MathJax testing

The following equations are represented in the HTML source code as LaTeX expressions.

The Lorenz Equations

\[\begin{aligned}

\dot{x} & = \sigma(y-x) \\


\dot{y} & = \rho x - y - xz \\

\dot{z} & = -\beta z + xy

\end{aligned} \]

The Cauchy-Schwarz Inequality

\[ \left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right) \]

A Cross Product Formula

\[\mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix}


\mathbf{i} & \mathbf{j} & \mathbf{k} \\

\frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\

\frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0


\end{vmatrix} \]

The probability of getting \(k\) heads when flipping \(n\) coins is:

\[P(E) = {n \choose k} p^k (1-p)^{ n-k} \]

An Identity of Ramanujan

\[ \frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) e^{\frac25 \pi}} =

1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}}

{1+\frac{e^{-8\pi}} {1+\ldots} } } } \]

A Rogers-Ramanujan Identity

\[ 1 + \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots =


\prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},

\quad\quad \text{for} |q|<1. \]

Maxwell’s Equations

\[ \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}

\]

Finally, while display equations look good for a page of samples, the ability to mix math and text in a paragraph is also important. This expression \(\sqrt{3x-1}+(1+x)^2\) is an example of an inline equation. As you see, MathJax equations can be used this way as well, without unduly disturbing the spacing between lines.