# mathjax test page

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Maxwell's equations in space plus time form:

\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}

conic section in polar coordinates:

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

Another example is here: cavatappo 2.0 geometry.