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.
Pitágoras\[ a^{2}+b^{2} = c^{2} \]
\[ \log xy = \log x + \log y \]
\[ \frac{df}{dt}=\lim_{h \to 0}\frac{f(t+h)-f(t)}{h} \]
\[ F = G \frac{m_{1}m_{2}}{d^{2}} \]
\[ i^{2}=-1 \]
\[ F - E + V = 2 \]
\[ \Phi (x)=\frac{1}{\sqrt[]{2\pi\sigma }}e^{^{\frac{(x-u)^{2}}{2\sigma ^{2}}}} \]
\[ \frac{\partial^{2} u}{\partial t^{2}}= c^{2}\frac{\partial^{2} u}{\partial x^{2}} \]
\[ X(\omega) = \int_{-\infty}^{\infty} x(t) e ^{-j \omega t} \mathrm{d}t = \int_{-\infty}^{\infty} x(t) e ^{-j 2 \pi f t} \mathrm{d}t \]
\[ \dfrac {\partial \mathbf u} {\partial t} + \mathbf u \cdot \nabla \mathbf u - \nu \nabla ^2 \mathbf u = - \nabla w + \mathbf g \]
\[ E = mc^{2} \]