#Markdown Math Formulas

In technical documentation, academic papers, and tutorials, mathematical formulas are often needed. While standard Markdown doesn't support math formulas, many Markdown editors and platforms support math formula display through LaTeX syntax. This chapter will detail how to write math formulas in Markdown.

#Math Formula Basics

#Two Types of Formulas

Math formulas are divided into two types:

1. Inline formulas: Formulas embedded in paragraphs
2. Block formulas: Formulas displayed independently

#Inline Formulas

Use single $ symbol to wrap:

Einstein's mass-energy equation is $E = mc^2$.

Effect:

Einstein's mass-energy equation is $E = mc^2$.

#Block Formulas

Use double $ symbols to wrap:

$$
E = mc^2
$$

Effect:

$$ E = mc^2 $$

#Supported Platforms

#GitHub

• Does not support math formulas (unless using special renderers)
• Needs MathJax or KaTeX

#GitLab

• Full support
• Uses KaTeX rendering

#Typora

• Full support
• Supports real-time preview

#Obsidian

• Full support
• Uses MathJax

#Jupyter Notebook

• Full support
• Native Markdown cells

#VS Code

• Needs plugin support
• Recommend Markdown Preview Enhanced

#LaTeX Syntax Basics

#Superscripts and Subscripts

Superscript: $x^2$, $x^n$
Subscript: $x_1$, $x_{ij}$
Combined: $x^{2n}$, $x_{i+j}$

Effect:

Superscript: $x^2$, $x^n$ Subscript: $x_1$, $x_{ij}$ Combined: $x^{2n}$, $x_{i+j}$

#Fractions

Simple fraction: $\frac{a}{b}$
Nested fraction: $\frac{\frac{a}{b}}{c}$
Complex fraction: $\frac{a + b}{c - d}$

Effect:

Simple fraction: $\frac{a}{b}$ Nested fraction: $\frac{\frac{a}{b}}{c}$ Complex fraction: $\frac{a + b}{c - d}$

#Roots

Square root: $\sqrt{x}$
nth root: $\sqrt[n]{x}$
Complex expression: $\sqrt{a^2 + b^2}$

Effect:

Square root: $\sqrt{x}$ nth root: $\sqrt[n]{x}$ Complex expression: $\sqrt{a^2 + b^2}$

#Summation and Integration

Summation: $\sum_{i=1}^{n} x_i$
Integration: $\int_{a}^{b} f(x) dx$
Indefinite integral: $\int f(x) dx$

Effect:

Summation: $\sum_{i=1}^{n} x_i$ Integration: $\int_{a}^{b} f(x) dx$ Indefinite integral: $\int f(x) dx$

#Limits

Limit: $\lim_{x \to \infty} f(x)$
One-sided limit: $\lim_{x \to 0^+} f(x)$

Effect:

Limit: $\lim_{x \to \infty} f(x)$ One-sided limit: $\lim_{x \to 0^+} f(x)$

#Greek Letters

#Lowercase Greek Letters

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

Effect:

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

#Uppercase Greek Letters

$\Gamma$  $\Delta$  $\Theta$  $\Lambda$  $\Sigma$  $\Phi$  $\Psi$  $\Omega$

Effect:

$\Gamma$ $\Delta$ $\Theta$ $\Lambda$ $\Sigma$ $\Phi$ $\Psi$ $\Omega$

#Math Symbols

#Relation Symbols

Equals: $=$
Not equals: $\neq$
Less than: $<$
Greater than: $>$
Less than or equal: $\leq$
Greater than or equal: $\geq$
Approximately equal: $\approx$
Identically equal: $\equiv$

Effect:

Equals: $=$ Not equals: $\neq$ Less than: $<$ Greater than: $>$ Less than or equal: $\leq$ Greater than or equal: $\geq$ Approximately equal: $\approx$ Identically equal: $\equiv$

