# Math Notes

LaTex is pretty messy when read textually and I don't use it very often so I copied a nice page from the MathJax website to use as reference whenever I need to write math equations on the blog.

### The Lorenz Equations

\begin{aligned} \dot{x} & = \sigma(y-x) \cr \dot{y} & = \rho x - y - xz \cr \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

$\mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \cr \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \cr \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \end{vmatrix}$

### 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}} \cr \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \cr \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \cr \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.