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LaTeX/MathJax Examples

This page demonstrates the comprehensive MathJax configuration for rendering mathematical notation in MkDocs with Jupyter integration.

Basic Inline Math

Simple inline equations using different delimiters:

  • Using \( and \): The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\)
  • Using $ delimiters: The area of a circle is \(A = \pi r^2\)
  • Einstein's famous equation: \(E = mc^2\)

Display Equations

Using \[ and \] delimiters:

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

Using $$ delimiters:

\[ \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} \]

Mathematical Sets and Notation

Using our custom macros:

  • Real numbers: \(\R\)
  • Complex numbers: \(\C\)
  • Natural numbers: \(\N\)
  • Integers: \(\Z\)
  • Rational numbers: \(\Q\)

Vectors and Matrices

Vector notation using custom macros: \(\(\vec{v} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}\)\)

Matrix operations: \(\(\mat{A} = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}\)\)

Matrix multiplication: \(\(\mat{C} = \mat{A} \mat{B} = \sum_{k=1}^{n} a_{ik} b_{kj}\)\)

Functions and Operators

Common mathematical functions using custom macros:

  • Minimum: \(\argmin_{x} f(x)\)
  • Maximum: \(\argmax_{x} f(x)\)
  • Trace: \(\trace(\mat{A})\)
  • Diagonal: \(\diag(\mat{A})\)
  • Rank: \(\rank(\mat{A})\)

Probability and Statistics

Statistical notation with custom macros:

  • Probability: \(\Pr(X = x)\)
  • Expected value: \(\E[X]\)
  • Variance: \(\Var(X)\)
  • Covariance: \(\Cov(X, Y)\)

Example probability density function: \(\(f(x|\mu,\sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}\)\)

Machine Learning Notation

Custom macros for ML:

  • Loss function: \(\loss(\params)\)
  • Dataset: \(\data = \{(x_i, y_i)\}_{i=1}^n\)
  • Model: \(\model_\params\)
  • Parameters: \(\params = (\theta_1, \theta_2, \ldots, \theta_d)\)

Gradient descent update: \(\(\params_{t+1} = \params_t - \alpha \nabla_\params \loss(\params_t)\)\)

Calculus and Derivatives

Using derivative macros:

  • Differential element: \(\dd x\)
  • Ordinary derivative: \(\frac{\dd y}{\dd x}\)
  • Partial derivative: \(\pdv{f}{x}\)

Example: Chain rule \(\(\frac{\dd}{\dd x} f(g(x)) = f'(g(x)) \cdot g'(x)\)\)

Multi-variable calculus: \(\(\pdv{f}{x} = \lim_{h \to 0} \frac{f(x+h,y) - f(x,y)}{h}\)\)

Advanced Mathematical Structures

Numbered Equations

Equation with automatic numbering: begin{equation} nabla times vec{E} = -frac{partial vec{B}}{partial t} end{equation}

Another numbered equation: begin{equation} nabla cdot vec{D} = rho_f end{equation}

Chemical Equations

Using mhchem package: \(\(\ce{H2O + NaCl -> Na+ + Cl- + H2O}\)\)

\[\ce{2H2 + O2 -> 2H2O}\]

Cancellation

Using cancel package: \(\(\frac{x^2 - 1}{x - 1} = \frac{(x-1)(x+1)}{x-1} = \frac{\cancel{(x-1)}(x+1)}{\cancel{x-1}} = x + 1\)\)

Color Support

Using color extensions: \(\(\color{red}{x^2} + \color{blue}{y^2} = \color{green}{r^2}\)\)

Complex Expressions

Fourier Transform: \(\(\mathcal{F}\{f(t)\} = F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} \dd t\)\)

Schrödinger equation: \(\(i\hbar \frac{\partial}{\partial t} \Psi(\vec{r},t) = \hat{H} \Psi(\vec{r},t)\)\)

Matrix Equations

Eigenvalue problem: \(\(\mat{A}\vec{v} = \lambda\vec{v}\)\)

System of linear equations: \(\(\begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} = \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_m \end{bmatrix}\)\)

Long Equations with Line Breaking

For very long equations, MathJax will automatically break lines:

\[ f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4 + a_5 x^5 + a_6 x^6 + a_7 x^7 + a_8 x^8 + a_9 x^9 + a_{10} x^{10} \]

Testing Jupyter Integration

This page will also be used to test how these equations render when this content is included in Jupyter notebooks or when Jupyter notebooks with similar content are rendered in MkDocs.

Edge Cases

Mixed Delimiters

Testing mixed inline and display math:

When we consider the function \(f(x) = x^2\), we can analyze its derivative:

\[\frac{\dd}{\dd x} f(x) = 2x\]

And its integral:

\[\int f(x) \dd x = \frac{x^3}{3} + C\]

Unicode Support

Mathematical symbols: ∫∑∏√∞≠≤≥±×÷∂∇∆

Greek letters: αβγδεζηθικλμνξοπρστυφχψω

Special Characters

Testing escaping and special characters: - Dollar sign in text: $100 - Backslash: \ - Percent: %