\magnification 1800
\parskip 15pt
\parindent 0pt
\hoffset -0.3truein
\hsize 7truein

\voffset -0.3truein
\vsize 10.7truein



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

\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\ep{\epsilon}
\def\f{\vskip 10pt}
\def\u{\vskip -5pt}

Defn:  Suppose $f: W \rightarrow N$ where $W \subset M$ and $N$ are smooth manifolds.  
$f$ is {\it smooth} if for all $p \in W$, $\exists$
 charts $(\phi, U)$
 and $(\p, V)$ and such that 
$p \in U$, $f(p) \in V$, $f(U) \subset V$ and $\p \circ f \circ \phi^{-1}$ is smooth.

\vfill

0.)  If  $f: W \rightarrow N$ is smooth, then given charts $(\phi, U)$ and $(\p, V)$  such that  $f(U) \subset V$, then $\p \circ f \circ \phi^{-1}$ is smooth.

1.)  If $f: W \rightarrow N$ is smooth, then $f$ is continuous.

2.) If  $f: W \rightarrow N$ is smooth, $V^{open} \subset W$, then $f: V \rightarrow N$ is smooth.



3.) $f: W \rightarrow N$, $W = \cup U_\alpha^{open}$, and $f: U_\alpha 
\rightarrow N$ smooth for all $\alpha$, then $f: W \rightarrow N$ is  smooth.

4.) If $f, g$ smooth, $f \circ g$ is smooth.

Defn:  $f: M \rightarrow N$ is a {\it diffeomorphism} if $f$ is a homeomorphism and if $f$ and 
$f^{-1}$ are smooth.  $M$ and $N$ are {\it diffeomorphic} if there exists a 
diffeomorphism $f: M \rightarrow N$.

Prop 1.2.9: Let $M$ and $N$ be smooth manifolds, and let $\{U_{\alpha},\phi_{\alpha}\}$
be the atlas for $N$. \ Suppose $f:M\rightarrow N$ is a diffeomorphism. \
Then $\{f^{-1}(U_{\alpha}),\phi_{\alpha}\circ f\}$ is the atlas for $M$.
 


\end