Proba latex-a

When $a\ne 0$, there are two solutions to $a{x}^2+bx+c=0$ and they are $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.$$

The Lorenz Equations

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

The Cauchy-Schwarz Inequality

$${\left(\sum _{k=1}^n{a}_k{b}_k\right)}^2\le \left(\sum _{k=1}^n{a}_k^2\right)\left(\sum _{k=1}^n{b}_k^2\right)$$

A Cross Product Formula

$${\mathbf{V}}_1\times {\mathbf{V}}_2=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ \frac{\partial X}{\partial u}& \frac{\partial Y}{\partial u}& 0\\ \frac{\partial X}{\partial v}& \frac{\partial Y}{\partial v}& 0\end{array}\right|$$

The probability of getting $k$ heads when flipping $n$ coins is:

$$P(E)=(\genfrac{}{}{0ex}{}{n}{k})p^k(1-p{)}^{n-k}$$

An Identity of Ramanujan

$$\frac{1}{(\sqrt{\varphi \sqrt{5}}-\varphi )e^{\frac{2}{5}\pi }}=1+\frac{e^{-2\pi }}{1+{\displaystyle \frac{e^{-4\pi }}{1+{\displaystyle \frac{e^{-6\pi }}{1+{\displaystyle \frac{e^{-8\pi }}{1+\dots }}}}}}}$$

A Rogers-Ramanujan Identity

$$1+\frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots =\prod _{j=0}^{\mathrm{\infty }}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},\text{for }|q|<1.$$

Maxwell’s Equations

$$\begin{array}{rl}\mathrm{\nabla }\times \overrightarrow{\mathbf{B}}-\frac{1}{c}\frac{\partial \overrightarrow{\mathbf{E}}}{\partial t}& =\frac{4\pi }{c}\overrightarrow{\mathbf{j}}\\ \mathrm{\nabla }\cdot \overrightarrow{\mathbf{E}}& =4\pi \rho \\ \mathrm{\nabla }\times \overrightarrow{\mathbf{E}}+\frac{1}{c}\frac{\partial \overrightarrow{\mathbf{B}}}{\partial t}& =\overrightarrow{\mathbf{0}}\\ \mathrm{\nabla }\cdot \overrightarrow{\mathbf{B}}& =0\end{array}$$

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.