#Operation Symbols

Addition: $+$
Subtraction: $-$
Multiplication: $\times$
Division: $\div$
Dot product: $\cdot$
Cross product: $\times$
Plus minus: $\pm$

Effect:

Addition: $+$ Subtraction: $-$ Multiplication: $\times$ Division: $\div$ Dot product: $\cdot$ Cross product: $\times$ Plus minus: $\pm$

#Set Symbols

Element of: $\in$
Not element of: $\notin$
Subset: $\subset$
Proper subset: $\subsetneq$
Union: $\cup$
Intersection: $\cap$
Empty set: $\emptyset$
Universal quantifier: $\forall$

Effect:

Element of: $\in$ Not element of: $\notin$ Subset: $\subset$ Proper subset: $\subsetneq$ Union: $\cup$ Intersection: $\cap$ Empty set: $\emptyset$ Universal quantifier: $\forall$

#Logic Symbols

Because: $\because$
Therefore: $\therefore$
And: $\land$
Or: $\lor$
Not: $\lnot$
Implies: $\implies$
If and only if: $\iff$

Effect:

Because: $\because$ Therefore: $\therefore$ And: $\land$ Or: $\lor$ Not: $\lnot$ Implies: $\implies$ If and only if: $\iff$

#Arrow Symbols

Right arrow: $\rightarrow$
Left arrow: $\leftarrow$
Left-right arrow: $\leftrightarrow$
Up arrow: $\uparrow$
Down arrow: $\downarrow$
Long right arrow: $\Longrightarrow$

Effect:

Right arrow: $\rightarrow$ Left arrow: $\leftarrow$ Left-right arrow: $\leftrightarrow$ Up arrow: $\uparrow$ Down arrow: $\downarrow$ Long right arrow: $\Longrightarrow$

#Matrices

#Basic Matrix

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

Effect:

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

#Bracketed Matrix

$$
\begin{bmatrix}
1 & 2 \\
3 & 4
\end{bmatrix}
$$

Effect:

$$ \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} $$

#Braced Matrix

$$
\begin{Bmatrix}
a & b \\
c & d
\end{Bmatrix}
$$

Effect:

$$ \begin{Bmatrix} a & b \ c & d \end{Bmatrix} $$

#Determinant

$$
\begin{vmatrix}
a & b \\
c & d
\end{vmatrix}
$$

Effect:

$$ \begin{vmatrix} a & b \ c & d \end{vmatrix} $$

#3x3 Matrix

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

Effect:

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

#Systems of Equations

#Basic System

$$
\begin{cases}
3x + 2y = 7 \\
x - y = 1
\end{cases}
$$

Effect:

$$ \begin{cases} 3x + 2y = 7 \ x - y = 1 \end{cases} $$

#Aligned Equations

$$
\begin{aligned}
f(x) &= (x + a)(x - a) \\
     &= x^2 - a^2
\end{aligned}
$$

Effect:

$$ \begin{aligned} f(x) &= (x + a)(x - a) \ &= x^2 - a^2 \end{aligned} $$

#Piecewise Functions

#Basic Piecewise Function

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

Effect:

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

#Complex Piecewise Function

$$
f(x) =
\begin{cases}
x^2 & x < 0 \\
2x & 0 \leq x < 1 \\
1 + x & x \geq 1
\end{cases}
$$

Effect:

$$ f(x) = \begin{cases} x^2 & x < 0 \ 2x & 0 \leq x < 1 \ 1 + x & x \geq 1 \end{cases} $$

#Advanced Formulas

#Differential Equations

$$
\frac{dy}{dx} = 2x + 3
$$

Effect:

$$ \frac{dy}{dx} = 2x + 3 $$

#Partial Derivatives

$$
\frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(x^2 + y^2) = 2x
$$

Effect:

$$ \frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(x^2 + y^2) = 2x $$

#Taylor Series

$$
f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n
$$

Effect:

$$ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n $$

#Euler's Formula

$$
e^{i\pi} + 1 = 0
$$

Effect:

$$ e^{i\pi} + 1 = 0 $$

#Normal Distribution

$$
f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}
$$

Effect:

$$ f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} $$

#Fourier Transform

$$
F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt
$$

Effect:

$$ F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt $$

#Common Formula Examples

#Pythagorean Theorem

$$
a^2 + b^2 = c^2
$$

Effect:

$$ a^2 + b^2 = c^2 $$

#Quadratic Formula

$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$

Effect:

