\documentclass[12pt]{exam}

%\printanswers

%Packages Used%
\usepackage{amsmath} 
\usepackage{amssymb}
\usepackage{latexsym}
\usepackage{amsfonts}
\usepackage{fullpage}


\usepackage[pagebackref,hypertexnames=false, colorlinks, citecolor=black, linkcolor=blue, urlcolor=red]{hyperref}

\setlength{\textwidth}{6.5in}
\setlength{\textheight}{9.5in}
\setlength{\topmargin}{0in}
\setlength{\evensidemargin}{0in}
\setlength{\oddsidemargin}{0in}
\setlength{\headheight}{0in}
\setlength{\headsep}{0in}
\setlength{\abovedisplayskip}{3mm}
\setlength{\belowdisplayskip}{3mm}
\setlength{\abovedisplayshortskip}{0mm}
\setlength{\belowdisplayshortskip}{2mm}
\setlength{\baselineskip}{12pt}
\setlength{\normalbaselineskip}{12pt}


\newcommand{\ci}[1]{_{ {}_{\scriptstyle #1}}}


\newcommand{\norm}[1]{\ensuremath{\left\|#1\right\|}}
\newcommand{\abs}[1]{\ensuremath{\left\vert#1\right\vert}}
\newcommand{\ip}[2]{\ensuremath{\left\langle#1,#2\right\rangle}}
\newcommand{\p}{\ensuremath{\partial}}
\newcommand{\pr}{\mathcal{P}}

\newcommand{\pbar}{\ensuremath{\bar{\partial}}}
\newcommand{\db}{\overline\partial}
\newcommand{\D}{\mathbb{D}}
\newcommand{\B}{\mathbb{B}}
\newcommand{\Sp}{\mathbb{S}}
\newcommand{\T}{\mathbb{T}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\scrH}{\mathcal{H}}
\newcommand{\scrL}{\mathcal{L}}
\newcommand{\td}{\widetilde\Delta}


\newcommand{\La}{\langle }
\newcommand{\Ra}{\rangle }
\newcommand{\rk}{\operatorname{rk}}
\newcommand{\card}{\operatorname{card}}
\newcommand{\ran}{\operatorname{Ran}}
\newcommand{\osc}{\operatorname{OSC}}
\newcommand{\im}{\operatorname{Im}}
\newcommand{\re}{\operatorname{Re}}
\newcommand{\tr}{\operatorname{tr}}
\newcommand{\vf}{\varphi}
\newcommand{\f}[2]{\ensuremath{\frac{#1}{#2}}}




\normalbaselines
\pagestyle{empty}
\raggedbottom





\begin{document}
\begin{center}
Homework 7
\end{center}

\noindent
Due Date: April 25, 2013.\\

\noindent
Page 212 of Rosenlicht: 11, 17.\\
Page 244 of Rosenlicht: 1, 7(a),(b), 24.\\

\begin{center}
Additional Problems:
\end{center}

\noindent
1. Let $M_n=\{A: A \textnormal{ is a } n\times n \textnormal{ matrix with entries in } \R\}$.  Define 
$$
f(A)=A^2
$$
\begin{itemize}
\item[(a)] Compute $Df(A)(H)$;
\item[(b)] When $n=2$ compute $f'(A)$.
\end{itemize}

\noindent
2.  Let $f:\mathbb{R}^n\to\R$.  For $x\in\R^n$, the limit
$$
\lim_{t\to 0}\frac{f(a+tx)-f(a)}{t}
$$
if it exists, is denoted by $D_x f(a)$ and is called the directional derivative of $f$ at $a$ in the direction $x$. 
\begin{itemize}
\item[(a)] For $e_i$ an element in the standard basis, show that $\partial_i f=D_{e_i}f$;
\item[(b)] Show that $D_{sx}f(a)=sD_x f(a)$;
\item[(c)] If $f$ is differentiable at $a$, show that $D_x f(a)=Df(a)x$.
\end{itemize}

\noindent
3.  Let $A\subset\R^n$ be an open set and $f:A\subset\R^n\to \R^n$ be a one-to-one continuously differentiable function such that $\det f'(x)\neq 0$ for all $x\in A$.  Show that $f(A)$ is an open set and $f^{-1}:f(A)\to A$ is differentiable.\\
%Marsden page 242, #3

\noindent
4.  Let $f:\R^2\to\R^2$ be smooth and suppose that
$$
\partial_x f_1=\partial_y f_2\quad \partial_y f_1=-\partial_x f_2.
$$
Show that $\det f'(x)=0$ if and only if $f'(x)=0$.\\
%Marsden page 246, #10

\noindent
5.  A function $f:\R^n\to\R$ is called homogeneous of degree $m$ if $f(tx)=t^mf(x)$ for all $x\in\R^n$ and $t\in\R$.  If $f$ is differentiable, show that for $x\in\R^n$ that
$$
Df(x)x=mf(x)\quad\textnormal{ i.e.,} \sum_{j=1}^n x_j \partial_j f=mf(x).
$$
Hint: Set $g(t)=f(tx)$ and compute $g'(1)$.

\end{document}