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@rhythmus rhythmus/samples.md
Created Apr 9, 2015

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TeX Samples

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

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

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 $\sqrt{3×-1}+(1+x)^2$ 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.

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