%Partial Derivatives
%If you modify this file, please indicate here and in the footnote
%Math263D

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\begin{center}
{\Large
Partial Derivatives\footnote{Copyright \copyright 2002 Steve Chapin.
All rights reserved. Please address comments to young@math.ohiou.edu.}}
\end{center}

\begin{enumerate}
\item Enter the following commands:\\
      \verb&   syms x y&   \\
      \verb&   f = x*y*(x^2-y^2)/(x^2+y^2) &\\
      \verb&   fx = diff(f, x) &\\
      \verb&   fx = simplify(fx) & \\
      \verb&   subs(fx,{x, y},{0, y}) & \dotfill This is $f_x(0, y)$.

\item Define $f(0, 0) = 0$ and compute, by hand,
$$
    f_x(0, 0) = \lim_{h \rightarrow 0} \frac{f(h, 0) - f(0, 0)}{h}.
$$   
Why is it necessary to use the definition to compute 
    $f_x(0, 0)$?

\item Try: \verb& fy = simplify(diff(f, y)) &\\
      \verb& subs(fy, {x, y}, {x, 0}) & \dotfill This is $f_y(x, 0)$.\\
Then, compute $f_y(0, 0)$ by hand.

\item Compute, by hand,
$$
	f_{xy}(0, 0) = (f_x)_y(0, 0) = \lim_{k \rightarrow 0}
	\frac{f_x(0, k) - f_x(0, 0)}{k}
$$
	and $\qquad$
$$
	f_{yx}(0, 0) = \lim_{h \rightarrow 0}
        \frac{f_y(h, 0) - f_y(0, 0)}{h}
$$
What do you notice about $f_{xy}(0, 0)$ and $f_{yx}(0, 0)$?

\item Try: \verb& fxy = diff(fx,y) &\\
      \verb& fxy = simplify(fxy) &\\
      \verb& ezmesh(fxy) &\\
What do you notice about the graph of $f_{xy}$?

\item Either obtain a printout of the graph, or,
      carefully sketch it by hand,
      making sure to clearly label axes.

\item Using complete sentences and standard mathematical notation, write a brief report,
      showing your hand calculations and answering all the questions.

\end{enumerate}

\vfill
\noindent
\textsf{
The user is reminded of the definition of derivative and
encounters a situation where it must be used. The user
also encounters a situation where second derivatives are not
continuous and $f_{xy} \ne f_{yx}$.
}
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