\magnification 1400
\parskip 20pt
\parindent 0pt
\hoffset -0.3truein
\hsize 7truein

\voffset -0.3truein
\vsize 9.8truein

\def\emph{}

\def\S{\Sigma_{i=1}^n}
\def\Sm{\Sigma_{i=1}^m}

\def\s{{\sigma}}
\def\ep{\epsilon}
\def\f{\vskip 10pt}
\def\h{\hskip 10pt}
\def\u{\vskip -8pt}
\def\bh{\hfil \break}

\def\N{{\bf N}}
\def\S{\Sigma_{i=1}^n}

\def\0{{\bf 0}}
\def\Z{{\bf Z}}
\def\R{{\bf R}}
\def\C{{\bf C}}
\def\x{{\bf x}}
\def\a{{\bf a}}
\def\y{{\bf y}}
\def\e{{\bf e}}
\def\i{{\bf i}}
\def\j{{\bf j}}
\def\k{{\bf k}}
\def\p{\psi}
\def\f{\phi}
\def\s{\sigma}
\def\ep{\epsilon}
\def\f{\vskip 10pt}
\def\u{ \vskip -5pt }
\def\h{\vskip -5pt \hskip 20pt}
\def\l{\vskip 5pt
\hrule
\vskip -5pt}



\centerline{133 Practice Exam}

This is a  40 minute exam. 
%%You are not allowed to use any written aids like
%%notes during this exam. 
 Work two questions from this practice exam.
Indicate on the front page of your exam which problems you
want graded. You may use results about functions in euclidean space (chain
rule, implicit or inverse function theorems for example) or "big theorems"
(such as Sard’s theorem, the regular value theorem) provided you give the
precise statement.

Due 5pm Thursday July 24 (slide it under my office door, B1H MLH).



1.)  Suppose $M, N$ are smooth manifolds and $f:  M \rightarrow N$ is 
smooth.  Show that $X = \{ (p, f(p)) ~|~ p \in M \}$ is a regular 
submanifold of $M \times N$.  





2.)  Let $G:  (\-{ \pi \over 2}, {3 \pi \over 2}) \rightarrow R^2$, $G(t) = 
(cos(t), sin (2t))$.  Show that $G$ is a 1-1 immersion, but not an 
embedding.





3.)  Let G be a Lie group. Show that G admits a never-zero vector field. 

\end