IB revision

Here is an aperiodically-updated source of revision sheets for Part IB of the Mathematical Tripos. The same principles apply as in IA revision.

Groups, Rings and Modules

• Examples of Groups
• Möbius Group
• First Isomorphism Theorem
• Cauchy’s Theorem
• Sylow’s Theorems
• Simple Groups
• EDs, PIDs and UFDs
• Fermat’s Christmas Theorem
• Gauss’s Lemma and Eisenstein’s Criterion
• Structure Theorem (proof and applications)

Linear Algebra

• Vector spaces and inner products
• Cauchy-Schwarz inequality
• Dot product
• Gram-Schmidt process
• Matrices glossary

Unofficial lecture notes (by Alex Chan) for Linear Algebra are available here.

Analysis II/Met ‘n’ Top

• Types of spaces
• Lipschitz equivalence
• Contraction mappings

Official lecture notes (by T. W. Körner) for Metric and Topological Spaces are available here. Unofficial lecture notes (by Christopher Williams) for Analysis II are available here.

Complex Analysis

• Complex analysis flowchart
• Complex differentiation
• Cauchy’s theorem and applications
• Singularities and Laurent series

Official lecture notes (by Keith Carne) are available here.

IB Geometry

• Riemannian metrics and surfaces
• Curvature and Gauss-Bonnet
• Sphere, plane and hyperbolic plane

Variational Principles

Lecture notes for a similar course (Calculus of Variations) at Edinburgh are available here.

Methods

• Sturm-Liouville theory
• Bessel functions and Legendre polynomials
• Fourier transforms

Official lecture notes (by Richard Jozsa) are available h e r e (each letter links to one of four separate PDF files).

Quantum Mechanics

• Observables
• Hydrogen atom

Electromagnetism

Fluid Dynamics

• Potential flows
• Unsteady potential flows
• Flows with a free surface
• Fluids in rotating frames

The course has changed somewhat. I recommend reading Grae Worster’s Understanding Fluid Flow to supplement Michael McIntyre’s lecture notes.