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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

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    This page was last modified on 2020 October 6 by Samuel Baltz