====== Mathematical Expression Formatting Guide ======

This guide explains how to format mathematical expressions in this DokuWiki using the MathJax plugin.

===== Basic Setup =====

To ensure mathematical expressions render correctly:

* Make sure the [[https://www.dokuwiki.org/plugin:mathjax|MathJax Plugin]] is installed and activated \\
* Use LaTeX syntax for mathematical notation

===== Inline Mathematics =====

For inline mathematical expressions, use single dollar signs: 

`$expression$`

Examples:
  * `$a^2 + b^2 = c^2$` renders as $a^2 + b^2 = c^2$
  * `$\alpha + \beta = \gamma$` renders as $\alpha + \beta = \gamma$
  * `$f(x) = \int_{a}^{b} g(x) dx$` renders as $f(x) = \int_{a}^{b} g(x) dx$

===== Display Mathematics =====

For centered mathematical expressions on their own line, use double dollar signs:

`$$expression$$`

Example:
```
$$
\frac{d}{dx}\left( \int_{a}^{x} f(t) dt \right) = f(x)
$$
```

Renders as:
$$
\frac{d}{dx}\left( \int_{a}^{x} f(t) dt \right) = f(x)
$$

===== Common Mathematical Notation =====

==== Sets and Logic ====

| Symbol | LaTeX Code | Description |
| ------ | ---------- | ----------- |
| $\in$ | `\in` | Element of |
| $\subset$ | `\subset` | Subset |
| $\cup$ | `\cup` | Union |
| $\cap$ | `\cap` | Intersection |
| $\emptyset$ | `\emptyset` | Empty set |
| $\mathbb{R}$ | `\mathbb{R}` | Real numbers |
| $\mathbb{Z}$ | `\mathbb{Z}` | Integers |
| $\mathbb{N}$ | `\mathbb{N}` | Natural numbers |
| $\forall$ | `\forall` | For all |
| $\exists$ | `\exists` | There exists |
| $\neg$ | `\neg` | Negation |
| $\wedge$ | `\wedge` | Logical and |
| $\vee$ | `\vee` | Logical or |
| $\implies$ | `\implies` | Implies |
| $\iff$ | `\iff` | If and only if |

==== Calculus and Analysis ====

| Symbol | LaTeX Code | Description |
| ------ | ---------- | ----------- |
| $\lim_{x \to a}$ | `\lim_{x \to a}` | Limit |
| $\sum_{i=1}^{n}$ | `\sum_{i=1}^{n}` | Summation |
| $\prod_{i=1}^{n}$ | `\prod_{i=1}^{n}` | Product |
| $\int_{a}^{b}$ | `\int_{a}^{b}` | Definite integral |
| $\frac{d}{dx}$ | `\frac{d}{dx}` | Derivative |
| $\partial$ | `\partial` | Partial derivative |
| $\nabla$ | `\nabla` | Gradient |
| $\infty$ | `\infty` | Infinity |

==== Algebra and Linear Algebra ====

| Symbol | LaTeX Code | Description |
| ------ | ---------- | ----------- |
| $\times$ | `\times` | Cross product |
| $\cdot$ | `\cdot` | Dot product |
| $\otimes$ | `\otimes` | Tensor product |
| $A^{-1}$ | `A^{-1}` | Matrix inverse |
| $A^T$ | `A^T` | Matrix transpose |
| $\det(A)$ | `\det(A)` | Determinant |
| $\text{tr}(A)$ | `\text{tr}(A)` | Trace |

===== Multi-line Equations =====

For aligned multi-line equations, use the `align` environment:

```
$$
\begin{align}
a &= b + c \\
  &= d + e \\
  &= f
\end{align}
$$
```

Renders as:
$$
\begin{align}
a &= b + c \\
  &= d + e \\
  &= f
\end{align}
$$

===== Cases =====

For definitions with multiple cases, use the `cases` environment:

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

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

===== Theorems, Lemmas, and Proofs =====

Use the following formatting for theorems and proofs:

```
**Theorem:** [Statement of theorem]

**Proof:**
[Proof steps...]

Therefore, [conclusion]. ■
```

The ■ symbol (black square) is typically used to denote the end of a proof, which you can create with `■`.

===== Additional Resources =====

  * [[https://en.wikibooks.org/wiki/LaTeX/Mathematics|LaTeX Mathematics Wikibook]]
  * [[https://oeis.org/wiki/List_of_LaTeX_mathematical_symbols|List of LaTeX mathematical symbols]]
  * [[https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference|MathJax tutorial and quick reference]]