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\begin{document}

\noindent Dror Bar-Natan: Classes: 2007-08: Math 1300 -- Geometry and Topology:

\vskip 10mm

\title{A Bit on Maxwell's Equations}

\maketitle

\par\noindent{\bf Prerequisites:}
\begin{itemize}
\item Poincar\'e's Lemma, which says that on $\bbR^n$, every closed form is
  exact. That is, if $d\omega=0$, then there exists $\eta$ with
  $d\eta=\omega$.
\item The Hodge star operator $\star$ which satisfies
  $\omega\wedge\star\omega=||\omega||^2dx_1\cdots dx_n$ and
  $\omega\wedge\star\eta=\eta\wedge\star\omega$ whenever $\omega$ and $\eta$
  are of the same degree.
\item Integration by parts: $\int\omega\wedge d\eta =
  -(-1)^{\deg\omega}\int(d\omega)\wedge\eta$ on domains that have no boundary.
\end{itemize}

\vskip 5mm \par\noindent{\bf The Action Principle: } The {\em Vector
Field} is a compactly supported 1-form $A$ on $\bbR^4$ which extremizes
the {\em action}
\[ \fbox{$\ds
  S_J(A) := \int_{\bbR^4}\frac12||dA||^2dtdxdydz + J\wedge A
$} \]
where the 3-form $J$ is the {\em charge-current}.

\vskip 5mm \par\noindent{\bf The Euler-Lagrange Equations} is this case
are $d\star dA=J$, meaning that there's no hope for a solution unless
$dJ=0$, and that we might as well (think Poincar\'e's Lemma!) change
variables to $F:=dA$.  We thus get
\[ \fbox{$\ds
  dJ=0
  \qquad dF=0
  \qquad d\star F=J
$} \]

\vskip 5mm \par\noindent{\bf These are the Maxwell equations! } Indeed,
writing $F=(E_xdxdt+E_ydydt+E_zdzdt)+(B_xdydz+B_ydzdx+B_zdxdy)$ and
$J=\rho dxdydz-j_xdydzdt-j_ydzdxdt-j_zdxdydt$, we find:

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\begin{center} \renewcommand{\arraystretch}{2}
  \begin{tabular}{|rcl|}
  \hline
    $\ds dJ=0 \Longrightarrow$ &
    $\ds \frac{\partial\rho}{\partial t} + \opdiv j = 0$ &
    \qquad ``conservation of charge''
  \\
    &
    $\ds \opdiv B=0$ &
    \qquad ``no magnetic monopoles''
  \\
    $\ds dF=0 \Longrightarrow$ &
    and &
  \\
    &
    $\ds \opcurl E = -\frac{\partial B}{\partial t}$ &
    \qquad that's how generators work!
  \\
    &
    $\ds \opdiv E = -\rho$ &
    \qquad ``electrostatics''
  \\
    $\ds d*F=J \Longrightarrow$ &
    and &
  \\
    &
    $\ds \opcurl B = -\frac{\partial E}{\partial t}+j$ &
    \qquad that's how electromagnets work!
  \\
  \hline
  \end{tabular}
\end{center}

\vskip 5mm \par\noindent{\bf Exercise: } Use the Lorentz metric to fix
the sign error!

\vskip 5mm \par\noindent{\footnotesize November 29, 2007;
http://www.math.toronto.edu/\~{}drorbn/classes/0708/GeomAndTop and
http://\linebreak[0]www.math.toronto.edu/\linebreak[0]\~{}drorbn/\linebreak[0]classes/\linebreak[0]0708/\linebreak[0]GeomAndTop/\linebreak[0]Maxwell.pdf}

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