$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

#Logarithmic Formula

$$
\log_a(x) = \frac{\ln x}{\ln a}
$$

Effect:

$$ \log_a(x) = \frac{\ln x}{\ln a} $$

#Probability Formula

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

Effect:

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

#Formula Numbering

#Basic Numbering

$$
E = mc^2 \tag{1}
$$

Effect:

$$ E = mc^2 \tag{1} $$

#Reference Formulas

According to formula (1), we can draw conclusions.

#Formatting Tips

#Spacing Adjustment

Wide spacing: $a \quad b$
Narrow spacing: $a \, b$
Negative spacing: $a \! b$

Effect:

Wide spacing: $a \quad b$ Narrow spacing: $a , b$ Negative spacing: $a ! b$

#Text in Formulas

Text in formula: $f(x) = x^2 \text{ if } x > 0$

Effect:

Text in formula: $f(x) = x^2 \text{ if } x > 0$

#Large Brackets

Large brackets: $\left( \frac{a}{b} \right)$
Adapts to content size

Effect:

Large brackets: $\left( \frac{a}{b} \right)$ Adapts to content size

#Practical Examples

#Physics Documentation

# Classical Mechanics

## Newton's Second Law

$$
F = ma
$$

## Energy Conservation

$$
E = K + U = \frac{1}{2}mv^2 + mgh
$$

## Circular Motion

$$
F_c = \frac{mv^2}{r} = m\omega^2r
$$

#Math Tutorial

# Calculus Basics

## Derivative Definition

$$
f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}
$$

## Integration Calculation

$$
\int x^n dx = \frac{x^{n+1}}{n+1} + C
$$

## Definite Integral

$$
\int_{0}^{\pi} \sin x dx = 2
$$

#Statistics Documentation

# Statistics Basics

## Expected Value

$$
E[X] = \sum_{i=1}^{n} x_i P(x_i)
$$

## Variance

$$
Var(X) = E[(X - \mu)^2] = E[X^2] - (E[X])^2
$$

## Standard Deviation

$$
\sigma = \sqrt{Var(X)}
$$

#Common Questions

#Q: Formulas not displaying on GitHub?

A: GitHub natively doesn't support math formulas, you can:

• Use GitLab
• Use static site generators (like Jekyll + MathJax)
• Use third-party renderers

#Q: How to quickly find LaTeX commands?

A:

• Use online tools: Detexify
• Check LaTeX reference manuals
• Use editor's auto-completion feature

#Q: How to wrap long formulas?

A: Use \\ for line breaks:

$$
f(x) = x^2 + 2x + 1 \\
     = (x + 1)^2
$$

#Q: How to display large brackets?

A: Use \left\{ and \right\}:

$$
f(x) = \left\{
\begin{array}{ll}
x^2 & x \geq 0 \\
-x^2 & x < 0
\end{array}
\right.
$$

#Summary

This chapter detailed Markdown math formula writing methods:

• Basic syntax: Inline formulas $ and block formulas $$
• Basic elements: Fractions, roots, superscripts, subscripts
• Greek letters: Common Greek letter symbols
• Math symbols: Relations, operations, sets, logic symbols
• Advanced features: Matrices, equation systems, piecewise functions
• Common formulas: Physics, mathematics, statistics formulas
• Formatting tips: Spacing, text, brackets, etc.

Mastering LaTeX syntax for math formulas allows you to precisely express mathematical concepts in technical documentation.

Next: Learn Markdown Diagram Drawing methods.