\magnification 1800
\parskip 10pt
\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\ep{\epsilon}
\def\f{\vskip 10pt}
\def\u{\vskip -5pt}

2.4 Higher order derivatives:

${\partial^2 f \over \partial x_{i_1} \partial x_{i_2}} = {\partial \over \partial x_{i_1} }(
{\partial \over \partial x_{i_2}}(f))$ 


Ex:  Let $f(x, y, z) =  x^2ln(yz)$

${\partial f \over \partial x}$ = \hfill
${\partial f \over \partial y}$ = \hfill
${\partial f \over \partial z}$ = \hfil~~~~~~~~

${\partial^2 f \over \partial x^2} = $ \hfill
${\partial^2 f \over \partial x \partial y} =$\hfill
${\partial^2 f \over \partial z \partial y} =$\hfil~~~~~~~~



${\partial^3 f \over \partial x^3} = $ \hfill
${\partial^3 f \over \partial x^2 \partial z} = $ \hfill
${\partial^3 f \over \partial x \partial y \partial z} =$\hfil~~~~~~~


\vskip 5pt
\hrule
Defn: Let  $V$ be a nonempty open subset of $R^n$, $f: V \rightarrow R^m$, 
$p \in {\N}$.
\u
i.)  $f$ is $C^p$ on $V$ is each partial derivative of order $k \leq p$ 
exists and is continuous on $V$.

ii.)  $f$ is $C^\infty$ on $V$ if  $f$ is $C^p$ on $V$ for all $p \in 
{\N}$ ($f$ is {\it smooth}).


Ex:  $g(x, y) = (x + y, x)$

\vfill
\vfill

\vfill

\vfill


Cor 1.7  If $f \in C^r$ on $U$, then   
${\partial^k f \over \partial x_{i_1} \partial x_{i_2} ...  \partial x_{i_k}
}$ = ${\partial^k f \over \partial x_{j_1} \partial x_{j_2} ...  \partial x_{j_k}
}$  where $(j_1, j_2, ... , j_k)$ is a permutation of $(i_1, i_2,  ... , i_k)$ 




\end