433 lines
7.5 KiB
Markdown
433 lines
7.5 KiB
Markdown
# 🧮 LaTeX/MathJax Reference
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NoteDiscovery supports **LaTeX mathematical notation** powered by MathJax 3. Write beautiful equations in your notes using familiar LaTeX syntax.
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## Syntax Overview
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### Inline Math (within text)
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Use `$...$` for inline equations:
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- `$E = mc^2$` renders as: $E = mc^2$
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- `$x^2 + y^2 = r^2$` renders as: $x^2 + y^2 = r^2$
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### Display Math (centered, on its own line)
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Use `$$...$$` for display equations:
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```markdown
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$$
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x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}
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$$
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```
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$$
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x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}
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$$
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---
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## Basic Examples
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### Superscripts and Subscripts
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**Superscripts** use `^`:
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- `$x^2$` → $x^2$
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- `$e^{i\pi}$` → $e^{i\pi}$
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**Subscripts** use `_`:
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- `$x_1$` → $x_1$
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- `$a_{ij}$` → $a_{ij}$
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**Combined**:
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- `$x_1^2$` → $x_1^2$
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- `$\sum_{i=1}^{n} i^2$` → $\sum_{i=1}^{n} i^2$
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### Fractions
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Simple fractions: `$\frac{a}{b}$` → $\frac{a}{b}$
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Complex fractions:
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$$
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\frac{\frac{1}{x}+\frac{1}{y}}{x+y} = \frac{x+y}{xy(x+y)} = \frac{1}{xy}
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$$
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### Square Roots
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- `$\sqrt{2}$` → $\sqrt{2}$
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- `$\sqrt[3]{8}$` → $\sqrt[3]{8}$ (cube root)
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- `$\sqrt{x^2 + y^2}$` → $\sqrt{x^2 + y^2}$
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---
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## Greek Letters
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### Lowercase
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`$\alpha, \beta, \gamma, \delta, \epsilon, \zeta, \eta, \theta, \lambda, \mu, \pi, \sigma, \tau, \phi, \chi, \psi, \omega$`
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$\alpha, \beta, \gamma, \delta, \epsilon, \zeta, \eta, \theta, \lambda, \mu, \pi, \sigma, \tau, \phi, \chi, \psi, \omega$
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### Uppercase
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`$\Gamma, \Delta, \Theta, \Lambda, \Xi, \Pi, \Sigma, \Phi, \Psi, \Omega$`
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$\Gamma, \Delta, \Theta, \Lambda, \Xi, \Pi, \Sigma, \Phi, \Psi, \Omega$
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---
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## Calculus
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### Integrals
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**Definite integral:**
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```
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$$
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\int_{0}^{\infty} e^{-x^2} dx = \frac{\sqrt{\pi}}{2}
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$$
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```
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$$
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\int_{0}^{\infty} e^{-x^2} dx = \frac{\sqrt{\pi}}{2}
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$$
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**Multiple integrals:**
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```
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$$
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\iiint_V f(x,y,z) \, dx \, dy \, dz
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$$
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```
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$$
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\iiint_V f(x,y,z) \, dx \, dy \, dz
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$$
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### Derivatives
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**First derivative:** `$\frac{df}{dx}$` → $\frac{df}{dx}$
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**Partial derivatives:** `$\frac{\partial f}{\partial x}$` → $\frac{\partial f}{\partial x}$
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**Gradient:**
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```
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$$
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\nabla f = \frac{\partial f}{\partial x}\mathbf{i} + \frac{\partial f}{\partial y}\mathbf{j} + \frac{\partial f}{\partial z}\mathbf{k}
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$$
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```
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$$
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\nabla f = \frac{\partial f}{\partial x}\mathbf{i} + \frac{\partial f}{\partial y}\mathbf{j} + \frac{\partial f}{\partial z}\mathbf{k}
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$$
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### Limits
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```
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$$
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\lim_{x \to \infty} \frac{1}{x} = 0
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$$
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```
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$$
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\lim_{x \to \infty} \frac{1}{x} = 0
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$$
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---
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## Summations and Products
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### Summation
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**Inline:** $\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$
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**Display:**
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```
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$$
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\sum_{k=1}^{\infty} \frac{1}{k^2} = \frac{\pi^2}{6}
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$$
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```
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$$
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\sum_{k=1}^{\infty} \frac{1}{k^2} = \frac{\pi^2}{6}
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$$
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### Product
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```
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$$
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\prod_{i=1}^{n} i = n!
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$$
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```
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$$
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\prod_{i=1}^{n} i = n!
