IA revision

Here you can find scanned PDFs of revision sheets that I compiled for Part IA of the Mathematical Tripos, together with other accumulated resources. There are a few things worth mentioning:

1. You’re allowed to print these off, and to link to this page, but not to distribute electronic copies of the PDFs. (The purpose of this is so that I don’t have to write ‘© 2013 Adam P. Goucher’ at the foot of each page.)
2. I’ve attempted to categorise them into courses, although certain sheets bridge more than one topic. For instance, Euler-Fermat belongs to both Groups and Numbers and Sets. At the foot of each section, there are links to miscellaneous resources.
3. I’ve subsumed IA Groups under GRM, which is technically a Part IB course. However, many first-year students attend the GRM lectures, so we consider it to be Part IA for our purposes. If you don’t like this, complain to the nearest person who actually cares.
4. These revision notes are more concise than the lecture notes, so ideal if you need to ‘cram’ immediately prior to the exams. If there are more substantial gaps in your understanding, you may need to consult the comprehensive lecture notes; I have included links and attributions.
5. There is no guarantee that these resources are free from errors; thanks go to John Howe for noticing and amending a few. Use them at your own risk!

Numbers and Sets

• Euler-Fermat and RSA
• Inclusion-exclusion
• Countability

Unofficial lecture notes (by Sebastian Pancratz) are available here. Official lecture notes (by J. Saxl, for a slightly different course) are available here.

Groups/GRM:

• 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)

Unofficial lecture notes (by Mark Jackson) are available here.

Analysis I

• Analysis flowchart
• Mean value theorems
• Taylor’s theorem
• Series tests

The official course page (by Vicky Neale) is located here. The lectures are detailed on her complementary blog. Official lecture notes (by T. W. Körner) are available here. Unofficial lecture notes (by Mark Jackson) are available here.

Probability

• Inequalities in Probability
• Multivariate normal distribution
• MGFs and Central Limit Theorem
• Basic probability
• Geometrical probability
• Discrete random variables
• Branching Processes

Unofficial lecture notes (by Mark Jackson) are available here. (Erratum: the formula for the bivariate normal distribution is wrong, containing an extraneous ² on the denominator of the quadratic form. The correct formula is in the revision sheet above, entitled ‘Multivariate normal distribution’.)

Vectors and Matrices

• Vector spaces and inner products
• Cauchy-Schwarz inequality
• Dot product
• Gram-Schmidt process
• Kronecker delta and Levi-Civita epsilon
• Matrices glossary

The official lecture notes (by P. F. Linden) for Vectors and Matrices are available here. Unofficial lecture notes (by Alex Chan) for Linear Algebra are available here.

Differential equations

• First order flowchart
• Second Order ODEs
• PDEs
• Predator-Prey systems

Official lecture notes (by Stuart Dalziel) are available here. Unofficial lecture notes (by Mark Jackson) are available here.

Vector calculus

• Curves in 3D space
• ∇
• Integral Theorems and Jacobians
• Gravity and Electromagnetism
• Polar Coordinates (from DAMTP)
• Supplementary Notes (from DAMTP)

The official lecture notes (by Stephen Cowley) are available here.

Dynamics and Relativity

• Kepler’s Laws
• Moments of Inertia
• Rotating frames
• Lorentz transformations
• Relativistic mechanics

Excellent lecture notes (by David Tong) are available here.