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\newcommand{\classnumber}{Math 20C -- Swanson }
\newcommand{\classiteration}{Fall 2020}

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\begin{itemize}
  \item The graded part of the homework is on WebAssign.
  \item The problems below are also assigned and you are responsible for doing them,
    but they will not be collected or graded.
\end{itemize}

\newpage

\begin{questions}
  \question We will study an example where mixed partial derivatives are
    \textit{not} equal at a particular point. This is based on problem
      $3.1.32$ in the book. Consider
        \[ f(x, y) =
            \begin{cases}
              \frac{xy(x^2 - y^2)}{x^2 + y^2} & \text{ if }(x, y) \neq (0, 0) \\
              0 & \text{ if }(x, y) = (0, 0).
            \end{cases} \]
    \begin{enumerate}[(a)]
      \item Compute $\frac{\partial f}{\partial x}$,
        $\frac{\partial f}{\partial y}$, and $\frac{\partial^2 f}{\partial y\, \partial x}$.
        For $(x, y) \neq (0, 0)$, the two mixed partials should be equal.
      \item Evaluate the limit as $(x, y) \to (0, 0)$ for each of
        $\frac{\partial f}{\partial x}$, $\frac{\partial f}{\partial y}$, and
        $\frac{\partial^2 f}{\partial y\, \partial x}$.
      \item It turns out that $\frac{\partial^2 f}{\partial x\, \partial y}(0, 0)$
        and $\frac{\partial^2 f}{\partial y\, \partial x}(0, 0)$ are both defined,
        but are not equal! Compute them both by the following
        procedure, which in general computes
          \[ \frac{\partial^2 f}{\partial x\, \partial y}
              = \left.\left(\frac{\partial}{\partial x}
                 \left(\left.\frac{\partial f}{\partial y}\right|_{y=b}\right)\right)\right|_{x=a} \]
        
        \begin{enumerate}[(i)]
          \item First compute the function $\frac{\partial f}{\partial y}$.
          \item In computing $\frac{\partial}{\partial x}$ of a function, we hold
            $y$ constant and then take the $x$-derivative. Hold
            $y$ constant by setting $y=b$ and simplifying, which evaluates
            $\frac{\partial f}{\partial y}(x, b)$.
          \item Compute the derivative $\frac{\partial}{\partial x}$ of
            $\frac{\partial f}{\partial y}(x, b)$ and evaluate at $x=a$.
        \end{enumerate}
        
        (To compute the other mixed partial, swap the roles of $x$ and $y$
        in the above procedure.)
    \end{enumerate}
    
    \begin{solution}
      
    \end{solution}
    \newpage
\end{questions}
\end{document}