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\title{Analysis II}
\author{Final Exam}
\date{May 31, 2019}
\maketitle

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\begin{enumerate}

\item Are the following statements true or false?  If true, give a
brief justification (in case the result is a named theorem or
otherwise known from class or from the homework, you may simply state
this).  If false, present a counter-example.

\begin{enumerate}

\item An arbitrary union of closed sets is closed.

\item An arbitrary union of open sets is open.

\item A subset of a compact set is compact.

\item A subset of $\R^n$ is compact if it is bounded and closed.

\item A convex set is connected.

\end{enumerate}
\hfill(2 pts.\ each)



\item Are the following statements true or false?  If true, give a
brief justification (in case the result is a named theorem or
otherwise known from class or from the homework, you may simply state
this).  If false, present a counter-example.

\begin{enumerate}

\item Let $f_n$ be a uniformly convergent sequence of continuous
functions on $I=[a,b]$.  Then
\[
  \lim_{n\to \infty} \int_a^b f_n(x) \, \d x
  = \int_a^b \lim_{n \to \infty} f_n(x) \, \d x \,.
\]

\item The statement from (a) with $I= [0,\infty)$.

\item Here and in the following, let $f \colon \R^n \to \R$.  Suppose
all partial derivatives of $f$ exist at some point $x \in \R^n$.  Then
all directional derivatives exist at $x$.

\item Suppose all directional derivatives exist at $x \in \R^n$.  Then
$f$ is differentiable at $x$.

\item Suppose $f$ is twice continuously differentiable.  Then the
Hessian of $f$ is symmetric.


\end{enumerate}
\hfill(2 pts.\ each)



\item Let $X$ be the vector space of all bounded sequences
$x = (x_1, x_2, \dots)$ endowed with the norm
\[
  \lVert x \rVert = \sup_{i \in \mathbb N} \, \lvert x_i \rvert \,.
\]

\begin{enumerate}
\item Show that $\lVert \cdot \rVert$ is indeed a norm.  

\item Show that the set
\[
  B = \{ x \in X \colon \lVert x \rVert \leq 1 \}
\]
is bounded and closed.

\item Show that $B$ is not compact.

\end{enumerate}
\hfill(5+5+5)




\item Find the power series expansion centered at $0$ for the function
\[
  f(x) = \frac1{x^2+4}
\]
and determine its radius of convergence.
\hfill (5)



\item Compute the derivative of the following maps.
\begin{enumerate}
\item $f \colon \R^n \to \R$ defined by
\[
  f(\boldsymbol v) = \boldsymbol v^T A \boldsymbol v
\]
where $A$ is a fixed $n \times n$ matrix, not necessarily symmetric.

\item $f \colon \R^{n \times n} \to \R$ defined by
\[
  f(A) = \boldsymbol v^T A \boldsymbol v
\]
where $\boldsymbol v \in \R^n$ is fixed.

\end{enumerate}
In each case, state the mapping properties (domain and range) of the
derivative explicitly.
\hfill(5+5)




\item Let $f \colon \R^2 \to \R$ be continuously differentiable.  Show
that $f$ cannot be injective.

\emph{Hint:} Implicit function theorem.
\hfill(10)




\item Maximize
\[
  f(x,y,z) = x y z
\]
subject to the constraint
\[
  g(x,y,z) = xy + xz + yz = 1 \,.
\]
\hfill(10)




\item Let
\[
  D = \{ (x,y) \colon x^2+y^2 \leq 1 \}
\]
denote the unit disk in $\R^2$.  Compute the integral
\[
  \int_D \cos (x^2 + y^2) \, \d x  \,.
\]
\emph{Hint:} Polar coordinates.
\hfill(5)



\item Compute the flux
\[
  \int_{\partial D} \boldsymbol F \cdot \boldsymbol n \, \d S \,,
\]
where $\boldsymbol n$ is the outward unit normal and
\[
  \boldsymbol F =
  \begin{pmatrix}
    z \, \cos x \, \sin y \\
    -z \, \cos x \, \sin y \\
    \tfrac12 \, z^2
  \end{pmatrix}
\]
through the surface of the unit ball in $\R^3$.

\emph{Hint:} Divergence theorem.

\end{enumerate}
\hfill(5)

\end{document}

