\documentclass[11pt]{amsart}
\usepackage{amssymb, amsmath, amsfonts, amsthm}
\usepackage[abbrev]{amsrefs}
%\usepackage[notref]{showkeys}
\usepackage[margin=0.8 in]{geometry}
\usepackage[colorlinks=true,
        linkcolor=blue]{hyperref}
        
\def \k {K\"ahler }
\def \s {Szeg\H{o} }
\def \d {\partial}
\def \y {{\tilde{y}}}
\def \dbar {\oo\partial}
\def \dd {\partial}
\def \dt {\frac{\partial}{\partial t}}
\def \eps {\varepsilon}
\DeclareMathOperator{\Tr}{Tr} 
\DeclareMathOperator{\Ric}{Ric}
\DeclareMathOperator{\Real}{Re}
\DeclareMathOperator{\Sym}{Sym}
\DeclareMathOperator{\vol}{vol}
\DeclareMathOperator{\Hess}{Hess}
\DeclareMathOperator{\Rm}{Rm}
\DeclareMathOperator{\osc}{osc}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\sn}{sn}
\DeclareMathOperator{\cs}{cs}
\DeclareMathOperator{\tn}{tn}

\newcommand{\oo}[1]{\overline{#1}}
\newcommand{\f}[2]{\frac{#1}{#2}}
\newtheorem{claim}{Claim}

\title{Math 147A Sample Midterm}
\date{}


\theoremstyle{definition} 
\newtheorem{definition}{Definition}[section]
\newtheorem{conjecture}{Conjecture}[section]
\newtheorem{example}{Example}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{remark}{Remark}[section]
\newtheorem{question}{Question}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem*{notation}{Notation}
\newtheorem{problem}{Problem}[section]

\begin{document}
\maketitle
\textbf{April 28, 2016}
The midterm will be 5 problems, some computational, some conceptual.  Feel free to bring a reasonable sized paper for notes and formulas. 

\begin{enumerate}
\item Compute the curvature of 
\begin{equation*}
\gamma(t) = (\cos^3(t),\sin^3(t)).
\end{equation*}
\item Compute the curvature and torsion (if it exists) of
\begin{equation*}
\gamma(t) = \left(\frac{4}{5}\cos(t),1-\sin(t),-\frac{3}{5}\cos(t)\right)
\end{equation*}
Show that $\gamma$ parametrizes a circle, find its center, radius and the plane in which it lies.
\item Show that a curve unit speed curve with $\kappa(s) >0$ for each $s\in [a,b]$ is a plane curve if and only if the torsion vanishes everywhere. 
\item State and prove the isoperimetric inequality, assuming Wirtinger's inequality and Green's theorem for area.
\item Show that a reparametrization by arc-length gives a unit-speed curve.
\item Derive the Frenet-Serret equation.
\item Let $\gamma:(a,b) \to \mathbb{R}^2$ be a regular plane curve and let $a \in \mathbb{R}^2$ such that $\gamma(t) \neq a$ for all $t$.  If there exists a $t_0 \in (a,b)$ such that 
\begin{equation*}
\|\gamma(t) - a\| \geq \|\gamma(t_0) -a\|
\end{equation*}
for all $t \in (a,b)$, show that the straight line joining the point $a$ with $\gamma(t_0)$ is the normal line of $\gamma$ at $t_0$.  The same is true if we reverse the inequality.  Draw a situation that illustrates both cases.
\item Let $\gamma:(a,b) \to \mathbb{R}^2$ be a regular plane curve and let $[\alpha,\beta] \subset (a,b)$ be such that $\gamma(\alpha) \neq \gamma(\beta)$.  Prove that there exists some $t_0 \in (\alpha,\beta)$ such that the tangent line of $\gamma$ at $t_0$ is parallel to the segment of the straight line joining $\gamma(\alpha)$ with $\gamma(\beta)$.  This is a curve version of mean value theorem.  Hint: Consider $f(t) = \det(\gamma(t), \gamma(\beta)-\gamma(\alpha))$.
\item Prove that a unit speed curve $\gamma:(a,b) \to \mathbb{R}^2$  is an arc of a circle if and only if all its normal lines pass through a given point.
\item Let $\gamma:(a,b) \to \mathbb{R}^3$ be a unit speed curve with positive curvature.  If $\|\gamma(s)\| =1$ for all $s$, i.e. $\gamma$ is a curve on a sphere, and it has constant torsion $\tau$, prove that there exists $b,c \in \mathbb{R}$ such that
\begin{equation*}
\kappa(s) = \frac{1}{b\cos(\tau s)+c\sin(\tau s)}.
\end{equation*}
\item Let $\gamma$ be a unit speed curve in $\mathbb{R}^3$ with constant curvature and zero torsion.  Show that $\gamma$ is a parametrization of a circle.
\end{enumerate}
\end{document}