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% Semester
\newcommand{\semester}{Fall 2003}
% Name of the class.
\newcommand{\class}{Ma235--Calculus Laboratory I}
% Title.
\newcommand{\labtitle}{Lab 3:  Clairaut's Theorem}

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\noindent \semester  \hfill \class
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\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.  If you haven't gotten
to that section yet in the Calculus course, look it up in the index.
Because most functions we work with are ``nice'', it is easy to
think that Clairaut's Theorem applies to {\it every} function.  One
purpose of the lab is to get you looking at functions of more than
one variable in Maple- feel free to do any appropriate graphing
to help you in this lab.  A lab checklist has been included on
the back of this sheet so that you can see what the grading
criteria will be.

\bigskip

\noindent Recall from your text\footnote{Stewart, 4th ed., page 902}

\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

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

\begin{itemize}

 \item Is $f$ continuous?  What are $f$'s domain and range?
Can looking at a contour plot help you guess if a function is
continuous?  Try comparing $f$ with something that is
continuous, like $g_1$ below, and with something that is not
continuous, like $g_2$ below:
$$
g_1(x,y) = \begin{cases} \frac{x^2y}{x^2+y^2} & (x,y)\neq 0 \\
0 & (x,y)=0 \end{cases} \quad
g_2(x,y) = \begin{cases} \frac{x^2}{x^2+y^2} & (x,y)\neq 0 \\
0 & (x,y)=0 \end{cases}
$$
 \item Show that $f_y(a,0)=a$ for all $a$ and $f_x(0,b)=-b$ for all $b$.
 \item Show that $f_{xy}(0,0) \neq f_{yx}(0,0)$.
 \item Discuss why Clairaut's Theorem does not apply here.
 \item Include in your write-up one substantive extension topic.

\end{itemize}




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