View source for Maxwell's equations

{{mytexdefs}}Our aim is to solve numerically Maxwell's equations for macroscopic
media:
\begin{align} 
\Curl{\vec{h}} - \partial_t \vec{d} &= \vec{j} , \label{eq:ampere}\\
\Curl{\vec{e}} + \partial_t \vec{b} &= 0 , \label{eq:faraday}\\
\Div{\vec{b}} &= 0 , \label{eq:gaussm}\\
\Div{\vec{d}} &= \rho . \label{eq:gausse}
\end{align}
Equations \eqref{eq:ampere}, \eqref{eq:faraday}, \eqref{eq:gaussm} and
\eqref{eq:gausse} are respectively the generalized Ampère law, Faraday's
law, the magnetic Gauss law and the electric Gauss law.  The four vector
fields $\vec{h}$, $\vec{e}$, $\vec{b}$, $\vec{d}$ are called the magnetic
field, the electric field, the magnetic flux density and the electric flux
density<ref>Many different names are used to denote these four vector fields (see e.g. http://en.wikipedia.org/wiki/Magnetic_field). We tend to use ''magnetic flux density'' and ''magnetic induction'' indifferently for $\vec{b}$. We also use ''electric flux density'' and ''electric displacement field'' indifferently for $\vec{d}$.</ref>. Taken together, they form a mathematical representation of the same
physical phenomenon: the electromagnetic field. To close the system, the following 
constitutive relations are added, which relate the magnetic 
induction to the magnetic field, and the electric displacement to the electric field:
\begin{align} 
\vec{b} &= \mu_0 ( \vec{h} + \vec{m} ) \label{eq:bhm} , \\
\vec{d} &= \epsilon_0 \vec{e} + \vec{p} \label{eq:dep} .
\end{align}
The electric charge density $\rho$, the current density $\vec{j}$, the
magnetization $\vec{m}$ and the electric polarization $\vec{p}$ are the
source terms in these equations. Given $\rho$, $\vec{j}$, $\vec{m}$, $\vec{p}$
and proper initial values for $\vec{e}$ and $\vec{h}$ at the initial time
instant $t=t_0$, the system \eqref{eq:ampere}&ndash;\eqref{eq:dep} determines
$\vec{h}$, $\vec{e}$, $\vec{b}$, $\vec{d}$ for any other time instant $t$.
Note that \eqref{eq:gausse} implies, by \eqref{eq:ampere}, the
equation of conservation of charge $\Div{\vec{j}} + \partial_t \rho = 0$,
so that, if $\vec{j}$ is given from the origin of time to the present, the
charge can be obtained by integrating this last equation with respect to
time. In the same way, Gauss's law \eqref{eq:gaussm} can be deduced from
\eqref{eq:faraday} if a zero divergence of the magnetic induction is
initially assumed.