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\lhead{Math 4441 Homework 2}
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\begin{document}
\title{Math 4441 Homework 2}
\author{}
\date{September 23, 2022}
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\begin{enumerate}
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\item Let $\vec{\alpha}$ be a unit-speed plane curve, and let $\{\vec{t}(s),\vec{n}(s)\}$ denote its planar Frenet frame, while $\{\vec{T}(s),\vec{N}(s),\vec{B}(s)\}$ denotes its Frenet frame as a space curve.
\begin{enumerate}
\item Prove that $\vec{t}(s)=\vec{T}(s)$ and $\vec{n}(s)=\pm\vec{N}(s)$, whenever $\vec{N}(s)$ is defined.
\begin{solution}
\end{solution}
\item Prove that $\kappa(s)=|k(s)|$, for all $s$ in the domain of $\vec{\alpha}$.
\item Prove that $\vec{n}'(s)=-k(s)\,\vec{t}(s)$, for all $s$ in the domain of $\vec{\alpha}$.
\end{enumerate}
\item Show that the evolute $\vec{\varepsilon}$ of a plane curve $\vec{\alpha}$ is uniquely determined by the condition that its tangent line at each point $\vec{\varepsilon}(t)$ is the normal line to $\vec{\alpha}$ at $\vec{\alpha}(t)$.
\item Let $\vec{\alpha}\colon[0,L]\to\mathbb{R}^2$ be a simple, closed, unit-speed plane curve. Prove that the \emph{tangent circular image} $\vec{t}\colon[0,L]\to S^1$ is onto. (Here $S^1\subset\mathbb{R}^2$ is the unit circle centered at the origin.)\\
\emph{Hint: Use the rotation index theorem.}
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\item Suppose $C$ is a convex, simple, closed plane curve with maximum curvature $\kappa_0$. Prove that the distance between any pair of parallel tangent lines of $C$ is at least $2/\kappa_0$.
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\item Let $\vec{\alpha}$ be a simple, closed plane curve whose image bounds a region $\mathcal{R}\subset\mathbb{R}^2$, and which is traversed counterclockwise. Prove that the area of $\mathcal{R}$ is given by
\[
\int_{\vec{\alpha}} x\, dy = -\int_{\vec{\alpha}}y\,dx,
\]
where $x$ and $y$ are the usual coordinates of $\mathbb{R}^2$.
\item Let $a$ and $b$ be positive numbers. Prove that
\[
\sqrt{ab}\leq \frac{1}{2}(a+b),
\]
with equality if and only if $a=b$. (This is called the {\bf arithmetic mean-geometric mean inequality}.)
\item Suppose $C$ is a simple, closed plane curve with $0<\kappa\leq c$, for some constant $c$. Prove that the length of $C$ is at least $2\pi/c$.
\item Consider distinct points $\vec{a},\vec{b}\in\mathbb{R}^2$, and fix some length $\ell>\|\vec{a}-\vec{b}\|$. Consider all possible regular curve segments in $\mathbb{R}^2$ which begin at $\vec{a}$, end at $\vec{b}$, and have length $\ell$. Which such curve will enclose the largest possible (signed) area when joined with the line segment from $\vec{b}$ to $\vec{a}$?\\
\emph{Note: Don't just recreate a proof of the isoperimetric inequality. Instead, assume the isoperimetric inequality and argue that your curve is the best one.}
\end{enumerate}
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