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\lhead{Math 4441 Homework 1}
\rhead{<your name>}


\begin{document}
\title{Math 4441 Homework 1}
\author{<your name>}
\date{September 9, 2022}
\maketitle
\vspace{5em}

\setcounter{section}{1}

\begin{enumerate}
	\item Suppose $A(t)$ is a differentiable family of orthogonal matrices.  Prove that the matrix $(A(t))^{-1}\,A'(t)$ must be skew-symmetric.
	
	\begin{solution}
	<your solution>
	\end{solution}
	
	\item Consider the following functions:
	\begin{align*}
	\vec{\alpha}(\theta) &= (\cos\theta,1-\cos\theta-\sin\theta,-\sin\theta)\\
	\vec{\beta}(\theta) &= (2\sin^2\theta,2\sin^2\theta\,\tan\theta,0)\\
	\vec{\gamma}(\theta) &= (\cos\theta,\cos^2\theta,\sin\theta).
	\end{align*}
	\begin{enumerate}[label=(\alph*)]
		\item Which of these functions give regular parametrized curves?
	
		\begin{solution}
		<your solution>
		\end{solution}
	
		\item For each curve which is regular at $\theta=\pi/4$, find an equation for the tangent line.
	
		\begin{solution}
		<your solution>
		\end{solution}
	\end{enumerate}
	
	\item Let $g\colon(0,\infty)\to(0,1)$ be given by $g(r)=r^2/(r^2+1)$.  Is this a reparametrization?
	
	\begin{solution}
	<your solution>
	\end{solution}
	
	\stepcounter{enumi}
	
	\item Let $\vec{\alpha}(t)$ be a regular curve.  Suppose there is a point $\vec{a}\in\mathbb{R}^3$ such that $\vec{\alpha}(t)-\vec{a}$ is orthogonal to $\vec{T}(t)$ for all $t$.  Prove that $\vec{\alpha}(t)$ lies on a sphere.
	
	\begin{solution}
	<your solution>
	\end{solution}
	
	\item Consider the right circular helix\label{hw1:helix}
	\[
	\vec{\alpha}(t) = (r\,\cos t,r\,\sin t, h\,t),
	\]
	where $r>0$ and $h>0$ are some constants.
	\begin{enumerate}[label=(\alph*)]
		\item Find the arc length of this helix for $0\leq t\leq 10$.
	
		\begin{solution}
		<your solution>
		\end{solution}
		
		\item Reparametrize the helix by arc length.
	
		\begin{solution}
		<your solution>
		\end{solution}
	\end{enumerate}
	
	\stepcounter{enumi}
	
	\stepcounter{enumi}
	
	\item Show that
	\[
	\vec{\alpha}(s) = \left(\dfrac{(1+s)^{3/2}}{3},\dfrac{(1-s)^{3/2}}{3},\dfrac{s}{\sqrt{2}}\right)
	\]
	is a unit speed curve and compute its Frenet-Serret apparatus.
	
	\begin{solution}
	<your solution>
	\end{solution}
	
	\stepcounter{enumi}
	
	\stepcounter{enumi}
	
	\item Prove that $\kappa\,\tau=-\langle\vec{T}',\vec{B}'\rangle$.
	
	\begin{solution}
	<your solution>
	\end{solution}
	
	\item Find the tangent, normal, and binormal spherical images of the helix in problem~\ref{hw1:helix}.
	
	\begin{solution}
	<your solution>
	\end{solution}
\end{enumerate}

\end{document}