These notes are based on lectures by Robert Hunt 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 1-4 are about the fundamental things: Div, grad, curl, and the laplacian, and proving basic properties of them using index notation.
Lectures 5-6 are about cylindrical and spherical coordinate systems, and here I prove the classical integral theorems (Green's, stokes and divergence theorem) in order to derive some formulae for changing variables.
Lectures 7-9 are about line integrals and curvature. We also discuss the idea that zero curl implies the integral along a loop is 0
Lectures 10-12 are about area and volume integrals, including change of variables.
Lectures 12-13 are about surface integrals.
Lectures 13-15 are about applications of the integral theorems.
Lectures 16-19 cover Green's identities, use integral theorems to talk about stuff working in simply-connected domains, and talk about Laplace's and Poisson's equations.
Lectures 20-24 are about Tensors, which are either higher dimensional matrices, or multilinear maps from several vectors to a number: We will see that these are the same thing.