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\begin{document}

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\section*{Maxwell&#39;s equations}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{differential_geoemtry}{}\paragraph*{{Differential geoemtry}}\label{differential_geoemtry}

[[!include synthetic differential geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{three_dimensional_formulation}{Three dimensional formulation}\dotfill \pageref*{three_dimensional_formulation} \linebreak
\noindent\hyperlink{integral_formulation_in_vacuum_in_si_units}{Integral formulation in vacuum, in SI units}\dotfill \pageref*{integral_formulation_in_vacuum_in_si_units} \linebreak
\noindent\hyperlink{differential_equations}{Differential equations}\dotfill \pageref*{differential_equations} \linebreak
\noindent\hyperlink{equations_in_terms_of_faraday_tensor_}{Equations in terms of Faraday tensor $F$}\dotfill \pageref*{equations_in_terms_of_faraday_tensor_} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In the context of [[electromagnetism]], \emph{Maxwell's equations} are the [[equations of motion]] for the electromagnetic [[field strength]] [[electric current]] and [[magnetic current]].

\hypertarget{three_dimensional_formulation}{}\subsection*{{Three dimensional formulation}}\label{three_dimensional_formulation}

$E$ is here the (vector of) strength of electric field and $B$ the strength of magnetic field; $Q$ is the charge and $j_{el}$ the density of the electrical current; $\epsilon_0$, $c$, $\mu_0$ are constants (electrical permeability, speed of the light, and magnetic permeability; all of/in vacuum: $\mu_0 \epsilon_0  = 1/c^2$).

\hypertarget{integral_formulation_in_vacuum_in_si_units}{}\subsubsection*{{Integral formulation in vacuum, in SI units}}\label{integral_formulation_in_vacuum_in_si_units}

\textbf{Gauss' law for electric fields}

\begin{displaymath}
\int_{\partial V} E\cdot d A =  \frac{Q}{\epsilon_0}
\end{displaymath}
where $\partial V$ is a closed surface which is a boundary of a 3d domain $V$ (physicists say ``volume'') and $Q = \int_V \rho d V$ the charge in the domain $V$; $\cdot$ denotes the scalar (dot) product. Surface element $d A$ is $\vec{n} d |A|$, i.e. it is the scalar surface measure times the unit vector of normal outwards.

\textbf{No magnetic monopoles} (Gauss' law for magnetic fields)

\begin{displaymath}
\int_\Sigma B\cdot d A =  0
\end{displaymath}
where $\Sigma$ is any closed surface.

\textbf{Faraday's law of induction}

\begin{displaymath}
\oint_{\partial \Sigma} E\cdot d s = - \frac{d}{d t} \int_\Sigma B\cdot d A
\end{displaymath}
The line element $d s$ is the differential (or 1-d measure on the boundary) of the length times the unit vector in counter-circle direction (or parametrize the curve with $s$ being a vector in 3d space, express magnetic field in the same parameter and calculate the integral as a function of parameter: $\cdot$ is a scalar (``dot'') product).

\textbf{Amp\`e{}re-Maxwell law} (or generalized Amp\`e{}re's law; Maxwell added the second term involving derivative of the flux of electric field to the Amp\`e{}re's law which described the magnetic field due electric current).

\begin{displaymath}
\oint_{\partial \Sigma} B\cdot d s = \mu_0 I + \mu_0 \epsilon_0 \frac{d}{d t} \int_\Sigma E\cdot d A
\end{displaymath}
where $\Sigma$ is a surface and $\partial \Sigma$ its boundary; $I$ is the total current through $\Sigma$ (integral of the component of $j_{el}$ normal to the surface).

\hypertarget{differential_equations}{}\subsubsection*{{Differential equations}}\label{differential_equations}

Here we put units with $c = 1$. By $\rho$ we denote the density of the charge.

\begin{itemize}%
\item \textbf{no magnetic charges} (magnetic Gauss law): $div B = 0$


\item \textbf{Faraday's law}: $\frac{d}{d t} B + rot E = 0$


\item \textbf{Gauss' law}: $div D = \rho$


\item \textbf{generalized Amp\`e{}re's law} $- \frac{d}{d t} D + rot H = j_{el}$



\end{itemize}
\hypertarget{equations_in_terms_of_faraday_tensor_}{}\subsection*{{Equations in terms of Faraday tensor $F$}}\label{equations_in_terms_of_faraday_tensor_}

This is adapted from \emph{\href{http://ncatlab.org/nlab/show/electromagnetic+field#MaxwellEquations}{electromagnetic field -- Maxwell's equations}}.

