%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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