Differential Forms in Electromagnetism 2020

Find here information on the ‘Differential Forms in Electromagnetism’ course for 2020.

We will cover Chapters 1-4 of Gauge Fields, Knots and Gravity by Baez & Muniain, followed by an independent project where you will apply this material to produce either a talk or report.

Please submit questions as you get them via the Google Form on Wattle. If I have time I will try and post some extra notes or clarifications on this page in-between meetings to try and answer these.

Schedule

There will be drop-ins Mondays 1-2pm on Zoom, which will continue over the teaching break. See Wattle for the Zoom link.

We want to finish Chapter 4 by the end of the third week after the mid-semester break. The final project will be due during the examination period, giving you three to four weeks to work on your final project.

Before the break we got half-way through Chapter 3 (everything before ‘Flows and the Lie Bracket’). We will hold drop-ins during the break where you can ask any questions you may have on that material. In the three weeks following the break our schedule will be

Week 1: The rest of Chapter 3, Chapter 4 ‘1-forms’ and ‘Cotangent Vectors’.

Week 2: Chapter 4 ‘Change of Coordinates’ and ‘p-forms’

Week 3: Chapter 4 ‘The Exterior Derivative’, review of chapters 1-4, and planning the project.

Further Reading

Differential Geometry

If you want to learn more about differential geometry I recommend reading the following books in order. Differential geometry is a very wide field; there are dozens of interconnected concepts, and you won’t feel like you understand any one of them until you understand all the others. So don’t feel bad about not understanding anything, just keep slowly building up your knowledge.

  1. (Prequel) Linear Algebra Done Right by Sheldon Axler. Differential geometry uses a lot of ideas from linear algebra, so a solid understanding of linear algebra will translate to a solid understanding of differential geometry. Axler’s book introduces linear algebra from an abstract viewpoint. Because of this abstraction, the ideas and intuition you gain will directly translate when you are trying to understand tensors in higher dimensions. This way of understanding linear algebra is also incredibly useful in quantum mechanics, computer science, and more. Bonus, the book is being offered for free in digital form for the duration of the coronavirus pandemic. Read at least chapters 1-8. It may seem like a lot to get through before you even start on manifolds, but if you really understand this material you will race through the later differential geometry textbooks.
  2. An Introduction to Manifolds by Loring Tu. This is a very gentle introduction to manifolds which gets you comfortable with all the big ideas and how to use the basic techniques. I recommend chapters 1-6 and 8. Note that this is different from Tu’s other book “Differential Geometry”.
  3. Introduction to Smooth Manifolds by John M Lee. Now you are ready to go through everything properly. This will repeat a lot of the material from Tu, but in more mathematical detail. Go through chapters 1-4, and then pick and choose chapters that you think will be interesting or useful. Since you’ve already read Tu you’ll have the basic ideas in your head, so you should be able to just jump in in most places.

Even having read all these books, you will just be at the beginning. There is so much more to look at, ideas like curvature, general relativity, symplectic manifolds which are used in classical mechanics, or applications of differential geometry to physics or robotics. But from here you should be able to pick up a textbook on any one of these areas and comfortably go through it.