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

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

A Rogers-Ramanujan Identity

$$1 + \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots = \prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})}, \quad\quad \text{for $|q|<1$}.$$