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% Semester
\newcommand{\semester}{Fall 2006}
% Name of the class.
\newcommand{\class}{M235--Calculus Laboratory I}
% Title.
\newcommand{\labtitle}{Lab 3:  Clairaut's Theorem}

\begin{document}


\noindent \semester  \hfill \class
\vspace{.1in}

\begin{center}
{\bf \large \em \labtitle}\footnote{Calculus On Manifolds by M.\ Spivak,
adaptation of exercise 24, page 29}
\end{center}
\vspace{.1in}

\noindent A famous theorem from Calculus is the theorem of Clairaut-
That mixed second partial derivatives are equal.  
Because most functions we work with are ``nice'', it is easy to
think that Clairaut's Theorem applies to {\it every} function-
In this lab, we will see that it does not.  Go through the
questions in this lab, using Maple for limits and graphs.
Try to incorporate the answers to the questions in a narrative
form.  The grading criteria are listed on our class website.

\begin{theorem}[Clairaut]
Suppose $f$ is defined on a disk $D$ that contains the point $(a,b)$.  If
the functions $f_{xy}$ and $f_{yx}$ are both continuous on $D$, then
$$f_{xy}(a,b) = f_{yx}(a,b).$$
\end{theorem}

\noindent Consider the function

$$f(x,y) = \begin{cases} \frac{xy(x^2-y^2)}{x^2+y^2} & (x,y)\neq 0 \\
0 & (x,y)=0 \end{cases} $$

\bigskip

\begin{enumerate}

 \item First get a feeling for what $f$ is by plotting it.  You
don't need to do a piecewise definition for the plot.
 \item Is $f$ continuous at the origin?
\begin{enumerate}
   \item Try using the definition (limits) in Maple.
   \item Try using the squeeze theorem.  (Hint:  You might look to
   see if $-|xy|\leq f(x,y)\leq |xy|$.  The absolute value
   function in Maple is \verb+abs()+)
\end{enumerate}
 \item Is $f$ differentiable?
\begin{enumerate}
  \item Compute $f_x$, $f_y$ using Maple (You might also use
  \verb+simplify+ here).
  \item Compute $f_x(0,0)$ and $f_y(0,0)$ by using the definition of the
  derivative.
  \item Show that $f_x$ and $f_y$ are continuous at the origin
 by seeing that $-2|y|\leq f_x(x,y) \leq 2|y|$, and similarly,
 $-2|x|\leq f_y(x,y)\leq 2|x|$.  Can you show these algebraically?
 Do they help prove continuity at the origin?  (How?)
 \end{enumerate}
 \item Show that $f_{xy}(0,0) \neq f_{yx}(0,0)$.
 (Note:  Compute $f_{xy}(0,0)$ using the {\it definition} of the
 derivative).
 \item Discuss why Clairaut's Theorem does not apply here.
\end{enumerate}

\end{document}

