These notes include all proofs for stuff not typically proven in this course, as I refuse to use and not prove stuff!
These notes are based on lectures by Christopher Thomas in the academic 2025-2026. Mistakes are almost surely mine and this is not official nor a recreation of what was in the lectures.
Summary of the course [Expand]
Lectures 1-3 are A level review. My notes include a proof of the mean value theorem, Rolle's theorem and Taylor's theorem.
Lectures 4-5 are an introduction to multivariable calculus: Partial derivatives, directional derivatives, the multivariate chain rule, symmetry of mixed partials, and differentiating under the integral sign. My notes provide proofs (under appropriate conditions on the functions in question).
Lectures 6-8 are mostly an A level review of differential equations stuff but we introduce some new perspectives.
Lecture 8 introduces exact equations and in my notes I provide a proof of a sufficient condition for an equation to be exact.
Lectures 8-10 are about the stability of solutions and how to analyze it.
Lectures 11-15 are about higher order equations and wronskians and I provide a proof in my notes that for linear equations of order up to 2, under appropriate conditions, we have a number of linearly independent solutions equal to the order.
Lectures 16-17 are about the dirac delta and heaviside step functions, and we take a detour to talk about a few recurrence relations.
Lectures 18-19 are about series solutions, a certain theorem (which my notes provide a proof of) and the frobenius method.
Lectures 20-21 are about multivariate functions and their countours and classifying their stationary points using the hessian matrix. My notes provide a proof of Sylvestor's criterion and the implicit function theorem, which are used in this section.
Lectures 22-23 are about solving systems of differential equations using matrix methods or analyzing the stability of points.
Lectures 23-24 are a brief introduction to partial differential equations and we show here how to solve the wave equation using the method of characteristics.