$$A_{m,n} =   \begin{pmatrix}   a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\   a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\   \vdots  & \vdots  & \ddots & \vdots  \\   a_{m,1} & a_{m,2} & \cdots & a_{m,n}   \end{pmatrix}$$

$$\begin{pmatrix} \begin{matrix} \begin{bmatrix} a_1 & a_2 & a_3 & a_4 \\ a_5 & a_6 & a_7 & a_8 \end{bmatrix} \\ 0 & \begin{Vmatrix} c_1 & c_2 \\ c_3 & c_4 \end{Vmatrix} \end{matrix} & \begin{Bmatrix} b_1 \\ b_2 \\ b_3 \\ b_4 \end{Bmatrix} \end{pmatrix}$$

$$f'(x) = \begin{cases}
\lim_{c \to x} \dfrac{f(c) - f(x)}{c-x} & \text{(A)} \\ 
\lim_{h \to 0} \frac{f(x+h) - f(x)}{h} & \text{(B)} \\ 
\lim_{t \to 1} \frac{f(tx) - f(x)}{tx - x} & \text{(C)}
\end{cases}$$

$$\begin{align}
\mathrm{Volume} & =\iiint_S\! \rho^2 \sin\theta \,\mathrm{d}\rho \,\mathrm{d}\theta \,\mathrm{d}\phi \\
& =\int_0^{2 \pi }\! \mathrm{d}\phi \,\int_0^{ \pi }\! \sin\theta \,\mathrm{d}\theta \,\int_0^R\! \rho^2 \mathrm{d}\rho \\
& =\phi |_0^{2\pi}\ (-\cos\theta) |_0^{ \pi }\ \dfrac 1 3 \rho^3 |_0^R \\
& =2\pi \times 2 \times \frac 1 3 R^3 \\
& =\frac 4 3 \pi R^3
\end{align}$$

$$\begin{array}{ccccccccccc}
                        &                 & \mathrm{cov}({\mathcal L})     & \longrightarrow &\mathrm{non}({\mathcal K}) & \longrightarrow & \mathrm{cof}({\mathcal K}) & \longrightarrow &\mathrm{cof}({\mathcal L})     & \longrightarrow & 2^{\aleph_0} \\
                        &                 & \multirow{3}{*}{\uparrow}      &                 &\uparrow                   &                 & \uparrow                   &                 &\multirow{3}{*}{\uparrow}       &                 & \\
                        &                 &                                &                 &{\mathfrak b}              & \longrightarrow & {\mathfrak d}              &                 &                               &                 & \\
                        &                 &                                &                 &\uparrow                   &                 & \uparrow                   &                 &                               &                 & \\
\aleph_1                & \longrightarrow & \mathrm{add}({\mathcal L})     & \longrightarrow &\mathrm{add}({\mathcal K}) & \longrightarrow & \mathrm{cov}({\mathcal K}) & \longrightarrow &\mathrm{non}({\mathcal L})     &                 &
\end{array}$$

$$\begin{array}{}
\multicolumn{3}{}{\rightarrow} & & & \multirow{3}{}{\downarrow} \\ 
\multirow{3}{}{\uparrow}&\multirow{2}{}{\multicolumn{2}{l}{\huge\circlearrowright}}&& \\
&&& \\
& \multicolumn{3}{}{\leftarrow}&&
\end{array}$$

$$\begin{array}{c}
{\large \text{Generalized Product Rule:}} \\
\displaystyle \dv{x} \left[\prod_{i=1}^k f_i(x)\right] = \sum_{j=1}^k\left(f^\prime_j(x)\prod_\substack{i=1\\i\not=j}^kf_i(x)\right)
\end{array}$$

$${(a}$$

$${x\not}$$

%this is a comment

$0123456789$

$$\forall A \, \exists P \, \forall B \, [B \in P \Leftrightarrow \forall C \, (C \in B \Rightarrow C \in A)]$$

