# Contents

## Idea

The field strength of the electromagnetic field.

## Details

Over Minkowski space ${ℝ}^{4}$:

$A=\varphi dt+{A}_{1}d{x}^{1}+{A}_{2}d{x}^{2}+{A}_{3}d{x}^{3}$A = \phi \mathbf{d}t + A_1 \mathbf{d}x^1 + A_2 \mathbf{d}x^2 + A_3 \mathbf{d}x^3

then field strength is the de Rham differential

$F≔dA={E}_{1}dt\wedge d{x}^{1}+{E}_{2}dt\wedge d{x}^{2}+{E}_{3}dt\wedge d{x}^{3}+{B}_{1}d{x}^{2}\wedge d{x}^{3}+{B}_{2}d{x}^{3}\wedge d{x}^{1}+{B}_{3}d{x}^{1}\wedge d{x}^{2}$F \coloneqq \mathbf{d}A = E_1 \mathbf{d}t \wedge \mathbf{d}x^1 + E_2 \mathbf{d}t \wedge \mathbf{d}x^2 + E_3 \mathbf{d}t \wedge \mathbf{d}x^3 + B_1 \mathbf{d}x^2 \wedge \mathbf{d}x^3 + B_2 \mathbf{d}x^3 \wedge \mathbf{d}x^1 + B_3 \mathbf{d}x^1 \wedge \mathbf{d}x^2

with

${E}_{i}=\frac{\partial \varphi }{\partial {x}^{i}}$E_i = \frac{\partial \phi}{\partial x^i}

the electric field strength

and

${B}_{1}=\frac{\partial {A}_{2}}{\partial {x}^{3}}-\frac{\partial {A}_{3}}{\partial {x}^{2}}$B_1 = \frac{\partial A_2}{\partial x^3} - \frac{\partial A_3}{\partial x^2}

etc

the magnetic field strength.

The field strength is closed, $dF=0$

this are the first 2 of 4 Maxwell equations

Created on November 9, 2012 17:34:27 by Urs Schreiber (80.187.201.44)