Mathematical Notation

Rhizome supports mathematical notation using LaTeX syntax, rendered with KaTeX. Both inline and block math are supported.

Inline Math

Wrap LaTeX in single dollar signs for inline equations:

The famous mass-energy equivalence $E = mc^2$ shows that energy and mass are related.

Other examples:

• The quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
• Euler's number: $e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$
• The golden ratio: $\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618$
• The area of a circle: $A = \pi r^2$

Block Math

Use double dollar signs for centered block equations:

The Gaussian integral:

$\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}$

Euler's identity, combining five fundamental constants:

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

Common Formulas

Calculus

Derivative definition:

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

Integration by parts:

$\int u \, dv = uv - \int v \, du$

Taylor series expansion:

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

Linear Algebra

Matrix multiplication:

$(AB)_{ij} = \sum_{k} A_{ik} B_{kj}$

The determinant of a 2×2 matrix:

$\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$

Probability

Bayes' theorem:

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

Expected value:

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

Physics

Newton's second law:

$F = ma$

Schrödinger equation:

$i\hbar\frac{\partial}{\partial t}\Psi = \hat{H}\Psi$

Maxwell's equations (differential form):

$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$

$\nabla \cdot \mathbf{B} = 0$

$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$

$\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0\epsilon_0\frac{\partial \mathbf{E}}{\partial t}$

Greek Letters

Syntax Reference

• Superscript: x^2 → $x^2$
• Subscript: x_i → $x_i$
• Fraction: \frac{a}{b} → $\frac{a}{b}$
• Square root: \sqrt{x} → $\sqrt{x}$
• Sum: \sum_{i=1}^{n} → $\sum_{i=1}^{n}$
• Product: \prod_{i=1}^{n} → $\prod_{i=1}^{n}$
• Integral: \int_{a}^{b} → $\int_{a}^{b}$
• Limit: \lim_{x \to \infty} → $\lim_{x \to \infty}$

Practical Examples

Compound Interest

$A = P\left(1 + \frac{r}{n}\right)^{nt}$

Where $A$ is the final amount, $P$ is principal, $r$ is annual rate, $n$ is compounds per year, and $t$ is years.

Machine Learning - Gradient Descent

$\theta_{j} := \theta_{j} - \alpha \frac{\partial}{\partial \theta_{j}} J(\theta)$

Cost function:

$J(\theta) = \frac{1}{2m} \sum_{i=1}^{m} (h_\theta(x^{(i)}) - y^{(i)})^2$

Shannon Entropy

$H(X) = -\sum_{i=1}^{n} p(x_i) \log_2 p(x_i)$

Fourier Transform

$\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i x \xi} \, dx$

Commutative Diagrams

In category theory, a commutative diagram shows that composing morphisms along different paths yields the same result:

$$\begin{CD} A @>f>> B \\ @VgVV @VVhV \\ C @>>k> D \end{CD}$$

This diagram states that $h \circ f = k \circ g$ (going right-then-down equals going down-then-right).

A classic example — the First Isomorphism Theorem:

$$\begin{CD} G @>\phi>> H \\ @V{\pi}VV @AA{i}A \\ G/\ker(\phi) @>>{\bar{\phi}}> \text{im}(\phi) \end{CD}$$

Where $i \circ \bar{\phi} \circ \pi = \phi$, showing that $G/\ker(\phi) \cong \text{im}(\phi)$.

CD Syntax Reference