$a' b'' c''' d'''' e''''' f'''''' g''''''' h'''''''' i''''''''' j''''''''''$

$$\forall n \in \mathbb{N} \exists \; x \; \in \mathbb{R} \; : \; n^x \not\in \mathbb{Q}$$

$$c = \overbrace {     \underbrace{\;\;\;\;\; a \;\;\;\;}_\text{real}     +     \underbrace{\;\;\;\;\; b\mathrm{i} \;\;\;\;}_\text{imaginary} }^\text{complex number}$$

$$\mathrm{\nabla} \cdot \vec {v} = \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z}$$

$$\left\langle\psi\left|\mathcal{T}\left\{\frac{\delta}{\delta\phi}F[\phi]\right\}\right|\psi\right\rangle = -\mathrm{i}\left\langle\psi\left|\mathcal{T}\left\{F[\phi]\frac{\delta}{\delta\phi}S[\phi]\right\}\right|\psi\right\rangle$$

$\mathscr{L} \text{ vs. } \mathcal{L}$

$$\binom{n}{k/2} = \frac{n!}{n!(n-k/2)!}$$

$a \not{=} b \quad \not{\alpha}=\not{b} \quad \not{abc}$

$$\frac{\sqrt{1 + \sqrt[3]{2 + \sqrt[5]{3 + \sqrt[7]{4 + \sqrt[11]{5 + \sqrt[13]{6 + \sqrt[17]{7 + \sqrt[19]{A}}}}}}}}}{\mathrm{e}^\pi} = x'''$$

$$\varphi=1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1+\cdots}}}}}=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}}}}}$$

$$\def\d{\mathrm{d}} \oint_C \vec{B}\circ \d\vec{l} = \mu_0 \left( I_{\text{enc}} + \varepsilon_0 \frac{\d}{\d t} \int_S {\vec{E} \circ \hat{n}}\; \d a \right)$$

$$\frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) e^{\frac25 \pi}} \equiv 1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}} {1+\frac{e^{-8\pi}} {1+\cdots} } } }$$

$$\left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)$$

$$\displaystyle\sum_{i=1}^{k+1}i$$

$$\displaystyle= \left(\sum_{i=1}^{k}i\right) +(k+1)$$

$$\displaystyle= \frac{k(k+1)}{2}+k+1$$

$$\displaystyle= \frac{k(k+1)+2(k+1)}{2}$$

$$\displaystyle= \frac{(k+1)(k+2)}{2}$$

$$\displaystyle= \frac{(k+1)((k+1)+1)}{2}$$

$$\displaystyle\text{ for }\lvert q\rvert < 1.$$

$$= \displaystyle \prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},$$

$$\displaystyle 1 + \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots$$

$k_{n+1} = n^2 + k_n^2 - k_{n-1}$

$\Gamma\ \Delta\ \Theta\ \Lambda\ \Xi\ \Pi\ \Sigma\ \Upsilon\ \Phi\ \Psi\ \Omega$

$\omicron\ \pi\ \rho\ \sigma\ \tau\ \upsilon\ \phi\ \chi\ \psi\ \omega\ \varepsilon\ \vartheta\ \varpi\ \varrho\ \varsigma\ \varphi$

$\alpha\ \beta\ \gamma\ \delta\ \epsilon\ \zeta\ \eta\ \theta\ \iota\ \kappa\ \lambda\ \mu\ \nu\ \xi$

$\gets\ \to\ \leftarrow\ \rightarrow\ \uparrow\ \Uparrow\ \downarrow\ \Downarrow\ \updownarrow\ \Updownarrow$

$\Leftarrow\ \Rightarrow\ \leftrightarrow\ \Leftrightarrow\ \mapsto\ \hookleftarrow$

$\leftharpoonup\ \leftharpoondown\ \rightleftharpoons\ \longleftarrow\ \Longleftarrow\ \longrightarrow$

$\Longrightarrow\ \longleftrightarrow\ \Longleftrightarrow\ \longmapsto\ \hookrightarrow\ \rightharpoonup$

$\rightharpoondown\ \leadsto\ \nearrow\ \searrow\ \swarrow\ \nwarrow$

$\surd\ \barwedge\ \veebar\ \odot\ \oplus\ \otimes\ \oslash\ \circledcirc\ \boxdot\ \bigtriangleup$

