\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: Homework 1}
\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{Due Thursday, April 14.}

Read Chapter 2
\begin{enumerate}
\item (2.1.1) Compute the curvature of $\gamma(t)=(t,\cosh t)$.
\item (2.2.1) Show that if $\gamma$ is a unit-speed plane curve,
\begin{equation*}
\dot{\mathbf{n}}_s = -\kappa_s \mathbf{t}.
\end{equation*}
\item (2.2.5) Let $\gamma$ be a regular plane curve and let $\lambda$ be a constant.  The parallel curve $\gamma^\lambda$ of $\gamma$ is defined by
\begin{equation*}
\gamma^\lambda (t) = \gamma(t) + \lambda\mathbf{n}_s(t).
\end{equation*}
Show that if $\lambda\kappa_s(t) \neq 1$ for all values of $t$, then $\gamma^\lambda$ is a regular curve and that its signed curvature is $\displaystyle \frac{\kappa_s}{|1-\lambda\kappa_s|}$.
\item (2.3.1) Compute $\kappa, \tau, \mathbf{t},\mathbf{n}, \mathbf{b}$ for 
\begin{equation*}
\gamma(t) = (\frac{1}{3}(1+t)^{\frac{3}{2}},\frac{1}{3}(1-t)^{\frac{3}{2}},\frac{t}{\sqrt{2}}).
\end{equation*}
and verify that the Frenet-Serret equations are satisfied.
\item (2.3.6) Let $(a_{ij})$ be a skew symmetric $3\times 3$ matrix.  Let $v_i$, $i=1,2,3$, be smooth vector functions of a parameter $s$ satisfying the differential equation
\begin{equation*}
\dot{v}_i = \sum_{j=1}^3 a_{ij}v_j
\end{equation*}
for all $i$, and suppose that for some $s_0$, the vectors $\{v_1(s_0), v_2(s_0), v_3(s_0)\}$ are orthonormal.  Show that the vectors $\{v_1,v_2,v_3\}$ are orthonormal for all $s$.
\item Give an example of two space curves with the same curvature but are not isometric to each other (there is no isometry between them).
\end{enumerate}

\end{document}