\documentclass[12pt]{article}
\usepackage{fullpage,amsmath,amsfonts,mathtools}

\def\d{{\mathrm d}}
\def\e{{\mathrm e}}
\def\R{{\mathbb R}}

\begin{document}

\title{Analysis II}
\author{Review Sheet for the Final Exam}
\date{Exam takes place May 31, 2019, 9:00-11:00, Lecture Hall Research
III}
\maketitle

The final exam will cover:
\begin{itemize}
\item Any topic from the midterm review problem set.
\item Differentiation in metric spaces.  Review, in particular,
Homework~7 Question~2.
\item Inverse and implicit function theorems.  Review the relevant
questions from the homework, in particular Homework~10 Question~2 and
Question~1 below.

\item Lagrange multipliers.  See Homework~11 and Question~2 below.

\item Iterated integrals.  See Homework~11.

\item Change-of-variable formula.  See Homework~12 and Question~3
below.

\item Divergence theorem.  See Homework~12 and Question 4 below.
\end{itemize}

\subsection*{Additional practice problems}

\begin{enumerate}



\item Let $A \subset \R^n$ be open, $f \colon A \to \R^n$ continuously
differentiable and injective, and $f'(x)$ invertible for every $x \in
A$.  Prove the following.
\begin{enumerate}
\item $f(A)$ is open,
\item $f^{-1}$ is differentiable on $f(A)$,
\item for every $B \subset A$ open, $f(B)$ is open.
\end{enumerate}

\emph{Hint:} inverse function theorem.



\item Let $p$ and $q$ be positive real numbers with
\[
  \frac1p+\frac1q = 1 \,.
\]
\begin{enumerate}
\item
Maximize
\[
  f(x,y) = x^{1/p} \, y^{1/q}
\]
subject to $x,y \geq 0$ and
\[
  g(x,y) = \frac{x}p + \frac{y}q = c
\]
with $c > 0$ fixed.

\item Conclude that, for $x,y \geq 0$ and $p,q$ as above,
\[
  x^{1/p} \, y^{1/q} \leq \frac{x}p + \frac{y}q \,.
\]
\end{enumerate}




\item Use the change of variables formula to compute the area of the
image of the unit square under the map
\[
  \begin{pmatrix}
    x \\ y
  \end{pmatrix}
  \mapsto
  \begin{pmatrix}
    x+y \\ y^2-x
  \end{pmatrix} \,.
\]



\item Compute the flux
\[
  \int_{\partial D} \boldsymbol F \cdot \boldsymbol n \, \d S \,,
\]
where $\boldsymbol n$ is the outward unit normal and
\[
  \boldsymbol F =
  \begin{pmatrix}
    x^2 \\ x^2 \, y \\ - x^2 \, z
  \end{pmatrix}
\]
through the surface of the tetrahedron $D$ with vertices $(0,0,0)$,
$(1,0,0)$, $(0,1,0)$, and $(0,0,1)$.

\begin{enumerate}
\item Use the divergence theorem.
\item Compute the flux integral directly by using a suitable
parametrization of the surface integral.
\end{enumerate}

\end{enumerate}

\end{document}

