Sample MathJax Equations

The Lorenz Equations

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

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:

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

An Identity of Ramanujan

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

An Identity of Ramanujan 2

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

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

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.

A test of Equation Numbering

Equation:
$$\begin{equation} E = mc^2 \end{equation}$$
Equation*:
$$\begin{equation} E = mc^2 \end{equation}$$
Brackets: $$E = mc^2$$ Brackets tagged: $$E = mc^2\tag{x}$$
Split:

$$ \begin{equation} \begin{split} a& =b+c-d\\ & \quad +e-f\\ & =g+h\\ & =i \end{split} \end{equation} $$

Multline: $$\begin{multline} a+b+c+d+e+f+g\\ M+N+O+P+Q\\ R+S+T\\ u+v+w+x+y+z \end{multline} $$ Multline*: $$\begin{multline} a+b+c+d+e+f+g\\ M+N+O+P+Q\\ R+S+T\\ u+v+w+x+y+z \end{multline}$$
Gather: $$\begin{gather} a_1=b_1+c_1\\ a_2=b_2+c_2-d_2+e_2 \end{gather}$$ Gather*: $$\begin{gather} a_1=b_1+c_1\\ a_2=b_2+c_2-d_2+e_2 \end{gather}$$
Align:

$$ \begin{align} a_1& =b_1+c_1\\ a_2& =b_2+c_2-d_2+e_2 \end{align} $$

Align*: $$\begin{align} a_1& =b_1+c_1\\ a_2& =b_2+c_2-d_2+e_2 \end{align}$$
Align: $$\begin{align} a_{11}& =b_{11}& a_{12}& =b_{12}\\ a_{21}& =b_{21}& a_{22}& =b_{22}+c_{22} \end{align}$$
Align with \notag and \tag: $$\begin{align} a_{11}& =b_{11}& a_{12}& =b_{12}\notag\\ a_{21}& =b_{21}& a_{22}& =b_{22}+c_{22} \tag{y} \end{align}$$
Align* with \tag: $$\begin{align} a_1& =b_1+c_1\tag{z}\\ a_2& =b_2+c_2-d_2+e_2 \end{align}$$

This page uses two different methods to define macros: either putting them in JavaScript notation in the MathJax configuration, or in TeX notation in the body of the document.

Some math that used the definitions: $$ f\colon\RR\to\RR^3 \hbox{ by } f(t)=\< t+1,{1\over 1+t^2}, \sqrt{t^2+1} > $$ and $$ \{\,z\in\CC \mid z^2 = \bold{\alpha}\,\} $$