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Sample MathJax Equations

The Lorenz Equations

$\begin{aligned} \dot{x} & = σ (y - x) \\ \dot{y} & = ρ x - y - x z \\ \dot{z} & = - β z + x y \end{aligned}$ $\begin{align} \dot{x} & = \sigma(y-x) \\ \dot{y} & = \rho x - y - xz \\ \dot{z} & = -\beta z + xy \end{align}$

The Cauchy-Schwarz Inequality

${(\sum_{k = 1}^{n} a_{k} b_{k})}^{2} \leq (\sum_{k = 1}^{n} a_{k}^{2}) (\sum_{k = 1}^{n} b_{k}^{2})$ $\left( \sum_{k=1}^n a_k b_k \right)^{\!\!2} \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)$

A Cross Product Formula

$V_{1} \times V_{2} = | \begin{matrix} i & j & k \\ \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\ \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \end{matrix} |$ $\mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\ \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \\ \end{vmatrix}$

The probability of getting $k$ heads when flipping $n$ coins is:

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

An Identity of Ramanujan

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

A Rogers-Ramanujan Identity

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

$\begin{aligned} \nabla \times \vec{B} - \frac{1}{c} \frac{\partial \vec{E}}{\partial t} & = \frac{4 π}{c} \vec{j} \\ \nabla \cdot \vec{E} & = 4 π ρ \\ \nabla \times \vec{E} + \frac{1}{c} \frac{\partial \vec{B}}{\partial t} & = \vec{0} \\ \nabla \cdot \vec{B} & = 0 \end{aligned}$ $\begin{align} \nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\ \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\ \nabla \cdot \vec{\mathbf{B}} & = 0 \end{align}$

In-line Mathematics

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.