TeX input, HTML output test

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

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.

With equation number

\[ \begin{equation*} E = mc^2 \end{equation*}\]

custom tagged equation

\[E = mc^2\tag{x}\]

multiline equation

\[\begin{multline}   a+b+c+d+e+f+g\\   M+N+O+P+Q\\   R+S+T\\   u+v+w+x+y+z \end{multline}\]

Split equation

\[\begin{equation} \begin{split}  a& =b+c-d\\  & \quad +e-f\\  & =g+h\\  & =i  \end{split}  \end{equation}\]

Align equation

\[\begin{align*}  a_1& =b_1+c_1\\  a_2& =b_2+c_2-d_2+e_2  \end{align*}\tag{101}\]

Long Equation:

$$\int_{G} \Theta\left(f_{\underline{\varepsilon}}(t)\right) d \mu(t)=-\int_{G} f_{\underline{\varepsilon}}^{4}(t) d \mu(t) + \left(b^{2}+2 a^{2}\right) \int_{G} f_{\underline{\varepsilon}}^{2}(t) d \mu(t)+2 a b^{2} \int_{G} f_{\underline{\varepsilon}}(t) d \mu(t)+a^{2} b^{2}-a^{4}$$

$$\begin{array}{r}\frac{\partial^{2}}{\partial t_{1}^{2}} f\left(t_{0}, t_{1}\right)=\left(\delta+2 t_{0}+2 t_{1}\right)^{\alpha\left(w-t_{0}+t_{1}\right)-1} \cdot\left(\frac{\partial^{2}}{\partial t_{1}^{2}} \alpha\left(w-t_{0}+t_{1}\right) \cdot\left(\delta+2 t_{0}+2 t_{1}\right) \cdot \log \left(\delta+2 t_{0}+2 t_{1}\right)+\right. \\\alpha^{\prime}\left(w-t_{0}+t_{1}\right) \cdot 2 \cdot \log \left(\delta+2 t_{0}+2 t_{1}\right)+\alpha^{\prime}\left(w-t_{0}+t_{1}\right) \cdot\left(\delta+2 t_{0}+2 t_{1}\right) \cdot \frac{2}{\delta+2 t_{0}+2 t_{1}}+ \\\left.2 \frac{\partial}{\partial t_{1}} \alpha\left(w-t_{0}+t_{1}\right)\right)+\left(\delta+2 t_{0}+2 t_{1}\right)^{\alpha\left(w-t_{0}+t_{1}\right)-2} \\\left(\frac{\partial}{\partial t_{1}} \alpha\left(w-t_{0}+t_{1}\right) \cdot\left(\delta+2 t_{0}+2 t_{1}\right) \cdot \log \left(\delta+2 t_{0}+2 t_{1}\right)+\left(\alpha\left(w-t_{0}+t_{1}\right)-2\right)\right) \\\left(\alpha^{\prime}\left(w-t_{0}+t_{1}\right) \cdot\left(\delta+2 t_{0}+2 t_{1}\right) \cdot \log \left(\delta+2 t_{0}+2 t_{1}\right)+2 \alpha\left(w-t_{0}+t_{1}\right)\right)= \\\left(\delta+2 t_{0}+2 t_{1}\right)^{\alpha\left(w-t_{0}+t_{1}\right)-1} \cdot\left(\frac{\partial^{2}}{\partial t_{1}^{2}} \alpha\left(w-t_{0}+t_{1}\right) \cdot\left(\delta+2 t_{0}+2 t_{1}\right) \cdot \log \left(\delta+2 t_{0}+2 t_{1}\right)+\right. \\\left.2 \cdot \alpha^{\prime}\left(w-t_{0}+t_{1}\right) \cdot\left(2+\log \left(\delta+2 t_{0}+2 t_{1}\right)\right)\right)+\left(\delta+2 t_{0}+2 t_{1}\right)^{\alpha\left(w-t_{0}+t_{1}\right)-2} \cdot( \\\left.\alpha^{\prime}\left(w-t_{0}+t_{1}\right) \cdot\left(\delta+2 t_{0}+2 t_{1}\right) \cdot \log \left(\delta+2 t_{0}+2 t_{1}\right)+\left(\alpha\left(w-t_{0}+t_{1}\right)-2\right)\right) \cdot( \\\left.\left.\alpha^{\prime}\left(w-t_{0}+t_{1}\right) \cdot\left(\delta+2 t_{0}+2 t_{1}\right) \cdot \log \left(\delta+2 t_{0}+2 t_{1}\right)+2 \alpha\left(w-t_{0}+t_{1}\right)\right)\right)<0\end{array}$$