A JEuler Sampler

Introduction

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Three equations

a x^2 + b x + c = 0

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

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

The derivative

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

An integral

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

Three cases

$\begin{cases} -x^{2}, &\text{if $x < 0$;}\\ \alpha + x, &\text{if $0 \leq x \leq 1$;}\\ x^{2}, &\text{otherwise.} \end{cases}$

General tensor

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

A big mess

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