JEuler Sampler

Introduction

The following examples are live demonstrations. The source page on the web server contains latex elements that are automatically rendered on demand.

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Quadratic

a x^2 + b x + c = 0

$x=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a}$

Schrödinger

$i \hbar \frac{\partial \psi}{\partial t}= - \frac{\hbar^2}{2 m} \frac{\partial^2\psi}{\partial x^2}$

Derivative

$f'(x)=\lim_{\epsilon \to 0} \frac{f(x+\epsilon) - f(x)}{\epsilon}$

Integral

$\int_{0}^{\pi} \sin x \, dx = 2$

Cases

$f(n) = \begin{cases} n/2, & \hspace{5mm} \mbox{if } n\mbox{ is even} \\ 3n+1, & \hspace{5mm} \mbox{if } n\mbox{ is odd} \end{cases}$

Tensor

$\mathbf{T}=T_{{j_1}\dots{j_m}}^{{i_1}\dots{i_n}}\partial_{i_1}\otimes \dotsb \otimes \partial_{i_n} \otimes dx^{j_1} \otimes \dotsb \otimes dx^{j_m}$

Differential form

$\mathbf{\omega}=\omega_{{j_1} \lt \dots \lt {j_m}} dx^{j_1} \wedge \dotsb \wedge dx^{j_m}$

Mess

$\sum_{i = 1}^{ \left[ \frac{n}{2} \right] } \binom{ x_{i, i + 1}^{i^{2}} } { \left[ \frac{i + 3}{3} \right] } \frac{ \sqrt{ \mu(i)^{ \frac{3}{2}} (i^{2} - 1) } } { \sqrt[3]{\rho(i) - 2} + \sqrt[3]{\rho(i) - 1} }$

Relativity

$L = \int_a^b \sqrt{ \left|\sum_{i,j=1}^ng_{ij}(\gamma(t))\left(\frac{d}{dt}x^i\circ\gamma(t)\right)\left(\frac{d}{dt}x^j\circ\gamma(t)\right)\right|}\,dt$