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\begin{document}

\title{Advanced Calculus and \\Methods of Mathematical Physics}
\author{Homework 3}
\date{Due in class Tuesday, February 25, 2020}
\maketitle

\emph{Note:} Assignments marked (*) will not be graded.  Do not turn
them in.  However, they will be discussed in the tutorial and example
solutions will be made available.

\begin{enumerate}

\item *Find a power series representation centered at $2$ for
\[
  \frac1{4\,x - x^2 - 3} \,.
\]
What is the radius of convergence?

\item *Let $X, Y$ be metric spaces, $E \subset X$ compact, and
$f \colon X \to Y$ continuous.  Show that $f(E)$ is a compact subset
of $Y$.

\item Recall the operator norm for maps $A$ between normed vector
spaces $X$ and $Y$, defined by
\[
  \lVert A \rVert
  = \sup_{x \neq 0} \frac{\lVert Ax \rVert}{\lVert x \rVert} \,.
\]
\begin{enumerate}
\item Verify that the operator norm is indeed a norm.
\item Let $X$, $Y$, and $Z$ be normed vector spaces, and let $B \colon
X \to Y$ and $A \colon Y \to Z$
be linear maps.  Show that
\[
  \lVert AB \rVert \leq \lVert A \rVert \, \lVert B \rVert \,.
\]
\end{enumerate}
\hfill(4)
  

\item \emph{Disconcerting Example 1.} Consider the function $f \colon
\R^2 \to \R$ defined by
\[
  f(x,y) =
  \begin{dcases}
    \frac{x^3}{x^2+y^2} & \text{when } (x,y) \neq (0,0) \,, \\
    0 & \text{when } (x,y) = (0,0) \,.
  \end{dcases}
\]
\begin{enumerate}
\item Compute the directional derivative $D_{\v}f (0,0)$ for every
$\v=(a,b) \in \R^2$.  Is $\v \mapsto D_{\v}f (0,0)$ linear?
\item Show that $f$ is not differentiable at the origin.
\end{enumerate}
\hfill(2)


\item \emph{Disconcerting Example 2.} Consider the function $f \colon
\R^2 \to \R$ defined by
\[
  f(x,y) =
  \begin{dcases}
    \frac{x^3\,y}{x^6+y^2} & \text{when } (x,y) \neq (0,0) \,, \\
    0 & \text{when } (x,y) = (0,0) \,.
  \end{dcases}
\]
\begin{enumerate}
\item Compute the directional derivative $D_{\v}f (0,0)$ for every
$\v=(a,b) \in \R^2$.  Is $\v \mapsto D_{\v}f (0,0)$ linear?
\item Show that $f$ is not continuous at the origin.
\end{enumerate}
\hfill(2)

\item \emph{Disconcerting Example 3.} Consider the function $f \colon
\R^2 \to \R$ defined by
\[
  f(x,y) =
  \begin{dcases}
    \frac{x^2\,y}{x^4+y^2} \, \sqrt{x^2+y^2} &
    \text{when } (x,y) \neq (0,0) \,, \\
    0 & \text{when } (x,y) = (0,0) \,.
  \end{dcases}
\]
\begin{enumerate}
\item Compute the directional derivative $D_{\v}f (0,0)$ for every
$\v=(a,b) \in \R^2$.  Is $\v \mapsto D_{\v}f (0,0)$ linear?
\item Show that $f$ is not differentiable at the origin.
\end{enumerate}
\hfill(2)

\end{enumerate}

\end{document}
