Mathjax Rendering

\frac{a}{b}       % Standard fraction
\sqrt{x}          % Square root
\sqrt[n]{x}       % n-th root
\binom{n}{k}      % Binomial coefficient

Examples:

$$
\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}
$$

$$
\sqrt{a^2 + b^2} = c \qquad \sqrt[3]{27} = 3
$$

3. Superscripts and Subscripts

$x^2$           % Superscript
$x_1$           % Subscript
$x_i^2$         % Both combined
$x^{10}$        % Multiple characters need braces
$x_{n+1}$       % Expression as subscript

Note: Use braces {} for multi-character exponents/subscripts.

4. Greek Letters

Common Letters

See reference.md for complete Greek alphabet.

5. Common Operators and Symbols

See reference.md for complete symbol tables.

6. Matrices

Matrix Environments

Examples

$$
A = \begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
$$

$$
\det(A) = \begin{vmatrix}
a & b \\
c & d
\end{vmatrix} = ad - bc
$$

$$
I = \begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}
$$

With Ellipsis

$$
\begin{pmatrix}
a_{11} & \cdots & a_{1n} \\
\vdots & \ddots & \vdots \\
a_{m1} & \cdots & a_{mn}
\end{pmatrix}
$$

7. Aligned Equations

Use aligned environment with & for alignment and \\ for line breaks:

$$
\begin{aligned}
(a+b)^2 &= (a+b)(a+b) \\
        &= a^2 + 2ab + b^2
\end{aligned}
$$

Conditional Definitions (cases)

$$
f(x) = \begin{cases}
x^2 & \text{if } x \geq 0 \\
-x  & \text{if } x < 0
\end{cases}
$$

Text in Math

Use \text{...} for regular text:

$$
x = 5 \text{ where } x \in \mathbb{N}
$$

8. Integrals, Sums, and Limits

Integrals

$$
\int_a^b f(x) \, dx \qquad \iint_D f \, dA \qquad \oint_C \mathbf{F} \cdot d\mathbf{r}
$$

Tip: Use \, before dx for proper spacing.

Sums and Products

$$
\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}
$$

$$
\prod_{i=1}^{n} a_i
$$

Limits

$$
\lim_{x \to 0} \frac{\sin x}{x} = 1
$$

$$
\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e
$$

9. Delimiters

Use \left and \right for auto-sizing:

$$
\left( \frac{a}{b} \right) \qquad \left[ \sum_{i=1}^{n} x_i \right] \qquad \left\{ x : x > 0 \right\}
$$

One-sided Delimiter

Use \left. or \right. for invisible delimiter:

$$
\left. \frac{df}{dx} \right|_{x=0}
$$

10. Font Styles

Number Sets

$$
\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}
$$

11. Decorations

Overbrace/Underbrace

$$
\overbrace{a + b + c}^{\text{sum}} = \underbrace{x + y + z}_{\text{total}}
$$

Arrows

$$
\overrightarrow{AB} \qquad \overleftarrow{CD}
$$

12. Common Patterns

Derivatives

$$
\frac{dy}{dx} \qquad \frac{\partial f}{\partial x} \qquad \nabla f
$$

Norm and Absolute Value

$$
\|x\| = \sqrt{\sum x_i^2} \qquad |x - y| \leq |x| + |y|
$$

Probability

$$
P(A \mid B) = \frac{P(B \mid A) P(A)}{P(B)}
$$

$$
\mathbb{E}[X] = \sum_{i} x_i P(X = x_i)
$$

Quick Reference

% Fractions and roots
\frac{a}{b}  \sqrt{x}  \sqrt[n]{x}

% Greek (common)
\alpha \beta \gamma \theta \lambda \pi \sigma \omega
\Gamma \Delta \Sigma \Omega

% Relations
= \neq \leq \geq \approx \equiv \in \subset

% Operations
+ - \times \div \cdot \pm

% Calculus
\int \sum \prod \lim \partial \nabla

% Sets
\mathbb{R} \mathbb{N} \mathbb{Z} \mathbb{Q} \mathbb{C}

% Decorations
\hat{x} \bar{x} \vec{x} \dot{x}