# Math

Mathematical notation rendered by MathJax. Enabled by the `dollarmath` and `amsmath` MyST extensions.

#### NOTE
See [MathJax](../extensions/mathjax.html.md) for setup instructions.

## Inline Math

Embed math expressions within a line of text.

The equation $E = mc^2$ describes mass-energy equivalence. The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

The MyST syntax:

```markdown
The equation $E = mc^2$ is inline math.
```

## Display Math

Standalone equations centered on their own line.

$$

\int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}
$$

$$

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

The MyST syntax:

```markdown
$$
\int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}
$$
```

## Labeled Equations

Add a label to reference an equation elsewhere in the document.

<a id="equation-euler"></a>
$$
e^{i\pi} + 1 = 0
$$

Euler’s identity is shown in equation [(1)](#equation-euler).

The MyST syntax:

```markdown
```{math}
:label: euler
e^{i\pi} + 1 = 0
```

See equation {eq}`euler`.
```

## Long Equations

Equations wider than the column scroll sideways inside their own box, so the page never scrolls. The equation number stays visible beside the formula. Narrow your browser window to try it.

<a id="equation-laplacian"></a>
$$
\nabla^2 f = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial f}{\partial r} \right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial f}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 f}{\partial \phi^2}
$$

## Multi-line Equations

Use the `amsmath` environment for aligned multi-line equations.

<a id="equation-maxwell"></a>
$$
\begin{aligned}
\nabla \cdot \mathbf{E} &= \frac{\rho}{\varepsilon_0} \\
\nabla \cdot \mathbf{B} &= 0 \\
\nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} \\
\nabla \times \mathbf{B} &= \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}
\end{aligned}
$$

Maxwell’s equations [(3)](#equation-maxwell) describe classical electromagnetism.

## Piecewise Functions

$$

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

The MyST syntax:

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

## Matrix Notation

$$

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

Different bracket styles:

| Syntax    | Brackets      | Result                                         |
|-----------|---------------|------------------------------------------------|
| `pmatrix` | Parentheses   | $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ |
| `bmatrix` | Square        | $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ |
| `vmatrix` | Vertical bars | $\begin{vmatrix} 1 & 2 \\ 3 & 4 \end{vmatrix}$ |

## Limits and Series

$$

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

$$

\prod_{i=1}^{n} x_i = x_1 \cdot x_2 \cdots x_n
$$

## Common Symbols Reference

### Greek Letters

| Lowercase   | Command    | Uppercase   | Command   |
|-------------|------------|-------------|-----------|
| $\alpha$    | `\alpha`   | $A$         | `A`       |
| $\beta$     | `\beta`    | $B$         | `B`       |
| $\gamma$    | `\gamma`   | $\Gamma$    | `\Gamma`  |
| $\delta$    | `\delta`   | $\Delta$    | `\Delta`  |
| $\epsilon$  | `\epsilon` | $E$         | `E`       |
| $\lambda$   | `\lambda`  | $\Lambda$   | `\Lambda` |
| $\mu$       | `\mu`      | $M$         | `M`       |
| $\pi$       | `\pi`      | $\Pi$       | `\Pi`     |
| $\sigma$    | `\sigma`   | $\Sigma$    | `\Sigma`  |
| $\omega$    | `\omega`   | $\Omega$    | `\Omega`  |

### Operators and Relations

| Symbol    | Command   | Symbol      | Command     |
|-----------|-----------|-------------|-------------|
| $\leq$    | `\leq`    | $\geq$      | `\geq`      |
| $\neq$    | `\neq`    | $\approx$   | `\approx`   |
| $\pm$     | `\pm`     | $\times$    | `\times`    |
| $\div$    | `\div`    | $\cdot$     | `\cdot`     |
| $\in$     | `\in`     | $\notin$    | `\notin`    |
| $\subset$ | `\subset` | $\subseteq$ | `\subseteq` |
| $\cup$    | `\cup`    | $\cap$      | `\cap`      |
| $\infty$  | `\infty`  | $\partial$  | `\partial`  |
| $\nabla$  | `\nabla`  | $\forall$   | `\forall`   |

### Arrows

| Symbol            | Command           | Symbol            | Command           |
|-------------------|-------------------|-------------------|-------------------|
| $\rightarrow$     | `\rightarrow`     | $\leftarrow$      | `\leftarrow`      |
| $\Rightarrow$     | `\Rightarrow`     | $\Leftarrow$      | `\Leftarrow`      |
| $\leftrightarrow$ | `\leftrightarrow` | $\Leftrightarrow$ | `\Leftrightarrow` |
| $\mapsto$         | `\mapsto`         | $\implies$        | `\implies`        |