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$$
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---
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## Matrices and Vectors
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### Basic Matrix
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```
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$$
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\begin{bmatrix}
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a & b \\\
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c & d
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\end{bmatrix}
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$$
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```
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$$
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\begin{bmatrix}
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a & b \\\
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c & d
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\end{bmatrix}
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$$
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### Larger Matrix
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```
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$$
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A = \begin{bmatrix}
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1 & 2 & 3 \\\
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4 & 5 & 6 \\\
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7 & 8 & 9
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\end{bmatrix}
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$$
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```
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$$
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A = \begin{bmatrix}
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1 & 2 & 3 \\\
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4 & 5 & 6 \\\
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7 & 8 & 9
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\end{bmatrix}
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$$
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### Identity Matrix
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```
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$$
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I = \begin{pmatrix}
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1 & 0 & 0 \\\
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0 & 1 & 0 \\\
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0 & 0 & 1
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\end{pmatrix}
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$$
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```
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$$
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I = \begin{pmatrix}
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1 & 0 & 0 \\\
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0 & 1 & 0 \\\
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0 & 0 & 1
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\end{pmatrix}
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$$
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### Determinant
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```
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$$
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\det(A) = \begin{vmatrix}
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a & b \\\
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c & d
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\end{vmatrix} = ad - bc
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$$
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```
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$$
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\det(A) = \begin{vmatrix}
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a & b \\\
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c & d
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\end{vmatrix} = ad - bc
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$$
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---
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## Advanced Features
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### Systems of Equations
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```
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$$
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\begin{cases}
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x + y = 5 \\\
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2x - y = 1
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\end{cases}
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$$
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```
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$$
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\begin{cases}
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x + y = 5 \\\
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2x - y = 1
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\end{cases}
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$$
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### Aligned Equations
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```
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$$
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\begin{aligned}
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f(x) &= (x+1)^2 \\\
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&= x^2 + 2x + 1
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\end{aligned}
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$$
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```
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$$
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\begin{aligned}
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f(x) &= (x+1)^2 \\\
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&= x^2 + 2x + 1
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\end{aligned}
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$$
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### Continued Fractions
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```
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$$
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\phi = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \cdots}}}
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$$
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```
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$$
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\phi = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \cdots}}}
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$$
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---
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## Mathematical Symbols
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### Operators
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| Symbol | LaTeX | Result |
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|--------|-------|--------|
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| Plus-minus | `$\pm$` | $\pm$ |
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| Multiply | `$\times$` | $\times$ |
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| Divide | `$\div$` | $\div$ |
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| Not equal | `$\neq$` | $\neq$ |
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| Less/Greater | `$\leq, \geq$` | $\leq, \geq$ |
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| Approx | `$\approx$` | $\approx$ |
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| Infinity | `$\infty$` | $\infty$ |
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### Set Theory
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| Symbol | LaTeX | Result |
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|--------|-------|--------|
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| Element of | `$\in$` | $\in$ |
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| Not element | `$\notin$` | $\notin$ |
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| Subset | `$\subset$` | $\subset$ |
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| Union | `$\cup$` | $\cup$ |
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| Intersection | `$\cap$` | $\cap$ |
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| Empty set | `$\emptyset$` | $\emptyset$ |
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### Logic
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| Symbol | LaTeX | Result |
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|--------|-------|--------|
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| And | `$\land$` | $\land$ |
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| Or | `$\lor$` | $\lor$ |
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| Not | `$\neg$` | $\neg$ |
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| Implies | `$\implies$` | $\implies$ |
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| If and only if | `$\iff$` | $\iff$ |
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| For all | `$\forall$` | $\forall$ |
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| Exists | `$\exists$` | $\exists$ |
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---
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## Famous Equations
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### Euler's Identity
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$$ e^{i\pi} + 1 = 0 $$
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### Einstein's Mass-Energy Equivalence
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$$ E = mc^2 $$
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### Pythagorean Theorem
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$$ a^2 + b^2 = c^2 $$
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### Schrödinger Equation
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$$ i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},t) = \hat{H}\Psi(\mathbf{r},t) $$
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### Maxwell's Equations
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```
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$$
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\begin{aligned}
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\nabla \cdot \mathbf{E} &= \frac{\rho}{\epsilon_0} \\\
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\nabla \cdot \mathbf{B} &= 0 \\\
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\nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} \\\
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\nabla \times \mathbf{B} &= \mu_0\mathbf{J} + \mu_0\epsilon_0\frac{\partial \mathbf{E}}{\partial t}
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\end{aligned}
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$$
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```
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$$
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\begin{aligned}
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\nabla \cdot \mathbf{E} &= \frac{\rho}{\epsilon_0} \\\
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\nabla \cdot \mathbf{B} &= 0 \\\
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\nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} \\\
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\nabla \times \mathbf{B} &= \mu_0\mathbf{J} + \mu_0\epsilon_0\frac{\partial \mathbf{E}}{\partial t}
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\end{aligned}
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$$
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---
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## Tips
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### 1. Preview Mode
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Always use **Split View** or **Preview Mode** to see your equations rendered in real-time.
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### 2. Escaping Dollar Signs
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If you need a literal dollar sign (not math), escape it: `$\\$100$` renders as $\\$100$
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### 3. Complex Expressions
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For very long equations, consider breaking them across multiple lines using `aligned` or `split` environments.
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### 4. Matrix & Multi-line Formatting
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**IMPORTANT**: Use **3 backslashes + space** (`\\\ `) for line breaks to enable multi-line formatting:
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```markdown
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✅ Good (readable multi-line format):
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$$
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\begin{bmatrix}
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a & b \\\
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c & d
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\end{bmatrix}
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$$
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❌ Bad (only 2 backslashes - won't work):
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$$
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\begin{bmatrix}
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a & b \\
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c & d
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\end{bmatrix}
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$$
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```
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**The Secret:** Use `\\\ ` (three backslashes + trailing space) at the end of each row, then add a newline. This allows for readable multi-line equations!
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### 5. Debugging
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If an equation doesn't render:
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- Check for matching delimiters (`$...$` or `$$...$$`)
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- Ensure backslashes are correct (`\frac` not `/frac`)
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- Look for unescaped special characters
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- For matrices/line breaks, use `\\\ ` (three backslashes + space) not `\\`
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- Make sure there's a trailing space after `\\\` before the newline
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### 6. Performance
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MathJax renders efficiently, but very equation-heavy notes (100+ equations) may take a moment to typeset.
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---
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## Resources
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For more LaTeX commands and symbols, see:
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- [MathJax Documentation](https://docs.mathjax.org/)
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- [LaTeX Math Symbols](http://tug.ctan.org/info/symbols/comprehensive/symbols-a4.pdf)
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- [Detexify](http://detexify.kirelabs.org/classify.html) - Draw a symbol to find its LaTeX command
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---
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💡 **Tip:** Copy and paste examples from this note to quickly start using math in your own notes! |