk is called a \emph{cycloid}.
\begin{enumerate}
\item Find a parametrization $\alpha(t)$ of the cycloid.
\item Compute the arclength of the portion of the cycloid corresponding to one complete rotation of the disk.
\end{enumerate}
\item (Do Carmo, 1-3, \#4) The curve
\begin{equation*}
\alpha(t) = \left( \sin t, \cos t + \log \tan \frac{t}{2} \right).
\end{equation*}
is called the tractrix. Show that
\begin{enumerate}
\item $\alpha$ is a differentiable parametrized curve, regular except at $t = \pi/2$.
\item The length of the portion of the tangent line to the tractrix between $\alpha(t)$ and the $y$-axis is always equal to $1$.
\end{enumerate}
\item (Based on Do Carmo, 1-5, \#3) Given a curve $\alpha(s)$ parametrized by arclength, consider the curve $T(s)$ on the unit sphere. This is called the \emph{tangent indicatrix} of $\alpha$. Prove that the speed of $T(s)$ is equal to the curvature of $\alpha$. The curve $N(s)$ is called the \emph{normal indicatrix} of $\alpha(s)$. Prove that the speed of $N(s)$ is equal to the length of the vector $(\kappa(s),\tau(s)) \in \R^2$. The curve $B(s)$ is called the \emph{binormal indicatrix}. Prove that the speed of $B(s)$ is $|\tau(s)|$.
\end{problems}
\section{Challenge Problems}
\noindent
\begin{problems}
\item (Do Carmo 1-3, \#8). Let $\alpha \co I \rightarrow \R^3$ be a differentiable (that is, $C^\infty$) regular curve and let $[a,b]$ be a closed interval. For every partition $P = a = t_0 < t_1 < \cdots < t_n = b$ of $[a,b]$, let
\begin{equation*}
\ell(P) = \sum |\alpha(t_{i+1}) - \alpha(t_i)|.
\end{equation*}
Let the \emph{mesh} of the partition be $|P| = \max t_{i+1} - t_i$. Prove that for any $\epsilon > 0$, there exists $\delta > 0$ so that if $|P| < \delta$ then
\begin{equation*}
\left| \int_{a}^b |\alpha'(t)| \,dt - \ell(P) \right| < \epsilon.
\end{equation*}
That is, the lengths of polygons inscribed in the curve converge to the length of the curve.
\item (Based on Do Carmo 1-3, \#10). Let $\alpha \co I \rightarrow \R^3$ be a a differentiable parametrized curve. Suppose $[a,b] \in I$ and $\alpha(a) = p$ while $\alpha(b) = q$.
\begin{enumerate}
\item Show that for any constant vector $v$ with $|v| = 1$,
\begin{equation*}
\left< q-p, v \right> \int_{a}^b \left< \alpha'(t), v \right> \,dt \leq \int_a^b |\alpha'(t)| \, dt.
\end{equation*}
\item Let
\begin{equation*}
v = \frac{q-p}{|q-p|}
\end{equation*}
and show that
\begin{equation*}
|\alpha(b) - \alpha(a)| \leq \int_a^b |\alpha'(t)| \, dt.
\end{equation*}
That is, the curve of shortest length joining two points is the straight line!
\end{enumerate}
\item Using the setup of the last problem, suppose that $p$ lies in the plane $z = 0$ (that is, $p = (p_1,p_2,0)$) and $q$ lies in the plane $z=1$ (that is, $q = (q_1,q_2,1)$). Prove that the shortest curve joining any such $p$ and $q$ is the straight line joining $p = (x,y,0)$ to $q = (x,y,1)$.
\item Prove that a nonplanar curve with curvature $\kappa(s)$ and torsion $\tau(s)$ lies entirely on a sphere if and only if
\begin{equation*}
\frac{\tau(s)}{\kappa(s)} = \frac{d}{ds} \left( \frac{\kappa'(s)}{\tau(s) \kappa^2(s)} \right)
\end{equation*}
\item If $\gamma(s)$ is an arclength-parametrized curve with nonzero curvature, find a vector $\omega(s)$, expressed as a linear combination of $T$, $N$, and $B$ so that
\begin{align*}
T'(s) &= \omega(s) \times T(s) \\
N'(s) &= \omega(s) \times N(s) \\
B'(s) &= \omega(s) \times B(s)
\end{align*}
This vector is called the \emph{Darboux vector}. Find a formula for the length of the Darboux vector in terms of the curvature $\kappa(s)$ and torsion $\tau(s)$ of the curve.
\end{problems}
\end{document}