x^2              % Superscript
x_i              % Subscript  
x^{10}           % Multi-character superscript
x_{ij}           % Multi-character subscript
x^2_i            % Both
x^{y^z}          % Nested superscripts
\sum_{i=1}^{n}   % Limits on operators

\frac{a}{b}           % Standard fraction
\dfrac{a}{b}          % Display style fraction (larger)
\tfrac{a}{b}          % Text style fraction (smaller)
\sfrac{1}{2}          % Slanted fraction (nicefrac package)
\frac{1}{2+\frac{1}{3}} % Nested fractions

\sqrt{x}              % Square root
\sqrt[3]{x}           % Cube root
\sqrt[n]{x}           % nth root
\sqrt{\frac{a}{b}}    % Root of fraction

\alpha    \beta     \gamma    \delta    
\epsilon  \zeta     \eta      \theta    
\iota     \kappa    \lambda   \mu       
\nu       \xi       \pi       \rho      
\sigma    \tau      \upsilon  \phi      
\chi      \psi      \omega              

% Variants
\varepsilon  \vartheta  \varpi  
\varrho      \varsigma  \varphi

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

+         -         \times    \div      
\pm       \mp       \cdot     \ast      
\star     \circ     \bullet   \oplus    
\ominus   \otimes   \oslash   \odot     

=         \neq      <         >         
\leq      \geq      \ll       \gg       
\approx   \sim      \simeq    \cong     
\equiv    \propto   \subset   \supset   
\in       \ni       \perp     \parallel 

\leftarrow     \rightarrow    \leftrightarrow
\Leftarrow     \Rightarrow    \Leftrightarrow
\uparrow       \downarrow     \updownarrow   
\nearrow       \searrow       \swarrow       
\nwarrow       \mapsto        \longmapsto    

\sum_{i=1}^{n}      % Summation
\prod_{i=1}^{n}     % Product
\int_{a}^{b}        % Integral
\oint               % Contour integral
\bigcup_{i=1}^{n}   % Union
\bigcap_{i=1}^{n}   % Intersection
\lim_{x \to \infty} % Limit

(x)        [x]        \{x\}      |x|       
\langle x \rangle    \lceil x \rceil        
\lfloor x \rfloor    \| x \|                

% Manual sizing
\big( \Big( \bigg( \Bigg(

% Automatic sizing (recommended)
\left( \frac{a}{b} \right)
\left[ \sum_{i=1}^{n} x_i \right]
\left\{ x : x > 0 \right\}
\left. \frac{df}{dx} \right|_{x=0}

% Numbered equation
\begin{equation}
E = mc^2
\label{eq:einstein}
\end{equation}

% Unnumbered equation
\begin{equation*}
E = mc^2
\end{equation*}

% Basic matrix
\begin{matrix}
a & b \\
c & d
\end{matrix}

% Matrix with delimiters
\begin{pmatrix}        % Parentheses
a & b \\
c & d
\end{pmatrix}

\begin{bmatrix}        % Brackets
a & b \\
c & d
\end{bmatrix}

\begin{vmatrix}        % Vertical lines (determinant)
a & b \\
c & d
\end{vmatrix}

\begin{Vmatrix}        % Double vertical lines
a & b \\
c & d
\end{Vmatrix}

% Basic text
\text{if } x > 0

% Text with math inside
\text{for all } x \in \mathbb{R}

% Common text operators
\sin x   \cos x   \tan x   \log x
\exp x   \max x   \min x   \det A

\mathbb{R}     % Blackboard bold (reals)
\mathbb{N}     % Natural numbers
\mathbb{Z}     % Integers
\mathbb{C}     % Complex numbers

\mathcal{L}    % Calligraphic
\mathscr{F}    % Script (mathrsfs package)
\mathfrak{g}   % Fraktur (amssymb package)
\mathbf{v}     % Bold (vectors)
\boldsymbol{\alpha} % Bold Greek

% Automatic spacing (usually best)
a + b

% Manual spacing
a \, b         % Thin space
a \: b         % Medium space  
a \; b         % Thick space
a \quad b      % Quad space
a \qquad b     % Double quad

% Negative space
a \! b         % Negative thin space

\usepackage{amsthm}

% Define theorem environments
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{proof}{Proof}

% Use in document
\begin{theorem}
For all $x \in \mathbb{R}$, we have $x^2 \geq 0$.
\end{theorem}

\begin{proof}
This follows from the definition of real numbers.
\end{proof}

% Basic derivative
\frac{df}{dx}

% Partial derivative  
\frac{\partial f}{\partial x}

% Higher derivatives
\frac{d^2f}{dx^2}

% At a point
\left.\frac{df}{dx}\right|_{x=0}

% Definite integral
\int_0^1 x^2 \, dx

% Indefinite integral
\int f(x) \, dx

% Multiple integrals
\iint_D f(x,y) \, dx \, dy
\iiint_V f(x,y,z) \, dV

% Sum
\sum_{n=1}^{\infty} \frac{1}{n^2}

% Product
\prod_{i=1}^{n} x_i

% Inline versions
$\sum\limits_{i=1}^{n}$ % Forces display style
$\sum\nolimits_{i=1}^{n}$ % Forces inline style