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