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# Notes

Here are some of my notes on special topics. They summarise other peoples' books or course notes on a given topic, and are mostly meant as reminder notes rather than as an introduction. Do feel free to use them, but please include the original sources in any attribution.

## Calculus in more than one real variable [474K pdf]

• Review of Euclidean space and linear mappings
• Multivariable derivatives and differentials
• Taylor's formula and quadratic forms
• Optimisation and critical points
• Area and review of 1-dimensional integrals
• Volume and n-dimensional integrals
• Step functions and Riemann sums
• Iterated integrals, Fubini's Theorem, Cavalieri's principle
• Change of variables
• Improper integrals and absolutely integrable functions
• Pathlength, line integrals, and differential forms
• Green's theorem

• ## Discrete state stochastic processes [314K pdf]

• Basic definitions in the study of Markov chains
• Return and stopping times
• Classification of states
• Existence, uniqueness, and methods for calculating stationary distributions
• Limit behaviour of finite Markov chains
• Markov chains with special characertistics
• Poisson processes
• Continuous-time Markov chains
• Martingales

• ## Notes on set theory and logic [180K pdf]

Sooner or later these will be notes on set theory and logic, but right now sadly the title is only half true, because I haven't typed up the "logic" part yet. Currently these notes are a quick ~10 pages covering the following:
• Some initial set theory definitions
• Operations on sets
• Intro to the algebra of sets
• Set relations
• Defining ordered n-tuples as sets
• Equivalence relations and equivalence classes
• Set theoretic definition of functions
• Function properties and related definitions
• Partial, simple, and well orderings, and isomorphisms