mathjax test page

the arrows over vectors are displaced left in IE and Google Chrome, but look good in Firefox! Yeah for open source!

Maxwell's equations in space plus time form

$\begin{array}{rl}\mathrm{\nabla }×\stackrel{⃗}{\mathbf{B}}-\phantom{\rule{thinmathspace}{0ex}}\frac{1}{c}\phantom{\rule{thinmathspace}{0ex}}\frac{\mathrm{\partial }\stackrel{⃗}{\mathbf{E}}}{\mathrm{\partial }t}& =\frac{4\pi }{c}\stackrel{⃗}{\mathbf{j}}\\ \mathrm{\nabla }\cdot \stackrel{⃗}{\mathbf{E}}& =4\pi \rho \\ \mathrm{\nabla }×\stackrel{⃗}{\mathbf{E}}\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}\frac{1}{c}\phantom{\rule{thinmathspace}{0ex}}\frac{\mathrm{\partial }\stackrel{⃗}{\mathbf{B}}}{\mathrm{\partial }t}& =\stackrel{⃗}{\mathbf{0}}\\ \mathrm{\nabla }\cdot \stackrel{⃗}{\mathbf{B}}& =0\end{array}$

\begin{aligned} \nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\ \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\ \nabla \cdot \vec{\mathbf{B}} & = 0 \end{aligned}

Electromagnetic field tensor and its dual

$(F^{\alpha\beta}) = \begin{pmatrix} 0&E^1&E^2&E^3\\ -E^1&0&B_3&-B_2\\ -E^2&-B^3&0&B_1\\ -E^3&B_2&-B_1&0 \end{pmatrix} \qquad (^* F^{\alpha\beta}) = \begin{pmatrix} 0&-B^1&-B^2&-B^3\\ B^1&0&E_3&-E_2\\ B^2&-E^3&0&E_1\\ B^3&E_2&-E_1&0 \end{pmatrix}$

Energy-momentum tensor

$$T_{\rm em}^{\mu\nu} = \frac{1}{4\pi} \left( F^{\mu\alpha} F^\nu{}_\alpha - \frac14 g^{\mu\nu} F^{\alpha\beta} F_{\alpha\beta} \right)$$

\begin{eqnarray} T_{\rm em}^{00} &=& \frac1{8\pi} (E^2+B^2) =U_{\rm em} \qquad \mbox{(energy-density)} \nonumber\\ T_{\rm em}^{0i} &=& \frac1{4\pi} (E\times B)^i = S^i \qquad \mbox{(momentum-density = Poynting vector)} \nonumber\\ T_{\rm em}^{ij} &=& \frac1{4\pi} [-E^i E^j - B^i B^j + \frac12 g^{ij} (E^2+B^2)] \qquad \mbox{(stress tensor)} \end{eqnarray}

$$r = \frac{p}{1+e \cos(\phi)}$$