$$E=mc^2$$

We find the value of an interesting integral:

\int_0^\infty \frac{x^3}{e^x-1}\,dx = \frac{\pi^4}{15}
\label{eq:sample}

See equation \eqref{eq:sample}.

## 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

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
\label{eq:NL}
\int_a^b f'(x) d x=f(b)-f(a)

In Theorem 1 the main part is \eqref{eq:NL}.