\magnification 1200
\parskip 15pt
\parindent 0pt
\hoffset -0.3truein
\hsize 7truein
\voffset -0.6truein
\vsize 10truein

\def\R{{\bf R}}
\def\x{{\bf x}}
\def\a{{\bf a}}
\def\S{\Sigma_{i=1}^n}
\def\Sk{\Sigma_{i, j=1}^n}
\def\Sjk{\Sigma_{i_1,..., i_k=1}^n}

\def\v{\vskip 8pt}



$p_2(\x) = f(\a) + Df(\a)(\x - \a) + {1 \over 2}(\x - \a)^T Hf(\a) (\x - \a)$

If $f:  \R^2 \rightarrow \R$,



$p_2(x, y) = f(a_1, a_2) + 
 \left(\matrix{ {\partial f \over \partial x}(a_1, a_2)  & 
{\partial f \over \partial y }(a_1, a_2)  }\right)
\left(\matrix{ x - a_1 \cr y - a_2}\right)  $ 

\v
\rightline{$+ {1 \over 2}(x - a_1, y -a_2)  \left(\matrix{ {\partial^2 f \over \partial^2 x}(a_1, a_2)  & 
{\partial^2 f \over \partial y \partial x}(a_1, a_2)    \cr {\partial^2 f \over \partial y \partial x}(a_1, a_2)   & {\partial^2 f \over \partial^2 y}(a_1, a_2)  }\right)
 \left(\matrix{ x - a_1 \cr y - a_2}\right)$}



$~~~~~~~~~~~ f(a_1, a_2) +  \left(\matrix{ f_{x} (a_1, a_2)  & 
f_{y}(a_1, a_2)   }\right)
\left(\matrix{ x - a_1 \cr y - a_2}\right)  $ 

\v
\rightline{$+ {1 \over 2}(x - a_1, y -a_2)  \left(\matrix{ f_{xx} (a_1, a_2)  & 
f_{xy}(a_1, a_2)    \cr f_{yx} (a_1, a_2)   & f_{yy}(a_1, a_2)  }\right)
 \left(\matrix{ x - a_1 \cr y - a_2}\right) $}


$~~~~~~~~~~~ = f(a_1, a_2) +  \left(\matrix{ f_{x} & 
f_{y} }\right)
\left(\matrix{ x - a_1 \cr y - a_2}\right)  $ 
{$+ {1 \over 2}(x - a_1, y -a_2)  \left(\matrix{ f_{xx} & 
f_{xy} \cr f_{yx} & f_{yy} }\right)
 \left(\matrix{ x - a_1 \cr y - a_2}\right) $}

\v

{$~~~~~~~~~~~ = f(a_1, a_2) +  f_{x} [x - a_1]   +
f_{y} [y - a_2]  $ 
$+ {1 \over 2}(x - a_1, y -a_2)  \left(\matrix{ f_{xx} [x - a_1]   +
f_{xy}[y - a_2]   \cr f_{yx} [x - a_1]   +
 f_{yy} [y - a_2]  }\right)$}


$~~~~~~~~~~~ = f(a_1, a_2) +  f_{x} [x - a_1]   +
f_{y} [y - a_2]  $ 

\v
\rightline{$+ {1 \over 2} (f_{xx} [x - a_1]^2   +
f_{xy}[x - a_1][y - a_2]   + f_{yx} [x - a_1][y - a_2]   +
 f_{yy} [y - a_2]^2) $}


$~~~~~~~~~~~ = f(a_1, a_2) +  f_{x} [x - a_1]   +
f_{y} [y - a_2]  $ 
{$+ {1 \over 2} (f_{xx} [x - a_1]^2   +
(f_{xy}+ f_{yx}) [x - a_1][y - a_2]  +
 f_{yy} [y - a_2]^2) $}

\v
$p_2(x, y) = f (a_1, a_2) +  f_{x}(a_1, a_2)  [x - a_1]   +
f_{y}(a_1, a_2)  [y - a_2]  $ 

\v
\rightline{~~$+ {1 \over 2} (f_{xx}(a_1, a_2)  [x - a_1]^2   +
(f_{xy}+ f_{yx})(a_1, a_2) [x - a_1][y - a_2]   +
 f_{yy}(a_1, a_2)  [y - a_2]^2) $}


\end