In modern language, the insight of (\hyperlink{Maxwell}{Maxwell, 1865}) is that locally, when physical [[spacetime]] is well approximated by a patch of its tangent space, i.e. by a patch of 4-dimensional [[Minkowski space]] $U \subset (\mathbb{R}^4, g = diag(-1,1,1,1))$, the electric field $\vec E = \left[ \itexarray{E_1 \\ E_2 \\ E_3} \right]$ and magnetic field $\vec B = \left[ \itexarray{B_1 \\ B_2 \\ B_3} \right]$ combine into a differential [[differential form|2-form]]

\begin{displaymath}
\begin{aligned}
    F & := E \wedge d t + B
      \\
      &:= 
        E_1 d x^1 \wedge d t + 
        E_2 d x^2 \wedge d t + 
        E_3 d x^3 \wedge d t  
     \\
      & +
        B_1 d x^2 \wedge d x^3 +
        B_2 d x^3 \wedge d x^1 +
        B_3 d x^1 \wedge d x^2
  \end{aligned}
\end{displaymath}
in $\Omega^2(U)$ and the [[electric charge]] density and current density combine to a differential 3-form

\begin{displaymath}
\begin{aligned}
    j_{el} &:= j\wedge dt - \rho d x^1 \wedge d x^2 \wedge d x^3 \\
& :=    j_1 d x^2 \wedge d x^3 \wedge d t +
        j_2 d x^3 \wedge d x^1 \wedge d t +
        j_3 d x^1 \wedge d x^2 \wedge d t
        -
        \rho \; d x^1 \wedge d x^2 \wedge d x^3
\end{aligned}
\end{displaymath}
in $\Omega^3(U)$ such that the following two equations of differential forms are satisfied

\begin{displaymath}
\begin{aligned}
    d F = 0
    \\
    d \star F  = j_{el}
  \end{aligned}
  \,,
\end{displaymath}
where $d$ is the de Rham differential operator and $\star$ the [[Hodge star]] operator. If we decompose $\star F$ into its components as before as

\begin{displaymath}
\begin{aligned}
\star F 
&= -D + H\wedge dt \\
&= -D_1 \; d x^2 \wedge d x^3 
   -D_2 \; d x^3 \wedge d x^1
   -D_3 \; d x^1 \wedge d x^2
     \\
     & +
     H_1 \; d x^1 \wedge d t
     +
     H_2 \; d x^2 \wedge d t
     +
     H_3 \; d x^3 \wedge d t
  \end{aligned}
\end{displaymath}
then in terms of these components the field equations -- called \textbf{Maxwell's equations} -- read as follows.

$d F = 0$ gives the magnetic Gauss law and Faraday's law

$d \star F = 0$ gives Gauss's law and Amp\`e{}re-Maxwell law

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Hodge-Maxwell theorem]]


\item [[Kirchhoff's laws]]


\item [[Yang-Mills equations]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

[[Maxwell's equations]] originate in

\begin{itemize}%
\item [[James Clerk Maxwell]], \emph{\href{http://en.wikipedia.org/wiki/A_Dynamical_Theory_of_the_Electromagnetic_Field}{A Dynamical Theory of the Electromagnetic Field},} Philosophical Transactions of the Royal Society of London 155, 459--512 (1865).

\end{itemize}
Discussion in terms of [[differential forms]] is for instance in

\begin{itemize}%
\item [[Theodore Frankel]], \emph{Maxwell's equations}, The American Mathematical Monthly, Vol 81, No 4 (1974) (\href{http://ocw.nctu.edu.tw/course/vanalysis/maxwell_amm.pdf}{pdf}, \href{http://links.jstor.org/sici?sici=0002-9890%28197404%2981%3A4%3C343%3AME%3E2.0.CO%3B2-D}{JSTOR})


\item [[Theodore Frankel]], section 3.5 in \emph{[[The Geometry of Physics - An Introduction]]}


\item Gregory L. Naber, \emph{Topology, geometry and gauge fields}, Appl. Math. Sciences vol. \textbf{141}, Springer 2000



\end{itemize}
Some history and reflection is in

\begin{itemize}%
\item [[Freeman Dyson]], \emph{Why is Maxwell’s Theory so hard to understand?}, Proceedings of \href{https://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=4446147}{The Second European Conference on Antennas and Propagation, EuCAP 2007} (\href{https://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=4446147}{doi: 10.1049/ic.2007.1146})

\end{itemize}
[[!redirects Maxwell equation]] [[!redirects Maxwell's equation]]

[[!redirects Maxwell equations]] [[!redirects Maxwell's equations]]



\end{document}