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\title{Advanced Calculus and \\Methods of Mathematical Physics}
\author{Homework 4}
\date{Due in class Tuesday, March 3, 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 *(Kantorovitz, p.~78, Exercise 6.) Let $f \colon \R^k \to \R^m$
be defined by
\[
  f(x) = \sum_{i=1}^k (x_i, x_i^2, \dots, x_i^m) \,.
\]
Compute the derivative $\d f (x)$.


\item *(Kantorovitz, p.~78, Exercise 8.) Let $f \colon \R^2 \to \R$ be
differentiable on $\R^2 \setminus \{0\}$.  Let
\[
  h(r,\theta) = (r \, \cos \theta, r \, \sin \theta) 
\]
denote the change from polar to Cartesian coordinates and set
$g = f \circ h$.  Prove that, for $r>0$,
\[
  \lVert (\nabla f) \circ h \rVert^2
  = \biggl( \frac{\partial g}{\partial r} \biggr)^2
    + \biggl( \frac1{r} \, \frac{\partial g}{\partial \theta}
    \biggr)^2 \,.
\]

\item *(Kantorovitz, p.~79, Exercise 9.) Prove or disprove the
differentiability of the following function $f \colon \R^2 \to \R$ at
$(0,0)$.
\begin{enumerate}
\item $\displaystyle 
  f(x,y) =
  \begin{dcases}
    \frac{x^2-y^2}{\lVert (x,y) \rVert} & \text{when } (x,y) \neq (0,0) \,, \\
    0 & \text{when } (x,y) = (0,0) \,.
  \end{dcases}$

\item $f(x,y) = (xy)^{2/3}$
\end{enumerate}


\item Show that the function
\[
  u (x) = \ln \lVert x \rVert
\]
solves the Laplace equation
\[
  \Delta u = 0
\]
on $\R^2 \setminus \{0\}$.

\item Let $V$ and $W$ be normed vector spaces and $f \colon V \to W$
be continuously differentiable and homogeneous of degree $\alpha>0$,
i.e., for every $x \in V$ and $t>0$,
\[
  f(tx) = t^\alpha \, f(x) \,.
\]
Show that
\[
  \d f(x)x = \alpha \, f(x) \,.
\]

\item Let $V$ be a normed vector space, $E \subset V$ open, and $f
\colon E \to \R$ differentiable on $E$.  Suppose that $f$ has a local
maximum at some point $x \in E$.  Show that, for every $\v \in V$,
\[
  D_{\v} f(x) = 0 \,.
\]
\emph{Remark:}  If $V \in \R^n$, this implies that
\[
  \nabla f(x) \equiv (\partial_1f(x), \dots, \partial_nf(x)) = 0 \,.
\]


\item Let $V$ and $W$ be normed vector spaces and let
$A \colon V \to W$ and $B \colon W \to V$ be bounded linear operators.
Furthermore, suppose that
\[
  \lVert I - BA \rVert_{L(V)} < 1 \,.
\]
\begin{enumerate}
\item Show that $F \colon L(W,V) \to L(W,V)$, defined for every $X \in
L(W,V)$ by 
\[
  FX = X + B - BAX \,,
\]
is a contraction.
\item Conclude that $F$ has a fixed point $X^*$.  State an upper bound
for the operator norm $\lVert X^* \rVert_{L(W,V)}$.
\end{enumerate}

\end{enumerate}
\end{document}
