\magnification 1800
\parskip 25pt
\parindent 0pt
\hoffset -0.3truein
\hsize 7truein

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

\def\R{{\bf R}}
\def\C{{\bf C}}
\def\x{{\bf x}}
\def\y{{\bf y}}
\def\e{{\bf e}}
\def\i{{\bf i}}
\def\j{{\bf j}}
\def\k{{\bf k}}
\def\ep{\epsilon}
\def\f{\vskip 10pt}
\def\u{\vskip -8pt}
\def\emph{{\it}}
\def\textit{{\it}}

Defn: $(\varphi, U)$ is  
a \emph{\it chart } or {\it coordinate neighborhood}   if $\varphi:U\rightarrow U^{\prime}$ is a homeomorphism,
where $U$ is open in $M$ and $U^{\prime}$ is open in $\R^n$.

Defn: Two charts, $(\varphi, U)$ and $(\psi, V)$ are {\it $C^{\infty}$ compatible} if the function $\varphi\psi^{-1}: \psi(U\cap V)\rightarrow\varphi(U\cap V)$ is a diffeomorphism.

Defn:  A \emph{\it (pre) atlas } or {\it differentiable or smooth structure} on $M$ is a collection of charts on $M$ satisfying the following two conditions:
\u\u
\hskip 30pt i.) the domains of the charts form an open cover of $M$
\u\u
 \hskip 30pt ii.)  Each pair of charts in the atlas is compatible.

 
Defn:  An atlas is a (maximal or complete) atlas if  it is maximal with respect to
properties i) and ii).


A {\it differential (or smooth or $C^{\infty}$) n-manifold} \emph{M}
is a topological n-manifold together with an atlas.

Theorem:  A (pre) atlas can be uniquely enlarged to a maximal atlas.
\end
\eject

Thm:  $f$ is differentiable at {\bf a} implies $f$ is continuous at {\bf a}.
\vfill

Thm:  Let $f:  \R^n \rightarrow \R^m$, $f = (f_1, ..., f_m)$. 
$f$ is differentiable at {\bf a} iff $f_i: \R^n \rightarrow \R$ is differentiable at {\bf a} for all $i = 1, ..., m$

Thm:  If $f$ is differentiable at {\bf a} then ${\partial f_i \over \partial x_j}$ exists for all $i, j$ and $Df({\bf a})$ = the Jacobian evaluated at {\bf a}.

Thm:  Let $f:  \R^n \rightarrow \R^m$, $f = (f_1, ..., f_m)$. If ${\partial f_i \over \partial x_j}$ exists and are continuous in a neighborhood of {\bf a} for all $i, j$, then $f$ is differentiable at {\bf a} 

Ex:  Is $f(x, y) = x^2y$ differentiable at (3, 1).   
\vskip 20pt

Find the equation of the tangent plane to $f(x, y) = x^2y$ at (3, 1).

Estimate f(3.1, .9)

\eject

2.4

Thm:   If $f, g:  \R^n \rightarrow \R^m$ is differentiable at {\bf a}, then $f + g$ is differentiable at {\bf a} and $D(f + g) = Df + Dg$.



Thm:  Let $c \in \R$.  If $f:  \R^n \rightarrow \R^m$ is differentiable at {\bf a}, then $cf$ is differentiable at {\bf a} and $D(cf) = cDf$.

Thm:   If $g:  \R^n \rightarrow \R^m$ is differentiable at {\bf a} and if
$f:  \R^m \rightarrow \R^k$ is differentiable at g({\bf a}), then $f \circ g$ is differentiable at {\bf a} and $D(f\circ g)({\bf a} ) = Df(g({\bf a}) ) Dg({\bf a} )$.



Note for the product and quotient rule, $f, g$ are real-valued functions, NOT vector valued.

Thm:   If $f, g:  \R^n \rightarrow \R$ is differentiable at {\bf a}, then $fg$ is differentiable at {\bf a} and $D(fg) = g( {\bf a})Df( {\bf a}) + f( {\bf a})Dg ( {\bf a})$.

Thm:   If $g({\bf a}) \not= 0$ and $f, g:  \R^n \rightarrow \R$ is differentiable at {\bf a}, then $f/g$ is differentiable at {\bf a} and $D(f/g) = 
{g( {\bf a})Df( {\bf a}) - f( {\bf a})Dg ( {\bf a}) \over g({\bf a})^2}$.






\end