Here is an introduction to manifolds (PDF, 23 MB) unlike any I’ve seen before. These notes by Brian Beckman devote a substantial amount of time to thinking about the problem of describing a location on a manifold, including an unexpected diversion into What3Words.

The notes are in the form of a Mathematica notebook. The link above is to a PDF rendering of the notebook. You can find the notebook itself here.

Here’s an excerpt from the abstract.

This tutorial offers a bridge between the abstract mathematics of manifolds and computational practice. Computational practice means writing simulation and control software in terms of matrices and vectors. We offer elementary examples fully explained and illustrated at length. …

Contemporary books and papers can be challenging. Many are written in highly abstract mathematical style (proofs without examples), with more generality than needed for applications (unphysical topologies), and with unfamiliar notation. If you didn’t learn fiber bundles, exterior calculus, and Lie theory in school and you want to catch up fast, this tutorial might be interesting to you. You’ll be equipped to read papers without choking on things you’ve never heard of, and get through to the juicy bits where we learn from the heavy math how to do calculations with matrices and vectors.

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