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\def\d{{\mathrm d}}
\def\e{{\mathrm e}}
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\begin{document}

\title{Analysis II}
\author{Homework 9}
\date{Due in class Tuesday, April 23, 2019}
\maketitle

Recall the \emph{inverse function theorem}.  For $E \subset \R^n$ open
and $f \in C^1(E,\R^n)$, suppose that $Df(a) \equiv f'(a)$ is
invertible for some $a \in E$.\footnote{Both notations, $Df$ and
$f'$, are commonly used to denote the derivative of a function between
normed vector spaces.  Here, in the context of the inverse and the
implicit function theorems, the $D$-notation has some advantages,
where we understand that $Df^{-1}(x)$ is the derivative of the inverse
map and $Df(x)^{-1}$ is the inverse of the derivative map at $x$.}  Then
\begin{itemize}
\item[(i)] There exist open sets $U,V \subset \R^n$ with $a \in U$
such that $f \colon U \to V$ is bijective,
\item[(ii)] $f^{-1} \in C^1 (V,U)$ and
\begin{equation}
  \tag{*}
  Df^{-1} (y) = Df(x)^{-1} 
\end{equation}
with $y = f(x)$.
\end{itemize}

\begin{enumerate}

\item (From Rudin, Exercise 9.17.) Let $f \colon \R^2 \to \R^2$ be
defined by
\[
  f(x) =
  \begin{pmatrix}
    \e^{x_1} \, \cos x_2 \\ \e^{x_1} \, \sin x_2
  \end{pmatrix} \,.
\]
\begin{enumerate}
\item What is the range of $f$?
\item Show that the \emph{Jacobian determinant}, $\operatorname{det}
Df(x)$, is non-zero for every $x \in \R^2$.  Thus every point in
$\R^2$ has a neighborhood in which $f$ is one-to-one.  Nevertheless,
$f$ is not one-to-one on $\R^2$.

\item Put $a = (0, \pi/3)$ and $b = f(a)$.  Find an explicit
formula for $f^{-1}$, compute $Df(a)$ and $Df^{-1}(b)$, and verify
formula (*) above.

\item What are the images under $f$ of lines parallel to the
coordinate axes?

\end{enumerate}

\item (From Rudin, Exercise 9.18.) Answer analogous questions for the
mapping defined by
\[
  f(x) =
  \begin{pmatrix}
    x_1^2 - x_2^2 \\ 2 \, x_1 \, x_2
  \end{pmatrix} \,.
\]

\item (From Rudin, Exercise 9.21.) Define $f \colon \R^2 \to \R$ by
\[
  f(x,y) = 2 \, x^3 - 3 \, x^2 + 2 \, y^3 + 3 \, y^2 \,.
\]
\begin{enumerate}
\item Find the four points in $\R^2$ at which the gradient $\nabla f =
(\partial_x f, \partial_y f)$ is zero.  Show that $f$ has exactly one
local maximum and one local minimum in $\R^2$.

\item Let
\[
  S = \{ (x,y) \in \R^2 \colon f(x,y) = 0 \} \,.
\]
Find these points of $S$ which have no neighborhoods in which the
equation $f(x,y)=0$ can be solved for $y$ in terms of $x$ (or for $x$
in terms of $y$).  Describe $S$ as precisely as you can.
\end{enumerate}


\item (From the \emph{inverse} to the \emph{implicit function
theorem}.)  Let $X \subset \R^n$ and $Y \in \R^m$ be open and $f \in
C^1(X \times Y, \R^n)$.  Define $F \colon X \times Y \to \R^n \times
\R^m$ as
\[
  F(x,y)
  =
  \begin{pmatrix}
    f(x,y) \\ y
  \end{pmatrix} \,.
\]
\begin{enumerate}
\item Show that $DF$ has the block-matrix structure\footnote{Here, we
identify the notion of derivative as a linear map with its matrix
representation with respect to the standard basis.  We make no attempt
to use distinct notation for these two technically separate concepts
because there is no useful general convention and, in practice, the
precise meaning can be inferred from context.}
\[
  DF =
  \begin{pmatrix}
    D_x f & D_y f \\
    0 & I_m
  \end{pmatrix} \,,
\]
where $I_m$ is the $m \times m$-identity matrix and $D_xf$ and $D_yf$
denote the generalized partial derivatives
\[
  D_x f =
  \begin{pmatrix}
    \partial_{x_1} f_1 & \dots & \partial_{x_n} f_1 \\
    \vdots & & \vdots \\
    \partial_{x_1} f_n & \dots & \partial_{x_n} f_n
  \end{pmatrix}
  \quad \text{and} \quad
  D_y f =
  \begin{pmatrix}
    \partial_{y_1} f_1 & \dots & \partial_{y_m} f_1 \\
    \vdots & & \vdots \\
    \partial_{y_1} f_n & \dots & \partial_{y_m} f_n
  \end{pmatrix} \,.
\]

\item Conclude that $DF$ is invertible if and only if $D_xf$ is
invertible.

\item Suppose that there exist $a \in X$ and $b \in Y$ such that
$f(a,b)=0$ and $D_xf(a,b)$ is invertible.  Argue that there exists an
open neighborhood $V$ of $(0,b)$ and an open neighborhood $U$ of
$(a,b)$ such that $F \colon U \to V$ is bijective with $F^{-1} \in
C^1(V,U)$.

\item Conclude that there exist an open neighborhood $A$ of $a$,
an open neighborhood $B$ of $b$, and a function $g \colon B \to A$
such that
\[
  f(g(y),y) = 0
\]
for every $y \in B$.

\item Finally, argue that $g \in C^1(B,A)$ with
\[
  Dg(y) = - D_xf(g(y),y)^{-1} \, D_y f(g(y),y) \,.
\]




\end{enumerate}

\end{enumerate}

\end{document}
