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

\begin{document}

\title{Advanced Calculus and \\Methods of Mathematical Physics}
\author{Final Exam}
\date{May 25, 9:00--11:00}
\maketitle

\begin{enumerate}

\item Let $A \subset \R^n$ be open, $f \in C^1(A,\R^n)$, and suppose
its Jacobian matrix $DF(x)$ is invertible for every $x \in A$.
\begin{enumerate}
\item 
Show that $B=f(A)$ is open, even if $f$ is not injective.
\item Give an example which shows that the assumption on the Jacobian
is necessary.
\end{enumerate}
\hfill(5+5)

\item Let $D \subset \R^n$ be a bounded domain with smooth boundary.
Let
\[
  \mathcal A
  = \{ u \in C^2(\overline D, \R) \colon u = 0 \text{ on } \partial D \} 
\]
and define a function $E \colon \mathcal A \to \R$ by
\[
  E(u) = \int_D \bigl( \tfrac12 \, \nabla u \cdot \nabla u + u \, f 
  \bigr) \, \d x 
\]
for some given function $f \in C(\overline D, \R)$.

\begin{enumerate}
\item Show that the derivative of $E$ at a point $u \in \mathcal A$ is
the linear map acting on arbitrary $v \in \mathcal A$ via
\[
  \d E(u) v = \int_D \bigl( -v \, \Delta u + v \, f 
  \bigr) \, \d x \,.
\]

\emph{Hint:} Divergence theorem.  And don't forget to argue why it is
possible to interchange integration and differentiation.

\item Use the result from part (a) to show that if $E$ has a local
minimum at some $u \in \mathcal A$, then $u$ solves Poisson's equation
\begin{gather*}
  \Delta u = f \quad \text{in } D \,, \\
  u = 0 \quad \text{on } \partial D \,.
\end{gather*}

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

\item Minimize $f \colon \R^5 \to \R$, 
\[
  f(x) = \lVert x \rVert^2
\]
subject to
\begin{gather*}
  x_1+2 \, x_2 + x_3 = 1 \,, \\
  x_3 - 2 \, x_4 + x_5 = 6 \,.
\end{gather*}
\hfill(10)

\item Find the power series expansion for the function
\[
  f(x) = \int_0^x \frac{\e^t - 1}t \, \d t
\]
and determine its radius of convergence.
\hfill(5)

\item Let $f$ be an odd (i.e., $f(x)=-f(-x)$), real-valued
$2\pi$-periodic function.  Show that its Fourier coefficients are also
odd (i.e., $f_k = -f_{-k}$) and purely imaginary.
\hfill(5)

\item Let $D \subset \R^2$ denote the diamond-shaped shaded region
sketched below.
\begin{center}
\includegraphics[width=0.4\textwidth]{sketch.pdf}
\end{center}
Compute
\[
  \int_D \frac{(x-y)^2}{(x+y+2)^2} \, \d x \, \d y \,.
\]
\emph{Hint:} use the change of variables $u=x+y$, $v=x-y$.
\hfill(10)

\item Let $D \subset \R^3$ be a simply connected domain.  Show that a
vector field $F \in C^1(D,\R^3)$ is conservative if and only if
$\nabla \times F = 0$.  \hfill(5+5)

\item Let
\[
  F =
  \biggl(
    \dfrac{-y}{x^2 + y^2}, \dfrac{x}{x^2+y^2}
  \biggr)
\]
be a vector field defined on $D=\R^2 \setminus \{0\}$.
\begin{enumerate}
\item Show that $F$ is locally conservative.
\item Compute the line integral
\[
  \int_\gamma F \cdot \d x
\]
(i) for any simple closed curve $\gamma$ encircling the origin, (ii)
for any simple closed curve $\gamma$ \emph{not} encircling the origin.

\item Set $z=x + \i y$.  Use the residue theorem to compute
\[
  \int_\gamma \frac1z \, \d z
\]
for any simple closed curve $\gamma$ encircling the origin in the
complex plane, (ii) for any simple closed curve $\gamma$ \emph{not}
encircling the origin.

\item State an identification between the real-variable computation
from (b) and the complex-variable computation from (c).

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

\item Let
\[
  F(x, y, z) = (y \, z, -x \,z, 1)
\]
be a vector field in $\R^3$.  Let $M$ be given by the surface of the
paraboloid
\[
  z = 4 - x^2 - y^2
\]
restricted to the first octant.  Let $\gamma_1$, $\gamma_2$, and
$\gamma_3$ be the part of the boundary of $M$ lying in the $xy$, $yz$,
and $xz$-planes, respectively and set $\gamma = \gamma_1 \cup \gamma_2
\cup \gamma_3$, in standard orientation.
\begin{enumerate}
\item Compute the line integral
\[
  \int_\gamma F \cdot \d x
\]
directly as line integrals along $\gamma_1$, $\gamma_2$ and $\gamma_3$
with suitable parameterizations.

\item Compute the same line integral via Stokes' theorem as a surface
integral over the capping surface $M$.
\end{enumerate}
\hfill(10+10)
\end{enumerate}

\end{document}
