# MathJax 2.0 Cheatsheet > **Note:** Use `$...$` for inline math and `$$...$$` or `\[...\]` for display math. --- ## Equations ### Simple equation ``` x = a + b ``` Result: $x = a + b$ ### Pythagorean theorem ``` x^2 + y^2 = z^2 ``` Result: $x^2 + y^2 = z^2$ ### Subscripts ``` x_1 + x_2 = x_{total} ``` Result: $x_1 + x_2 = x_{total}$ ### Simple fraction ``` \frac{a}{b} = \frac{c}{d} ``` Result: $\frac{a}{b} = \frac{c}{d}$ --- ## Multi-line Equations ### Using align environment (with alignment) ``` \begin{align} x &= a + b \\ y &= c + d \\ z &= e + f \end{align} ``` `&` is used to mark align points. For the equations below `&` is before `=`, so alignment is done at the `=` sign See the result below: Result: $\begin{align}x &= a + b \\y &= c + d \\z &= e + f\end{align}$ ### Using gather environment (no alignment) ``` \begin{gather} x = a + b \\ y = c + d \end{gather} ``` Result: $\begin{gather}x = a + b \\y = c + d\end{gather}$ --- ## Fractions and Roots ### Basic fraction ``` \frac{numerator}{denominator} ``` Result: $\frac{numerator}{denominator}$ ### Square root ``` \sqrt{x} ``` Result: $\sqrt{x}$ ### nth root ``` \sqrt[n]{x} ``` Result: $\sqrt[n]{x}$ ### Complex fraction ``` \frac{x^2 + 1}{x - 1} ``` Result: $\frac{x^2 + 1}{x - 1}$ --- ## Matrices ### Basic matrix (no brackets) ``` \begin{matrix} a & b \\ c & d \end{matrix} ``` Result: $\begin{matrix}a & b \\ c & d \end{matrix}$ ### Matrix with parentheses ``` \begin{pmatrix} a & b \\ c & d \end{pmatrix} ``` Result: $\begin{pmatrix}a & b \\c & d\end{pmatrix}$ ### Matrix with square brackets ``` \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} ``` Result: $\begin{bmatrix}1 & 2 & 3 \\4 & 5 & 6 \\7 & 8 & 9\end{bmatrix}$ ### Determinant (vertical bars) ``` \begin{vmatrix} a & b \\ c & d \end{vmatrix} ``` Result: $\begin{vmatrix}a & b \\c & d\end{vmatrix}$ --- ## Vectors **Arrow vector:** ``` \vec{v} ``` Result: $\vec{v}$ **Vector from A to B:** ``` \overrightarrow{AB} ``` Result: $\overrightarrow{AB}$ **Bold vector:** ``` \mathbf{v} ``` Result: $\mathbf{v}$ **Unit vector:** ``` \hat{i} ``` Result: $\hat{i}$ **Column vector:** ``` \vec{v} = \begin{pmatrix} x \\ y \\ z \end{pmatrix} ``` **Dot product:** ``` \vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta ``` --- ## Summation **Basic summation:** ``` \sum_{i=1}^{n} x_i ``` Result: $\sum_{i=1}^{n} x_i$ **Infinite series:** ``` \sum_{i=1}^{\infty} \frac{1}{i^2} ``` Result: $\sum_{i=1}^{\infty} \frac{1}{i^2}$ **Product notation:** ``` \prod_{i=1}^{n} x_i ``` Result: $\prod_{i=1}^{n} x_i$ --- ## Limits ### Basic limit ``` \lim_{x \to 0} f(x) ``` Result: $\lim_{x \to 0} f(x)$ ### Limit to infinity ``` \lim_{x \to \infty} \frac{1}{x} = 0 ``` Result: $\lim_{x \to \infty} \frac{1}{x} = 0$ ### Derivative definition ``` \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} ``` Result: $\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ ## Integration ### Indefinite integral ``` \int f(x) \, dx ``` Result: $\int f(x) , dx$ ### Definite integral ``` \int_{a}^{b} f(x) \, dx ``` Result: $\int_{a}^{b} f(x) , dx$ ### Double integral ``` \iint_{D} f(x,y) \, dx \, dy ``` Result: $\iint_{D} f(x,y) , dx , dy$ ### Line integral ``` \oint_{C} \vec{F} \cdot d\vec{r} ``` Result: $\oint_{C} \vec{F} \cdot d\vec{r}$ --- ## Differentiation ### Basic derivative ``` \frac{d}{dx} f(x) ``` Result: $\frac{d}{dx} f(x)$ ### Leibniz notation ``` \frac{df}{dx} ``` Result: $\frac{df}{dx}$ ### Prime notation ``` f'(x) ``` Result: $f'(x)$ ### Partial derivative ``` \frac{\partial f}{\partial x} ``` Result: $\frac{\partial f}{\partial x}$ ### Second derivative ``` \frac{d^2y}{dx^2} ``` Result: $\frac{d^2y}{dx^2}$ --- ## Common Mathematical Symbols ### Greek letters (lowercase) ``` \alpha, \beta, \gamma, \delta, \epsilon, \theta, \lambda, \mu, \pi, \sigma, \phi, \omega ``` Result: $\alpha, \beta, \gamma, \delta, \epsilon, \theta, \lambda, \mu, \pi, \sigma, \phi, \omega$ ### Greek letters (uppercase) ``` \Delta, \Gamma, \Lambda, \Sigma, \Phi, \Omega ``` Result: $\Delta, \Gamma, \Lambda, \Sigma, \Phi, \Omega$ ### Infinity and plus/minus ``` \infty, \pm, \mp ``` Result: $\infty, \pm, \mp$ ### Inequalities ``` \leq, \geq, \neq, <, > ``` Result: $\leq, \geq, \neq, <, >$ ### Set theory ``` \in, \notin, \subset, \supset, \subseteq, \supseteq ``` Result: $\in, \notin, \subset, \supset, \subseteq, \supseteq$ ### Set operations ``` \cup, \cap, \emptyset, \setminus ``` Result: $\cup, \cap, \emptyset, \setminus$ ### Logic ``` \land, \lor, \neg, \implies, \iff ``` Result: $\land, \lor, \neg, \implies, \iff$ --- ## Special Functions ### Trigonometric functions ``` \sin x, \cos x, \tan x, \sec x, \csc x, \cot x ``` Result: $\sin x, \cos x, \tan x, \sec x, \csc x, \cot x$ ### Inverse trigonometric functions ``` \arcsin x, \arccos x, \arctan x ``` Result: $\arcsin x, \arccos x, \arctan x$ ### Logarithmic functions ``` \log x, \ln x, \log_a x ``` Result: $\log x, \ln x, \log_a x$ ### Exponential functions ``` e^x, a^b, \exp(x) ``` Result: $e^x, a^b, \exp(x)$ ### Hyperbolic functions ``` \sinh x, \cosh x, \tanh x ``` Result: $\sinh x, \cosh x, \tanh x$ --- ## Spacing and Formatting ### Manual spacing ``` a \, b \quad c \qquad d ``` Result: $a , b \quad c \qquad d$ ### Text in math mode ``` \text{This is text in math mode} ``` Result: $\text{This is text in math mode}$ ### Special fonts ``` \mathbb{R}, \mathcal{F}, \mathbf{x}, \mathrm{d}x ``` Result: $\mathbb{R}, \mathcal{F}, \mathbf{x}, \mathrm{d}x$ ### Overlines and underlines ``` \overline{x}, \underline{y}, \hat{z}, \tilde{a} ``` Result: $\overline{x}, \underline{y}, \hat{z}, \tilde{a}$ --- ## Advanced Examples ### Quadratic formula ``` x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ``` Result: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ ### Euler's formula ``` e^{i\theta} = \cos\theta + i\sin\theta ``` Result: $e^{i\theta} = \cos\theta + i\sin\theta$ ### Gaussian integral ``` \int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi} ``` Result: $\int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}$ ### Matrix equation ``` \mathbf{A}\mathbf{x} = \mathbf{b} ``` Result: $\mathbf{A}\mathbf{x} = \mathbf{b}$ ### System of equations ``` \begin{cases} x + y = 1 \\ 2x - y = 0 \end{cases} ``` Result: $\begin{cases}x + y = 1 \\2x - y = 0\end{cases}$ --- ## Pro Tips 1. **Build complex expressions gradually** - Start with simple parts and combine them 2. **Use proper spacing** - `\,` for thin space, `\quad` for medium, `\qquad` for large 3. **Align equations properly** - Use `&` in `align` environment for clean alignment 4. **Use `\text{}` for words** - Don't put regular text directly in math mode 5. **Test your expressions** - Always preview to ensure they render correctly 6. **Use `\left` and `\right`** - For automatically sized parentheses: `\left( \frac{a}{b} \right)` --- ## Common Environments Summary - `align` - For aligned equations with `&` alignment points - `gather` - For centered equations without alignment - `matrix` - Basic matrix without brackets - `pmatrix` - Matrix with parentheses ( ) - `bmatrix` - Matrix with square brackets [ ] - `vmatrix` - Matrix with vertical bars | | (determinant) - `cases` - For piecewise functions and systems