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\title{Perspectives of Mathematics I}
\author{Homework 2}
\date{due November 5, 2011}
\maketitle

\begin{enumerate}

\item Consider a mechanical setup according the the following figure
(from T.F. Tokieda, \textit{Mechanical ideas in geometry}, The
American Mathematical Monthly \textbf{105} (1998), 697--703) where the
strings can slide freely through the holes.

\begin{center}
\includegraphics{table.pdf}
\end{center}

\begin{enumerate}
\item Argue that, in equilibrium, the length of string above the table
is minimal.

\item Argue that the angles around the knot are all $120^\circ$.
\end{enumerate}
Remark: the location of the knot is called the \emph{short-center} or
\emph{brachycenter} of the triangle $ABC$.

\item (From M. Levi, \textit{The Mathematical Mechanic}, Princeton
University Press, 2009, p.\ 83.)  Prove the inequality
\[
  \frac1{\frac1{a+b}+\frac1{c+d}} 
  \geq \frac1{\frac1a+\frac1c} + \frac1{\frac1b+\frac1d}
\]
for positive real numbers $a,b,c,d$.  Hint: consider the following
circuit from Levi's book and change some resistances:

\begin{center}
\includegraphics{network.pdf}
\end{center}


\item Consider the flow of a planar vector field $(u,v)$ and denote
points of the plane by $(x,y)$.
\begin{enumerate}
\item  A stick of infinitesimal length, initially parallel to the
$y$-axis, is carried along by the flow.  Show that its initial angular
velocity is given by $-\partial u/\partial y$.

\item Conclude that the planar curl of the vector field, $\partial
v/\partial x - \partial u/\partial y$, is twice the average angular
velocity of the flow.
\end{enumerate}
(Without loss of generality, you may translate the point of interest
into the origin.)

\item 
\begin{enumerate}
\item Draw a careful sketch of the vector field the flow lines
corresponding to the complex-valued function
\[
  f(z) = \frac1{z^2} \,.
\]

\item What are flux and circulation about a small loop enclosing the
origin?  Argue by symmetry.

\item What are flux and circulation about a small loop enclosing the
origin for the vector field corresponding to $f(z) = z^{-n}$ for
integers $n\geq 3$?  (You may argue by symmetry, or show a
computation.) 

\end{enumerate}

\item Show, by looking at the flow of the vector field corresponding
to the function
\[
  f(z) = \frac{\cot (\pi z)}{2 \, z^4}
\]
that
\[
  \frac{\pi^4}{90}
  = \frac1{1^4} + \frac1{2^4} + \frac1{3^4} + \dots \,.
\]



\end{enumerate}


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