# ๐Ÿงฎ LaTeX/MathJax Reference NoteDiscovery supports **LaTeX mathematical notation** powered by MathJax 3. Write beautiful equations in your notes using familiar LaTeX syntax. ## Syntax Overview | Delimiter | Type | Behavior | |-----------|------|----------| | `$...$` | Inline | Flows with text, not centered | | `\(...\)` | Inline | Same as `$...$` (LaTeX standard) | | `$$...$$` | Display | Own paragraph, centered, larger | | `\[...\]` | Display | Same as `$$...$$` (LaTeX standard) | ### Inline Math (within text) Inline math flows with your text: `$E = mc^2$` renders as $E = mc^2$ ### Display Math (centered, on its own line) Display math gets its own centered paragraph: ```markdown $$ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} $$ ``` Or using LaTeX-style delimiters: ```markdown \[ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \] ``` $$ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} $$ --- ## Basic Examples ### Superscripts and Subscripts **Superscripts** use `^`: - `$x^2$` โ†’ $x^2$ - `$e^{i\pi}$` โ†’ $e^{i\pi}$ **Subscripts** use `_`: - `$x_1$` โ†’ $x_1$ - `$a_{ij}$` โ†’ $a_{ij}$ **Combined**: - `$x_1^2$` โ†’ $x_1^2$ - `$\sum_{i=1}^{n} i^2$` โ†’ $\sum_{i=1}^{n} i^2$ ### Fractions Simple fractions: `$\frac{a}{b}$` โ†’ $\frac{a}{b}$ Complex fractions: $$ \frac{\frac{1}{x}+\frac{1}{y}}{x+y} = \frac{x+y}{xy(x+y)} = \frac{1}{xy} $$ ### Square Roots - `$\sqrt{2}$` โ†’ $\sqrt{2}$ - `$\sqrt[3]{8}$` โ†’ $\sqrt[3]{8}$ (cube root) - `$\sqrt{x^2 + y^2}$` โ†’ $\sqrt{x^2 + y^2}$ --- ## Greek Letters ### Lowercase `$\alpha, \beta, \gamma, \delta, \epsilon, \zeta, \eta, \theta, \lambda, \mu, \pi, \sigma, \tau, \phi, \chi, \psi, \omega$` $\alpha, \beta, \gamma, \delta, \epsilon, \zeta, \eta, \theta, \lambda, \mu, \pi, \sigma, \tau, \phi, \chi, \psi, \omega$ ### Uppercase `$\Gamma, \Delta, \Theta, \Lambda, \Xi, \Pi, \Sigma, \Phi, \Psi, \Omega$` $\Gamma, \Delta, \Theta, \Lambda, \Xi, \Pi, \Sigma, \Phi, \Psi, \Omega$ --- ## Calculus ### Integrals **Definite integral:** ``` $$ \int_{0}^{\infty} e^{-x^2} dx = \frac{\sqrt{\pi}}{2} $$ ``` $$ \int_{0}^{\infty} e^{-x^2} dx = \frac{\sqrt{\pi}}{2} $$ **Multiple integrals:** ``` $$ \iiint_V f(x,y,z) \, dx \, dy \, dz $$ ``` $$ \iiint_V f(x,y,z) \, dx \, dy \, dz $$ ### Derivatives **First derivative:** `$\frac{df}{dx}$` โ†’ $\frac{df}{dx}$ **Partial derivatives:** `$\frac{\partial f}{\partial x}$` โ†’ $\frac{\partial f}{\partial x}$ **Gradient:** ``` $$ \nabla f = \frac{\partial f}{\partial x}\mathbf{i} + \frac{\partial f}{\partial y}\mathbf{j} + \frac{\partial f}{\partial z}\mathbf{k} $$ ``` $$ \nabla f = \frac{\partial f}{\partial x}\mathbf{i} + \frac{\partial f}{\partial y}\mathbf{j} + \frac{\partial f}{\partial z}\mathbf{k} $$ ### Limits ``` $$ \lim_{x \to \infty} \frac{1}{x} = 0 $$ ``` $$ \lim_{x \to \infty} \frac{1}{x} = 0 $$ --- ## Summations and Products ### Summation **Inline:** $\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$ **Display:** ``` $$ \sum_{k=1}^{\infty} \frac{1}{k^2} = \frac{\pi^2}{6} $$ ``` $$ \sum_{k=1}^{\infty} \frac{1}{k^2} = \frac{\pi^2}{6} $$ ### Product ``` $$ \prod_{i=1}^{n} i = n! $$ ``` $$ \prod_{i=1}^{n} i = n! $$ --- ## Matrices and Vectors ### Basic Matrix ``` $$ \begin{bmatrix} a & b \\\ c & d \end{bmatrix} $$ ``` $$ \begin{bmatrix} a & b \\\ c & d \end{bmatrix} $$ ### Larger Matrix ``` $$ A = \begin{bmatrix} 1 & 2 & 3 \\\ 4 & 5 & 6 \\\ 7 & 8 & 9 \end{bmatrix} $$ ``` $$ A = \begin{bmatrix} 1 & 2 & 3 \\\ 4 & 5 & 6 \\\ 7 & 8 & 9 \end{bmatrix} $$ ### Identity Matrix ``` $$ I = \begin{pmatrix} 1 & 0 & 0 \\\ 0 & 1 & 0 \\\ 0 & 0 & 1 \end{pmatrix} $$ ``` $$ I = \begin{pmatrix} 1 & 0 & 0 \\\ 0 & 1 & 0 \\\ 0 & 0 & 1 \end{pmatrix} $$ ### Determinant ``` $$ \det(A) = \begin{vmatrix} a & b \\\ c & d \end{vmatrix} = ad - bc $$ ``` $$ \det(A) = \begin{vmatrix} a & b \\\ c & d \end{vmatrix} = ad - bc $$ --- ## Advanced Features ### Systems of Equations ``` $$ \begin{cases} x + y = 5 \\\ 2x - y = 1 \end{cases} $$ ``` $$ \begin{cases} x + y = 5 \\\ 2x - y = 1 \end{cases} $$ ### Aligned Equations ``` $$ \begin{aligned} f(x) &= (x+1)^2 \\\ &= x^2 + 2x + 1 \end{aligned} $$ ``` $$ \begin{aligned} f(x) &= (x+1)^2 \\\ &= x^2 + 2x + 1 \end{aligned} $$ ### Continued Fractions ``` $$ \phi = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \cdots}}} $$ ``` $$ \phi = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \cdots}}} $$ --- ## Mathematical Symbols ### Operators | Symbol | LaTeX | Result | |--------|-------|--------| | Plus-minus | `$\pm$` | $\pm$ | | Multiply | `$\times$` | $\times$ | | Divide | `$\div$` | $\div$ | | Not equal | `$\neq$` | $\neq$ | | Less/Greater | `$\leq, \geq$` | $\leq, \geq$ | | Approx | `$\approx$` | $\approx$ | | Infinity | `$\infty$` | $\infty$ | ### Set Theory | Symbol | LaTeX | Result | |--------|-------|--------| | Element of | `$\in$` | $\in$ | | Not element | `$\notin$` | $\notin$ | | Subset | `$\subset$` | $\subset$ | | Union | `$\cup$` | $\cup$ | | Intersection | `$\cap$` | $\cap$ | | Empty set | `$\emptyset$` | $\emptyset$ | ### Logic | Symbol | LaTeX | Result | |--------|-------|--------| | And | `$\land$` | $\land$ | | Or | `$\lor$` | $\lor$ | | Not | `$\neg$` | $\neg$ | | Implies | `$\implies$` | $\implies$ | | If and only if | `$\iff$` | $\iff$ | | For all | `$\forall$` | $\forall$ | | Exists | `$\exists$` | $\exists$ | --- ## Famous Equations ### Euler's Identity $$ e^{i\pi} + 1 = 0 $$ ### Einstein's Mass-Energy Equivalence $$ E = mc^2 $$ ### Pythagorean Theorem $$ a^2 + b^2 = c^2 $$ ### Schrรถdinger Equation $$ i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},t) = \hat{H}\Psi(\mathbf{r},t) $$ ### Maxwell's Equations ``` $$ \begin{aligned} \nabla \cdot \mathbf{E} &= \frac{\rho}{\epsilon_0} \\\ \nabla \cdot \mathbf{B} &= 0 \\\ \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} \\\ \nabla \times \mathbf{B} &= \mu_0\mathbf{J} + \mu_0\epsilon_0\frac{\partial \mathbf{E}}{\partial t} \end{aligned} $$ ``` $$ \begin{aligned} \nabla \cdot \mathbf{E} &= \frac{\rho}{\epsilon_0} \\\ \nabla \cdot \mathbf{B} &= 0 \\\ \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} \\\ \nabla \times \mathbf{B} &= \mu_0\mathbf{J} + \mu_0\epsilon_0\frac{\partial \mathbf{E}}{\partial t} \end{aligned} $$ --- ## Tips ### 1. Preview Mode Always use **Split View** or **Preview Mode** to see your equations rendered in real-time. ### 2. Escaping Dollar Signs If you need a literal dollar sign (not math), escape it: `$\\$100$` renders as $\\$100$ ### 3. Complex Expressions For very long equations, consider breaking them across multiple lines using `aligned` or `split` environments. ### 4. Matrix & Multi-line Formatting **IMPORTANT**: Use **3 backslashes + space** (`\\\ `) for line breaks to enable multi-line formatting: ```markdown โœ… Good (readable multi-line format): $$ \begin{bmatrix} a & b \\\ c & d \end{bmatrix} $$ โŒ Bad (only 2 backslashes - won't work): $$ \begin{bmatrix} a & b \\ c & d \end{bmatrix} $$ ``` **The Secret:** Use `\\\ ` (three backslashes + trailing space) at the end of each row, then add a newline. This allows for readable multi-line equations! ### 5. Debugging If an equation doesn't render: - Check for matching delimiters (`$...$` or `$$...$$`) - Ensure backslashes are correct (`\frac` not `/frac`) - Look for unescaped special characters - For matrices/line breaks, use `\\\ ` (three backslashes + space) not `\\` - Make sure there's a trailing space after `\\\` before the newline ### 6. Performance MathJax renders efficiently, but very equation-heavy notes (100+ equations) may take a moment to typeset. --- ## Resources For more LaTeX commands and symbols, see: - [MathJax Documentation](https://docs.mathjax.org/) - [LaTeX Math Symbols](http://tug.ctan.org/info/symbols/comprehensive/symbols-a4.pdf) - [Detexify](http://detexify.kirelabs.org/classify.html) - Draw a symbol to find its LaTeX command --- ๐Ÿ’ก **Tip:** Copy and paste examples from this note to quickly start using math in your own notes!