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\begin{document}

\title{Analysis II}
\author{Homework 10}
\date{Due in class Tuesday, April 30, 2019}
\maketitle

\begin{enumerate}


\item Recall the implicit function theorem: Let $X \subset \R^n$ and
$Y \in \R^m$ be open and $f \in C^1(X \times Y, \R^n)$.  Suppose there
exist $a \in X$ and $b \in Y$ such that $f(a,b)=0$ and $D_xf(a,b)$ is
invertible.  Then there exist an open neighborhood $A$ of $a$, an open
neighborhood $B$ of $b$, and a function $g \in C^1(B,A)$ such that
\[
  f(g(y),y) = 0
\]
for every $y \in B$, and 
\[
  Dg(y) = - D_xf(g(y),y)^{-1} \, D_y f(g(y),y) \,.
\]

\begin{enumerate}
\item Argue that if $f \in C^k(X \times Y, \R^n)$ for $k \geq 1$, then
$g \in C^k(B,A)$.

\emph{Hint:} Refer to Cramer's rule for the inverse matrix.
\item Show that if $r$ is a simple root of the polynomial
\[
  p(x) = a_0 + a_1 \, x + \dots + a_n \, x^n \,,
\]
then $r$ is a smooth, i.e.\ $C^\infty$ function of the coefficients
$a_0, \dots, a_n$.
\end{enumerate}

\item Consider the equation
\[
  \sqrt{x^2 + y^2 + 2 \, z^2} = \cos z
\]
near $(0,1,0)$.  Can you solve for $x$ in terms of $y$ and $z$?  For
$z$ in terms of $x$ and $y$?


\end{enumerate}

\end{document}
