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\title{Math 147A: Homework 1}
\date{}


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\begin{document}
\maketitle
\textbf{Due Thursday, March 7.}

Read sections 1.1 to 1.5
\begin{enumerate}
\item (1.1.8) Show that $\gamma(t) = (\cos^2(t) - \frac{1}{2}, \sin(t)\cos(t),\sin(t))$ is a parametrization of the curve of intersection of the circular cylinder of radius $\frac{1}{2}$ and the $z$-axis with the sphere of radius 1 and center $(-\tfrac{1}{2},0,0).$  This is called \textit{Viviani's Curve}.
\item (1.2.3) A plane curve is given (in polar coordinates) by 
\begin{equation*}
\gamma(t) = (r(t)\cos(t), r(t)\sin(t)),
\end{equation*}
where $r(t)$ is a smooth function of $t$.  Under what conditions is $\gamma$ regular? Find all functions $r(t)$ for which $\gamma$ is unit-speed.  Show that, if $\gamma$ is unit-speed, the image of $r(t)$ is a circle; what is its radius?
\item Compute the arc-length formula in polar coordinates (defined above).
\item (1.3.2) The \textit{cissoid of Diocles} is the curve whose equation in temrs of polar coordinates $(r,\theta)$ is 
\begin{equation*}
r=\sin\theta\tan\theta, \quad -\frac{\pi}{2} < \theta < \frac{\pi}{2}.
\end{equation*}
Note that this is a level curve description, versus the previous problems had a parametric curve description.  Write down a parametrization of the cissoid using $t$ as a parameter and show that
\begin{equation*}
\gamma(t) = \left(t^2,\frac{t^3}{\sqrt{1-t^2}} \right), \quad -1 < t < 1
\end{equation*}
is a reparametrization of it.
\item (1.4.2) Give an example to show that a reparametrization of a closed curve need not be closed.
\item (1.4.5) Suppose that a non-constant function $\gamma: \mathbb{R} \to \mathbb{R}$ is $T$-periodic for some $T\neq 0$. Show that there is a smallest positive $T_0$ such that $\gamma$ is $T_0$-periodic.
\end{enumerate}
\end{document}