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\begin{document}

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

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\begin{enumerate}

\item Let $f \colon \R^2 \to \R^2$ be given by
\[
  f (x, y) = (\e^x \, \cos y, \e^x \, \sin y) \,.
\]
\begin{enumerate}
\item Show that $f$ is locally invertible at every point $(x,y) \in
\R^2$.
\item Show that $f$ is not globally one-to-one.  Why does this not
contradict the inverse function theorem?
\end{enumerate}
\hfill(5+5)
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\item
Let $a,b \in \R^n$ fixed.  Consider arbitrary smooth curves $\gamma
\colon [0,1] \to \R^n$ that connect $a$ and $b$, i.e., satisfying
$\gamma(0)=a$ and $\gamma(1)=b$.  Recall that the length of the curve
is given by
\[
  L = \int_0^1 \lVert \gamma'(t) \rVert \, \d t \,.
\]

\begin{enumerate}
\item Show that the derivative of $L$ at a particular curve $\gamma$
is the linear map acting on an abitrary smooth curve
$\phi \colon [0,1] \to \R^n$ satisfying $\phi(0)=\phi(1)=0$
\[
  \d L(\gamma) \phi = \int_D \frac\d{\d t} \biggl(
  \frac{\gamma'(t)}{\lVert \gamma'(t) \rVert} \biggr) \cdot \phi \, \d
  t \,.
\]

\item Conclude from (a) that the length is minimized only if $\gamma$
is a straight line segment.

\end{enumerate}
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\item Minimize $f \colon \R^3 \to \R$, 
\[
  f(x) = x + y + z
\]
subject to
\begin{gather*}
  x^2 + y^2 + z^2 = 1 \,, \\
  x - y - z = 1 \,.
\end{gather*}
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\item Find the power series expansion for the function
\[
  f(x) = \frac{\ln(1+x)}x
\]
about the point $x=0$ and determine its radius of convergence.
\hfill(5)
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\item Recall that the Fourier series of a $2\pi$-periodic
complex-valued continuous function is given by
\[
  f(x) = \sum_{k=-\infty}^\infty f_k \, \e^{\i k x}
\]
where
\[
  f_k = \frac1{2\pi} \int_0^{2\pi} \e^{-\i k x} \, f(x) \, \d x \,.
\]
Show that for $2\pi$-periodic complex-valued functions $f$ and $g$,
\[
  \frac1{2\pi} \int_0^{2\pi} \overline{f(x)} \, g(x) \, \d x
  = \sum_{k=-\infty}^\infty \overline{f_k} \, g_k \,.
\]
(You may assume without further discussion that all integrals exist in
a suitable sense.)
\mbox{~~~}\hfill(5)
%\newpage

\item 
Convert to an integral in polar coordinates and evaluate:
\[
  \int_0^2 \int_0^{\sqrt{4-x^2}} \d y \, \d x \,.
\]
\hfill(10)
%\newpage

\item Determine whether or not
\[
  F = ( z/\cos^2 x, z, y + \tan x )
\]
is conservative on $(-\pi/2,\pi/2) \times \R^2$.  If $F$ is
conservative, find a potential function for $F$.
\hfill(5+5)
%\newpage

\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)
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%\mbox{}
%\newpage

\item Let
\[
  F(x, y, z) =  (y, x\,z, 1)
\]
be a vector field in $\R^3$.  Let $M$ be the upper hemisphere
\[
  x^2 + y^2 + z^2 = 1\,, \quad z \geq 0 \,. 
\]
Let $\gamma = \partial M$ be the unit circle in the $xy$-plane,
oriented counter-clockwise when viewed from above.
\begin{enumerate}
\item Compute the line integral
\[
  \int_\gamma F \cdot \d x
\]
directly.

\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}
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\end{document}
