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

$ \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) $

$ \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} $

$ \begin{aligned} \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{aligned} $

$ \frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) 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} } } } $ $$ \newcommand{\abs}[1]{\lvert#1\rvert} \newcommand{\norm}[1]{\lVert#1\rVert} \abs{z}\\ \norm{v} $$ $$ \genfrac{[}{)}{1pt}{}{1}{2} $$ $$ \cfrac{1}{\sqrt{2}+ \cfrac{1}{\sqrt{2}+ \cfrac{1}{\sqrt{2}+\dotsb }}} $$ $$ \newcommand{\binom}[2]{\genfrac{(}{)}{0pt}{}{#1}{#2}} 2^k-\binom{k}{1}2^{k-1}+\binom{k}{2}2^{k-2}\\ \biggl[\sum_i a_i\Bigl\lvert\sum_j x_{ij}\Bigr\rvert^p\biggr]^{1/p} $$ $$ \sum_{\substack{ 0\le i\le m\\ 0< j< n}} P(i,j) $$ $$ \sideset{_*^*}{_*^*}\prod $$ $$ \sideset{}{’} \sum_{n< k,\;\text{$n$ odd}} nE_n $$ $$ \frac{{\displaystyle\sum_{n > 0} z^n}} {{\displaystyle\prod_{1\leq k\leq n} (1-q^k)}} $$ \begin{equation} \Re{z} =\frac{n\pi \dfrac{\theta +\psi}{2}}{ \left(\dfrac{\theta +\psi}{2}\right)^2 + \left( \dfrac{1}{2} \log \left\lvert\dfrac{B}{A}\right\rvert\right)^2} \end{equation} \begin{equation} E = mc^2 \end{equation} $$ \unicode{5127} \\\\ \unicode{5404} \\\\ \unicode{5770} \\\\ \unicode{8752}_x^y $$ $$ \begin{CD} A @>a>b> B\\ @VlVrV @AlArA\\ C @<a<b< D \end{CD} $$ $$ \begin{CD} A @<<< B @>>> C\\ @. @| @AAA\\ @. D @= E \end{CD} $$ $$ \ce{C6H5-CHO} \\ \ce{$A$ ->[\ce{+H2O}] $B$} \\ \ce{SO4^2- + Ba^2+ -> BaSO4 v} $$ $ \enclose{circle}[mathcolor=red]{x} \enclose{circle}[mathcolor=red]{\color{black}{x}} \enclose{circle,box}{x} \enclose{circle}{\enclose{box}{x}} $ $ \text{a} \, {\color{red} \text{b}} \, \colorbox{red}{c} \, \fcolorbox{red}{blue}{d} \, {\color{white} \fcolorbox{blue}{black}{e}} $ $ \definecolor{mycolor}{rgb}{0.3,0.1,0.4} \color{mycolor} \text{my color} $ $ \definecolor{gray25}{gray}{0.75} \color{gray} gray \,\color{gray25} gray25 $ $ \fbox{text} $ $ \boxed{\vec{a} = \vec{b} +\vec{c}} $ $ \newcommand{\textcolorbox}[4] {{ \color{#1} \fcolorbox{#2}{#3}{#4} }} \textcolorbox{white}{blue}{Cyan}{text} $ $ \newcommand{\mathcolorbox}[4] {{ \color{#1} \fcolorbox{#2}{#3}{$#4$} }} \mathcolorbox{white}{blue}{Cyan}{\vec{a} = \vec{b}} = \vec{c} $ $ \vec{a} = {\color{red}(x_a,y_a,z_a)} \qquad \vec{b} = {\color{blue}(x_b,y_b,z_b)} $ $ \newcommand{\highlight}[2][yellow]{{\colorbox{#1}{$#2$}}} \begin{alignat*}{2} & \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = \ell \\ \iff & \forall \epsilon > 0, \, \exists \delta > 0 \text{ s.t. if } 0 < \abs{\Delta x - 0} < \delta, \text{ then } \abs{\frac{\Delta y}{\Delta x} - \ell} < \epsilon \\ \iff & \forall \epsilon > 0, \, \exists \delta > 0 \text{ s.t. if } 0 < \highlight{\abs{x - x_0}} < \delta, \text{ then } \highlight[green]{\abs{\frac{f(x) - f(x_0)}{x-x_0} - \ell}} < \epsilon \end{alignat*} $