Sample Equations

The Lorenz Equations

$$\begin{align} \dot{x} & = \sigma(y-x) \label{lorenz}\\ \dot{y} & = \rho x - y - xz \\ \dot{z} & = -\beta z + xy \end{align}$$

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:

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

An Identity of Ramanujan

$$\frac{1}{(\sqrt{\phi \sqrt{5}}-\phi) e^{\frac25 \pi}} = 1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}} {1+\frac{e^{-8\pi}} {1+\ldots} } } }$$

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

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

In-line Mathematics

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, equations can be used this way as well, without unduly
disturbing the spacing between lines.

References to equations

Here is a reference to the Lorenz Equations ($\ref{lorenz}$). Clicking on the equation number will take you back to the equation.