A comprehensive syllabus for ultralearning pure mathematics, based on Cambridge's curriculum. This includes core topics, recommended textbooks, and resources for self-study.
This is a curriculum designed for ultralearning. For more info on my modified methodology for ultralearning, see the post
. To learn about it's foundation read the book Ultralearning by Scott Young.
Introduction
This is a curriculum designed for those who are interested in pursuing a advanced understanding of pure mathematics, particularly one based on the
Cambride Mathematical Tripos. It is designed as a accelerated learning path, allowing you to cover the material in a much shorter time frame than tradtional methods.
If done right it is intended to be a full replacement for a university degree in pure mathematics as far as the core material is concerned.
Core Philosophy
Prerequisites
Core Pure Mathematics
- Proof Writing: Understanding and constructing mathematical proofs (direct proof, proof by contradiction, proof by induction).
- Algebra: Polynomials, algebraic division, factor theorem, inequalities, functions (domain, range, inverse, composite), graph sketching.
- Coordinate Geometry: Equations of lines and circles, parametric equations.
- Sequences and Series: Arithmetic and geometric series, binomial expansion.
- Trigonometry: Identities, equations, inverse trigonometric functions, sum and product formulae.
- Exponentials and Logarithms: Properties and graphs.
- Calculus: Differentiation (from first principles, rules for differentiation, applications to gradients, tangents, normals, stationary points), Integration (as a limit of a sum, fundamental theorem, techniques of integration, applications to areas and volumes).
- Numerical Methods: Iteration, locating roots.
Further Pure Mathematics
- Complex Numbers:
- Matrices:
- Further Algebra & Functions:
- Further Calculus:
- Vectors:
- Hyperbolic Functions:
- Differential Equations:
- Polar Coordinates:
General Resources to Gather
