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\title{Math 147A Midterm}
\date{April 28, 2016}


\theoremstyle{definition} 
\newtheorem{definition}{Definition}[section]
\newtheorem{conjecture}{Conjecture}[section]
\newtheorem{example}{Example}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{remark}{Remark}[section]
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\newtheorem*{notation}{Notation}
\newtheorem{problem}{Problem}[section]

\begin{document}
\maketitle
Choose 5 of the 6.  Each problem is worth 10 points for a total of 50 points.  No extra credit for doing more problems.  Please circle your choice of 5.

\begin{enumerate}
\item Compute the curvature and torsion of
\begin{equation*}
\gamma(t) = \left(2\cos\left(\frac{t}{\sqrt{5}}\right), 2\sin\left(\frac{t}{\sqrt{5}}\right),\frac{t}{\sqrt{5}}\right).
\end{equation*}
What curve is this.
\item Compute the curvature and torsion (if it exists) of
\begin{equation*}
\gamma(t) = \left(\alpha\cos(t),1-\sin(t),\beta\cos(t)\right),
\end{equation*}
where $\alpha^2 + \beta^2 = 1$.  Show that $\gamma$ parametrizes a circle, find its center, radius and the plane in which it lies.
\item Show that a reparametrization by arc-length of a regular curve gives a unit-speed curve.  Show this for the curve $\gamma(t) = (\cos(3t),\sin(3t))$.
\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\| \leq \|\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 $t_0$.  
\item Let $\gamma:(a,b) \to \mathbb{R}^3$ be a unit speed-curve such that $\kappa > 0$. Consider the Frenet-Serret frame $\{T,N,B\}$ centered at $\gamma(t_0)$ and use coordinates $\gamma(t) = (x(t),y(t),z(t))$ so that $T = (1,0,0)$, $N=(0,1,0)$ and $B= (0,0,1)$, and $\gamma(t_0) = (0,0,0)$.  Prove that
\begin{equation*}
\tau(t_0) = \lim_{s\to 0}\frac{6z(s)}{\kappa(t_0)s^3}.
\end{equation*}
\item Let $\gamma: (a,b) \to \mathbb{R}^3$ be a unit-speed curve whose curvature $\kappa$ and torsion $\tau$ do not vanish.  Prove that if $\gamma$ lies on a unit-sphere, then
\begin{equation*}
\frac{1}{\kappa^2} + \left(\frac{\kappa'}{\kappa^2\tau}\right)^2 = 1
\end{equation*}
Hint: For an orthonormal frame $\{T,N,B\}$, $\gamma = (\gamma\cdot T)T + (\gamma\cdot N)N + (\gamma\cdot B)B$.

\noindent The converse is true if we assume that $\kappa' \neq 0$.  Try to prove this for fun.
\end{enumerate}
\end{document}