$$E=mc^2$$

We find the value of an interesting integral: \begin{equation} \int_0^\infty \frac{x^3}{e^x-1}\,dx = \frac{\pi^4}{15} \label{eq:sample} \end{equation}

See equation \eqref{eq:sample}.

Contents

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

\[ \mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \frac{tial X}{tial u} & \frac{tial Y}{tial u} & 0 \frac{tial X}{tial v} & \frac{tial Y}{tial v} & 0 \end{vmatrix} \]

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{tial\vec{\mathbf{E}}}{tial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{tial\vec{\mathbf{B}}}{tial 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.

Hello LaTeX2HTML

Theorem 1 (Newton-Leibnitz). If $f\in C^1([a,b])$ then \begin{equation}\label{eq:NL} \int_a^b f'(x) d x=f(b)-f(a) \end{equation} In Theorem 1 the main part is \eqref{eq:NL}.