# Math

<p>
\begin{align}
\dot{x} & = \sigma(y-x) \\
\dot{y} & = \rho x - y - xz \\
\dot{z} & = -\beta z + xy
\end{align}
</p>

<p><b>The Cauchy-Schwarz Inequality</b></p>

<p>$\left( \sum_{k=1}^n a_k b_k \right)^{\!\!2} \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)$</p>

<p><b>A Cross Product Formula</b></p>

<p>$\mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\ \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \\ \end{vmatrix}$</p>

<p><b>The probability of getting $$k$$ heads when flipping $$n$$ coins is:</b></p>

<p>$P(E) = {n \choose k} p^k (1-p)^{ n-k}$</p>

<p><b>An Identity of Ramanujan</b></p>

<p>$\frac{1}{(\sqrt{\phi \sqrt{5}}-\phi) e^{\frac25 \pi}} = 1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}} {1+\frac{e^{-8\pi}} {1+\ldots} } } }$</p>

<p><b>A Rogers-Ramanujan Identity</b></p>

<p>$1 + \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots = \prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})}, \quad\quad \text{for |q|<1}.$</p>

<p><b>Maxwell's Equations</b></p>

<p>
\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}
</p>

<p>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.</p>