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Wyatt's Notes — University

Mathematics

Mathematics

This section presents proof-based undergraduate mathematics at the rigour expected of a university programme. The treatment emphasises precise definitions, complete proofs, and the logical dependencies between topics: each subject begins with its axioms and builds upward, so that no result is invoked before it has been established.

The material spans the core analytical and algebraic strands. Real analysis reconstructs the calculus from the topology of the real line; linear algebra treats vector spaces and linear maps in full generality; multivariable calculus extends differentiation and integration to functions of several variables. Where a result admits multiple proofs, the most instructive is given; where a computation is required, every step is shown.

Readers should be comfortable with single-variable calculus and with reading and constructing proofs. Cross-references link related results across subjects.

Overview

University-level mathematics notes covering analysis, algebra, and geometry.

Subjects Covered

  • Analysis: Real analysis, complex analysis, measure theory, functional analysis
  • Algebra: Linear algebra, abstract algebra, number theory
  • Applied Mathematics: Differential equations, probability, statistics
  • Geometry and Topology: Differential geometry, topology, algebraic topology

Prerequisites

  • Single-variable calculus
  • Basic linear algebra
  • Mathematical proofs and logic
  • Mathematical maturity

How to Use These Notes

Start with the foundational subjects (real analysis, linear algebra) to build the necessary background, then progress to more advanced topics. Each section includes worked examples and practice problems.

Use the sidebar to browse topics, or start with the introductory pages linked from the sidebar.

Additional Resources

Each section includes:

  • Detailed explanations of key concepts
  • Worked examples with step-by-step solutions
  • Practice problems with answers
  • Common pitfalls and how to avoid them
  • Connections to other areas of mathematics

Study Tips

  1. Master the foundations: Ensure you understand real analysis and linear algebra before moving to advanced topics
  2. Practise proofs: Mathematics requires active proof writing, not just reading
  3. Build connections: Understand how different areas of mathematics relate to each other
  4. Learn standard examples: Know the properties of common mathematical objects
  5. Connect to applications: Relate abstract concepts to physics, computer science, and engineering