test-math
Math Formulas Test Document (Markdown + LaTeX)
This document demonstrates various mathematical formulas and symbols using LaTeX syntax in Markdown.
1. Inline Math
Simple inline equations: $E = mc^2$, $a^2 + b^2 = c^2$, and $\alpha + \beta = \gamma$.
The quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ solves $ax^2 + bx + c = 0$.
2. Display Math
$$ \int_{-\infty}^{\infty} e^{-x^2} , dx = \sqrt{\pi} $$
$$ \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e $$
$$ \sum_{k=1}^{n} k = \frac{n(n+1)}{2} $$
3. Fractions & Roots
Inline: $\frac{1}{2}$, $\dfrac{a+b}{c+d}$
Nested: $\frac{1}{1 + \frac{1}{x}}$
Square root: $\sqrt{x}$, Cube root: $\sqrt[3]{x}$
General: $\sqrt[n]{x^n + y^n}$
4. Superscripts & Subscripts
- Simple: $x^2$, $x_i$, $x_i^2$
- Multi-character: $e^{i\pi}$, $a_{ij}^{kl}$
- Nested: $x^{y^z}$, $a_{b_{c}}$
5. Greek Letters & Symbols
Lowercase
$\alpha$ $\beta$ $\gamma$ $\delta$ $\epsilon$ $\zeta$ $\eta$ $\theta$ $\iota$ $\kappa$ $\lambda$ $\mu$ $\nu$ $\xi$ $\pi$ $\rho$ $\sigma$ $\tau$ $\upsilon$ $\phi$ $\chi$ $\psi$ $\omega$
Uppercase
$\Gamma$ $\Delta$ $\Theta$ $\Lambda$ $\Xi$ $\Pi$ $\Sigma$ $\Upsilon$ $\Phi$ $\Psi$ $\Omega$
Common Symbols
$\infty$ $\nabla$ $\partial$ $\forall$ $\exists$ $\neg$ $\land$ $\lor$ $\oplus$ $\otimes$
6. Relations & Operators
Binary Operators
$+$ $-$ $\times$ $\div$ $\pm$ $\mp$ $\cdot$ $\ast$ $\star$ $\circ$ $\bullet$
Relations
$=$ $\neq$ $\approx$ $\equiv$ $\leq$ $\geq$ $\ll$ $\gg$ $\sim$ $\propto$
Set Theory
$\in$ $\notin$ $\subset$ $\subseteq$ $\supset$ $\cup$ $\cap$ $\emptyset$ $\mathbb{R}$ $\mathbb{Z}$ $\mathbb{Q}$ $\mathbb{C}$
Arrows
$\rightarrow$ $\leftarrow$ $\leftrightarrow$ $\Rightarrow$ $\Leftarrow$ $\Leftrightarrow$ $\mapsto$
7. Calculus
Derivatives
Inline: $\frac{dy}{dx}$, $\frac{d^2y}{dx^2}$, $f’(x)$, $\partial_t u$
Display: $$ \frac{d}{dx}\left[\int_0^x f(t),dt\right] = f(x) $$
Integrals
Single: $\int f(x),dx$
Definite: $\int_a^b f(x),dx$
Double/Triple: $\iint_R f,dA$, $\iiint_V f,dV$
Contour: $\oint_C \mathbf{F} \cdot d\mathbf{r}$
8. Matrices & Arrays
Basic Matrix
$$ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} $$
Bracket Variants
$$ \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \quad \begin{Bmatrix} 1 & 2 \\ 3 & 4 \end{Bmatrix} \quad \begin{vmatrix} a & b \\ c & d \end{vmatrix} $$
Large Matrix
$$ \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix} $$
Augmented Matrix
$$ \left[\begin{array}{cc|c} 1 & 2 & 5 \\ 3 & 4 & 6 \end{array}\right] $$
9. Piecewise Functions
$$ f(x) = \begin{cases} x^2 & \text{if } x < 0 \\ 0 & \text{if } x = 0 \\ \sqrt{x} & \text{if } x > 0 \end{cases} $$
10. Aligning Equations
$$ \begin{align} (a+b)^2 &= (a+b)(a+b) \\ &= a^2 + ab + ba + b^2 \\ &= a^2 + 2ab + b^2 \end{align} $$
11. Special Functions
- Trigonometric: $\sin(x)$, $\cos(x)$, $\tan(x)$, $\sec(x)$
- Inverse: $\arcsin(x)$, $\arctan(x)$
- Hyperbolic: $\sinh(x)$, $\cosh(x)$, $\tanh(x)$
- Logarithmic: $\log(x)$, $\ln(x)$, $\log_b(x)$
- Special: $\Gamma(z)$, $\zeta(s)$, $\text{erf}(x)$
12. Accents & Diacritics
$\hat{x}$ $\check{x}$ $\tilde{x}$ $\bar{x}$ $\vec{x}$ $\dot{x}$ $\ddot{x}$ $\breve{x}$
13. Spacing & Formatting
Tight: $ab$
Small space: $a,b$
Medium: $a;b$
Large: $a\quad b$
Extra large: $a\qquad b$
14. Complex Example: Maxwell’s Equations
$$ \begin{align} \nabla \cdot \mathbf{E} &= \frac{\rho}{\varepsilon_0} \\ \nabla \cdot \mathbf{B} &= 0 \\ \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} \\ \nabla \times \mathbf{B} &= \mu_0\mathbf{J} + \mu_0\varepsilon_0\frac{\partial \mathbf{E}}{\partial t} \end{align} $$
15. Common Pitfalls & Tips
✅ Use \, before differentials: $\int f(x),dx$
✅ Use \left/\right for auto-sizing: $\left(\frac{a}{b}\right)$
✅ Use \text{} for words in math: $x_{\text{max}}$
❌ Avoid: $sin(x)$ → ✅ Use: $\sin(x)$
End of Math Formulas Test Document