\magnification 1800
\parskip 10pt
\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\a{{\bf a}}
\def\t{{\theta}}
\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}

Let $p: \R^2 \rightarrow \R^2$, $p(r, \t) = (r cos \t, r sin \t)$.

Then $Dp(r, \t) = \left(\matrix{cos \t & -rsin \t \cr sin \t & r cos \t }\right) $

Let $f: \R^2 \rightarrow \R$, $f(x, y) = x^2 + y^2 + 3y$

$(f \circ p)(r, \t) = f(p(r, \t)) = f(r cos \t, r sin \t) $

\rightline{$= r^2cos^2 \t + r^2 sin^2\t + 3rsin\t = r^2 + 3rsin\t $}

{\bf Note we used the function $p$ to convert $x^2 + y^2 + 3y$
into polar coordinates.}

\vskip 5pt
\hrule

Use the chain rule to calculate $D(f \circ p)(r, \t)$:

$f(x, y) = x^2 + y^2 + 3y$. 
Thus,
$Df(x, y) = \left(\matrix{2x & 2y +3}\right) $

$D(f \circ p)(r, \t) = (Df)(p(r, \t)) Dp(r, \t)$

\rightline{$= \left(\matrix{2rcos\t & 2rsin\t + 3}\right) 
\left(\matrix{cos \t & -rsin \t \cr sin \t & r cos \t }\right) =$}
 

{$\left(\matrix{2rcos^2\t + 2rsin^2\t + 3sin\t~ &
~-2r^2sin\t cos\t + 2r^2sin\t cos \t + 3r cos\t 
}\right) $}
\vskip 10pt

\rightline{$= \left(\matrix{2r+ 3sin\t ~&
~ 3r cos\t 
}\right) $}

\vskip 5pt
\hrule

$D(f \circ p)(r, \t) = (Df)(p(r, \t)) Dp(r, \t)$

\rightline{$= \left(\matrix{ {\partial f \over \partial x} & {\partial f \over \partial y} }\right) 
\left(\matrix{cos \t & -rsin \t \cr sin \t & r cos \t }\right) $}

\vskip 10pt

\centerline{$= \left(\matrix{ {\partial f \over \partial x}cos \t + {\partial f \over \partial y}  sin \t &
-{\partial f \over \partial x} rsin \t + {\partial f \over \partial y} r cos \t }\right) $}

\eject

\centerline{ $ {\partial \over \partial r} = cos \t {\partial  \over \partial x} + sin \t {\partial  \over \partial y} $}

\centerline{$ {\partial \over \partial \t} =
- rsin \t {\partial  \over \partial x}  + r cos \t {\partial  \over \partial y} $}

\vfill

$\left(\matrix{cos \t & sin \t & {\partial \over \partial r} \cr
-rsin \t  & r cos \t & {\partial \over \partial \t}
}\right) $
$\rightarrow$
$\left(\matrix{rcos \t sin \t & rsin^2 \t & rsin \t{\partial \over \partial r} \cr
-rcos \t sin \t  & r cos^2 \t & cos \t{\partial \over \partial \t}
}\right) $


$\rightarrow$$\left(\matrix{rcos \t sin \t & rsin^2 \t & rsin \t{\partial \over \partial r} \cr
0 & r & cos \t{\partial \over \partial \t} + rsin \t{\partial \over \partial r} 
}\right) $

$\rightarrow$$\left(\matrix{cos \t  & sin \t & {\partial \over \partial r} \cr
0 & 1 & {cos \t \over r}{\partial \over \partial \t} + sin \t{\partial \over \partial r} 
}\right) $

$\rightarrow$$\left(\matrix{cos \t  & 0 & {\partial \over \partial r}
-sin \t [{cos \t \over r}{\partial \over \partial \t} + sin \t{\partial \over \partial r} ] \cr
0 & 1 & {cos \t \over r}{\partial \over \partial \t} + sin \t{\partial \over \partial r} 
}\right) $

$\rightarrow$$\left(\matrix{cos \t  & 0 & {\partial \over \partial r}
-sin \t {cos \t \over r}{\partial \over \partial \t} - sin^2 \t{\partial \over \partial r}  \cr
0 & 1 & {cos \t \over r}{\partial \over \partial \t} + sin \t{\partial \over \partial r} 
}\right) $

$\rightarrow$$\left(\matrix{cos \t  & 0 & -sin \t {cos \t \over r}{\partial \over \partial \t}+ (1 - sin^2 \t){\partial \over \partial r}  \cr
0 & 1 & {cos \t \over r}{\partial \over \partial \t} + sin \t{\partial \over \partial r} 
}\right) $$\rightarrow$


 $\left(\matrix{cos \t  & 0 & -sin \t {cos \t \over r}{\partial \over \partial \t}+ cos^2 \t{\partial \over \partial r}  \cr
0 & 1 & {cos \t \over r}{\partial \over \partial \t} + sin \t{\partial \over \partial r} 
}\right) $$\rightarrow$
 $\left(\matrix{1  & 0 & {-sin \t  \over r}{\partial \over \partial \t}+ cos \t{\partial \over \partial r}  \cr
0 & 1 & {cos \t \over r} {\partial \over \partial \t} + sin \t{\partial \over \partial r} 
}\right) $

\vfill

\centerline{${\partial \over \partial x} = cos \t{\partial \over \partial r} - {sin \t  \over r}{\partial \over \partial \t}$}

\centerline{${\partial \over \partial y} = sin \t{\partial \over \partial r} + {cos \t \over r} {\partial \over \partial \t} 
 $}
\end

\left(\matrix{}\right)