\magnification 1800
\parskip 16pt
\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 12pt}

$f(x, y) = ln({xy \over 3})$.

$\nabla f(x, y) = Df(x, y) = ({1 \over x}, {1 \over y})$ 

Hessian of $f$ = $Hf (x, y) =
\left(\matrix{ {\partial[\nabla f(x, y)] \over \partial x} \cr {\partial[\nabla f(x, y)] \over \partial y} }\right)=
 \left(\matrix{ {\partial^2 f \over \partial^2 x} & 
{\partial^2 f \over \partial y \partial x}   \cr {\partial^2 f \over \partial y \partial x}  & {\partial^2 f \over \partial^2 y} }\right)$

Thus  $Hf (x, y) =
\left(\matrix{-x^{-2} & 0 \cr 0 & -y^{-2}}\right)$

Let $\a = (3, 1)$.  Then $f(3, 1) = ln({(3)(1) \over 3}) = ln(1) = 0$.

 $\nabla f(3, 1) = Df(3, 1) = ({1 \over 3}, {1})$ 

 $Hf (3, 1) =
\left(\matrix{-3^{-2} & 0 \cr 0 & -1^{-2}}\right)
=
\left(\matrix{-{1 \over 9} & 0 \cr 0 & -1}\right)$



{\bf First order approximation of $f$ near $\a = (3, 1)$: }

Tangent plane to $f(x, y) = ln(x, y)$ at $\a = (3, 1)$ is 
\v
\centerline{$p_1(x, y) = f(\a) + Df(\a)(\x - \a)$}

$p_1(x, y)  = f(3, 1) + Df(3, 1) 
\left(\matrix{ x - 3 \cr y - 1}\right)$
$= 0 + ({1 \over 3}, 1) \left(\matrix{ x - 3 \cr y - 1}\right)$
\v
\rightline{
$= {1 \over 3}(x - 3) + y - 1 = {1 \over 3}x - 1 + y - 1$.}

\v
\centerline{Thus $p_1(x, y)  = {1 \over 3}x + y - 2 $.}
\eject

{\bf Second order approximation of $f$ near $\a = (3, 1)$: }

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

\v
\v

 $p_2(x, y) = {1 \over 3}x + y - 2 + {1 \over 2}(x-3, y-1) \left(\matrix{-{1 \over 9} & 0 \cr 0 & -1}\right)\left(\matrix{ x - 3 \cr y - 1}\right)$

$p_2(x, y) = {1 \over 3}x + y - 2 + {1 \over 2}(x-3, y-1) \left(\matrix{  -{1 \over 9} (x - 3) \cr -(y - 1)}\right)$


$p_2(x, y) = {1 \over 3}x + y - 2  - {1 \over 18} (x - 3)^2 - {1 \over 2} (y - 1)^2$
\end
\v
\hrule

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

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



$p_2(x, y) = 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) $}


$p_2(x, y) = f(a_1, a_2) +  \left(\matrix{ f_{x} & 
f_{y} }\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} & 
f_{xy} \cr f_{yx} & f_{yy} }\right)
 \left(\matrix{ x - a_1 \cr y - a_2}\right) $}


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

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


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

\v
\line{$+ {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) $}

$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]  $ 


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


\end