\documentclass[11pt]{article}
\usepackage{amssymb}
\usepackage{epsfig}

\pagestyle{empty}
\setlength{\textwidth}{6.5in} 
\setlength{\textheight}{9in} 
\setlength{\headheight}{0in} 
\setlength{\evensidemargin}{0in}
\setlength{\oddsidemargin}{0in}
\setlength{\topmargin}{0in}
\newcommand{\qed}{}
\input{/home/jeff/tex/shortcuts.tex}

\begin{document}
\begin{center}
{\bf Math 366, Winter `03 \\ Homework 5}
\end{center}

\vspace*{.3in}
\noindent{{\bf Profs Personal Problems:}}
\vspace*{.2in}

\noindent{\bf 1.} Consider the set 
$C = \{(x,y,z)\in\R^3: x^2 - y^2 - z^2 = 0\}$
At which points does $C$ fail to be an embedded submanifold?  What is
the dimension and codimension of $C$?  Draw a picture of $C$.

\vspace*{.1in}
\noindent{\bf 2.} Consider the function $h:\R\to\R^2$ given by
$h(t) = (t^2,t^3)$ and the set $C = h(\R)\subset\R^2$.  
\begin{itemize}
\item At which points does $C$ fail to be a submanifold of $\R^2$?  
\item Describe the tangent space to $C$ at the point $(1,-1)$.
\item Draw a picture of $C$.
\end{itemize}

\vspace*{.1in}
\noindent{\bf 3.} Consider the following set
$$
M\eqdef \{(x,y,z,w)\in\R^4: x^2 - y^2 - zw = 1, zy+wx =0\}
$$
\begin{itemize}
\item  At which points does $M$ fail to be an
       embedded submanifold of $\R^4$.  
\item What is the dimension and codimension of $M$?  
\item Describe (i.e. give a basis for) the tangent space of $M$ at the point 
$(1,1,0,0)$.
\item Draw a picture of $M$.  Full color.  Stereo sound.
\end{itemize}

\vspace*{.2in}
\noindent{\bf 4.} (One more time) Consider the function 
$$
f(x,y) = (x^3 - 2x^2 y + y, y^3 - 2x^2).
$$
Observe that $f(1,1) = (0,-1)$ and verify that $f$ has a local inverse
satisfying $f^{-1}(0,-1) = (1,1)$.  Starting with a guess of 
$(x_0,y_0) = (1,1)$, compute two (increasingly better) approximations of the
point $(x,y) = f^{-1}(.1,-.8)$.
\end{document}
\end