$\bigtriangledown\ \dagger\ \diamond\ \star\ \triangleleft\ \triangleright\ \angle\ \infty\ \prime\ \triangle$

$$\int u \frac{dv}{dx}\,dx=uv-\int \frac{du}{dx}v\,dx$$

$$f(x) = \int_{-\infty}^\infty \hat f(\xi)\,e^{2 \pi i \xi x}$$

$$\oint \vec{F} \cdot d\vec{s}=0$$

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

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

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

$\hat{x}\ \vec{x}\ \ddot{x}$

$$\left(\frac{x^2}{y^3}\right)$$

$$\left.\frac{x^3}{3}\right|_0^1$$

$$f(n) = \begin{cases} \frac{n}{2}, & \text{if } n\text{ is even} \\ 3n+1, & \text{if } n\text{ is odd} \end{cases}$$

$$\begin{aligned}\nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\\nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\\nabla \cdot \vec{\mathbf{B}} & = 0 \end{aligned}$$

$$\begin{aligned}\nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\[1em] \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\[0.5em]\nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\[1em]\nabla \cdot \vec{\mathbf{B}} & = 0 \end{aligned}$$

$$\frac{n!}{k!(n-k)!} = {^n}C_k$$

$n \choose k$

$$\frac{\frac{1}{x}+\frac{1}{y}}{y-z}$$

$$\sqrt[n]{1+x+x^2+x^3+\ldots}$$

$$\begin{pmatrix}a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33}\end{pmatrix}$$

$$\begin{bmatrix} 0 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 0 \end{bmatrix}$$

$f(x) = \sqrt{1+x} \quad (x \ge -1)$

$f(x) \sim x^2 \quad (x\to\infty)$

$f(x) = \sqrt{1+x}, \quad x \ge -1$

$f(x) \sim x^2, \quad x\to\infty$

$$\mathcal L_{\mathcal T}(\vec{\lambda}) = \sum_{(\mathbf{x},\mathbf{s})\in \mathcal T} \log P(\mathbf{s}\mid\mathbf{x}) - \sum_{i=1}^m \frac{\lambda_i^2}{2\sigma^2}$$

$$S (\omega)=\frac{\alpha g^2}{\omega^5} \,e ^{[-0.74\bigl\{\frac{\omega U_\omega 19.5}{g}\bigr\}^{-4}]}$$

$$\sigma = \sqrt{\frac 1 n \sum_{i=1}^n (\mu - x_i)^2}$$

$$\left.\int_a^b f(x) dx =  F(x) \right\mid_a^b$$

$$\lim_{b\to\infty}\int_0^{b}e^{-x^2} dx = \frac{\sqrt{\pi}}{2}$$

$$\int_{\int_a^b f(x) dx}^{\int_c^d g(x) dx}h(x)dx$$

$$\Gamma(t) = \int_0^{+\infty} x^{t-1}e^{-x}\dv*{x}!/ = \frac 1 t \prod_{n=1}^{\infty}\frac{(1+\frac 1 t)^t}{1+\frac 1 t} \sim\sqrt{\frac{2\pi}{t}}{\left(\frac t e \right)^t}$$

$$\int_0^1 x^x\,\mathrm{d}x = \sum_{n = 1}^\infty{(-1)^{n + 1}\,n^{-n}}$$

$$\sideset{_{{}_\alpha^\beta\mathfrak{A}_\delta^\gamma}^{{}_\epsilon^\zeta\mathfrak{B}_\theta^\eta}}{_{{}_\rho^\sigma\mathfrak{E}_\upsilon^\tau}^{{}_\nu^\xi\mathfrak{D}_\pi^o}}\prod_{{}_\phi^\chi\mathfrak{F}_\omega^\psi}^{{}_\iota^\kappa\mathfrak{C}_\mu^\lambda}$$

$$\sum_{\substack{1\lt i\lt 3 \\ 1\le j\lt 5}}a_{ij}=\prod^{\substack{1\lt i\lt 3 \\ 1\le j\lt 5}}b_{ij}$$

$\lim_{x\to a} \quad \int_a^b f(x) dx \quad \sum_{i=0}^n a_i$

$\lim\limits_{x\to a} \quad \int\limits_a^b f(x) dx \quad \sum\limits_{i=0}^n a_i$

$\lim\nolimits_{x\to a} \quad \int\nolimits_a^b f(x) dx \quad \sum\nolimits_{i=0}^n a_i$

$^x$

$_x$

$x^y$

$x_y$

$x_y^z$

$x^y_z$

$^x_y$

$_x^y$

$x^y^z$

$x_y_z$

$x^{y^z}$

${x^y}^z$

$x_{y_z}$

${x_y}_z$

$x^{y_z}$

${x^y}_z$

$x_{y^z}$

${x_y}^z$

$x_{y_a^b}^{z_c^d}$

$x^\text{hello world}$

$x^{\text{hello world}}$