Number Systems

• The Real Numbers

Analysis

• Real Analysis
• Complex Functions
• Limits and Continuity
• Complex Differentiation
• Complex Integration
• Infinite, Taylor, and Laurent Series
• Residues

Calculus

• Coordinate Systems
• Vector Calculus
• Operator theory

Differential equations

• Ordinary Differential Equations
• Partial Differential Equations
• Solving Partial Differential Equations

Special functions

• Legendre Polynomials
• Spherical Harmonics
• Spherical Harmonics and the Schrodinger Equation
• Bessel Functions
• Hermite Polynomials
• Stirling’s Approximation
• The Gamma Function

Abstract algebra

• Sets
• Operations
• Mappings and Functions
• Group Theory
• Fields of Scalars

Linear algebra

• Vector Spaces
• Bases
• Inner Product Spaces
• Lie Brackets
• Matrices
• Linear Mappings
• Multilinear Algebra and Tensors
• Tensor Rotations
• Extensions of Rotations to rank-2 Tensors
• One-forms

Geometry

• Manifolds
• Differential forms

Probability

• Probability
• Information
• Prior knowledge
• Feature spaces and Kernels
• Structured probability distributions
• Inference
• Stochastic processes
• Approximate inference methods
• Hierarchical modelling
• Atlas of Probability Distributions