\documentclass{math25}
% By Tom Braden
% Minor editing by Nils Barth

\title{Math 25b -- Problem Set 10, due Friday, May 1.}
\begin{document}

Reminder: the final exam is Thursday, May 14 at 9:15am in Science Center E.

\begin{enumerate}
\item CS, p.\ 523 \#\,3, \#\,5
\item We will use the winding number $w(\gamma)$ to give another proof
of the fundamental theorem of algebra.  Let $n>1$ and take a polynomial
\[f(z) = z^n + a_{n-1}z^{n-1} +\dots  + a_0\]
with complex coefficients.
Suppose that $f(z) \ne 0$ for all $z \in \C$.
  \begin{enumerate}
  \item Show that there is an $r_0>0$ so that for all $z\in \C$
  with $|z| = r_0$ we have $|z^n| > |f(z) - z^n|$.
  \item For each $r \ge 0$ define a closed path $\gamma_r\colon [0, 1]
  \to \C\setminus\set{0}$ by \[\gamma_r(t) = f(re^{(2\pi t)i}) = f(r\cos(2\pi t) +
  ir\sin(2\pi t)).\]  (Recall the definition of $e^{i\theta}$ from exercise 4,
  p.\ 307 of CS)

  We can identify $\C\setminus \set{0}$ with
  $\R^2 \setminus\set{(0,0)}.$  Show that the winding number
  $w(\gamma_r)$ doesn't depend on $r$.
  \item Show that $\gamma_{r_0}$ is homotopic to the path given by
  $\gamma(t) = e^{(2n\pi t)i}$, and that $w(\gamma) = n$.  Thus
  $w(\gamma_0) = n$, a contradiction.
  \end{enumerate}

\item Let $f(x, y)$ be a $\cC^2$ function on $\R^2$.  Suppose that $f$
is \emph{harmonic;} this means that the Laplacian
\[\Delta f = \frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}\]
is zero everywhere.
  \begin{enumerate}
  \item Define a path $\gamma_r(t) = (r\cos t, r\sin t)$ for $t\in [0, 2\pi]$,
  $r>0$. Define a function $g$ by
  \[g(r) = \frac{1}{2\pi}\int_0^{2\pi} f(\gamma_r(t))\,dt\]
  Differentiate under the integral sign to get
  \[g'(r) = \frac{1}{2\pi r}\int_{\gamma_r} - \frac{\partial f}{\partial y}\,dx +
  \frac{\partial f}{\partial x}\,dy\]
  \item Use Stokes' theorem to show that $g'(r) = 0$, and hence
  \[g(r) = f(0,0)\]
  Thus the average value of a harmonic function $f$ on a circle with center $p$
  is $f(p)$.
  \end{enumerate}
\pagebreak

\item (The volume form on a surface).  Let $S \subset \R^3$ be a
smooth surface defined by the equation $f = 0$.  Define a
vector field $\vec n\colon \R^3 \to \R^3$ by
$\vec n(p) = \grad f/\lnorm \grad f\rnorm$.  Then for $p\in S$, $\vec n(p)$
is a unit vector perpendicular to $T_pS$.

Let $dA$ be the $2$-form on $\R^3$ defined by $\vec n$; i.e. let
$dA = n_1\,dy\,dz - n_2\,dx\,dz + n_3\,dx\,dy$.
We define the surface area of $S$ to be  $|\int_S dA|$.
 \begin{enumerate}

  \item Let $S = S_r$ be the sphere of radius $r$ centered at the origin;
  use the equation $x^2 + y^2 + z^2 - r^2 = 0$ to define $S$.  Using the
  divergence theorem, show that the surface area of $S$ is $4\pi r^2$.

  \item Do the same calculation by parametrizing $S$ in spherical coordinates: let
  $h(\theta, \phi) = (\cos \theta \sin \phi, \sin \theta \sin \phi, \cos \phi)$
  and calculate $h^*dA$ explicitly.

  \item Show that the form $\omega = 1/r^2 dA$ (where $r^2 = x^2 + y^2 + z^2$)
  satisfies $d\omega = 0$.  Use the previous parts of this problem to show that there
  is no $1$-form $\eta$ with $d\eta = \omega$.  This form is often called
  the ``solid angle form'' in $\R^3$.
  \end{enumerate}

\end{enumerate}

\end{document}
