Sample Math 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

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

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

\(E=mc^2\)

$$E=mc^2$$

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.

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
\[
\{\,z\in\CC \mid z^2 = \bold{\alpha}\,\}
\]