What is the inverse of a vector?

This may be true in some places, but my undergraduate physics education spent a lot of time on standard Gibbs-style vector calculus. Taylor Classical Mechanics and Griffiths Electrodynamics especially depend on them. Maybe there is a case to be made that first years should start with differential forms, but until that happens I think geometric algebra could be a big improvement.

To put it in concrete terms, where does GA really fit into the story of undergraduate physics (or mathematics)?

Suppose I want to teach first-semester mechanics. I can get through this fine with the usual vector notation. Vectors and dot products are intuitive when taught well (the latter just being projections), and while cross products are a little hairy, they don't play a major role in the course. There's no time for GA, and it would confuse more than illuminate in any case.

Next, I want to teach E&M. Here, I'd probably lead with the usual vector calculus notation (because even if it's ugly, it's standard and students should know it), and then follow with an explanation in terms of differential forms. [I assume this is a more theoretical, or honors, class; I might stick with vector calculus if it's more computational.] So now students know differential forms, they can do everything in a coordinate-free way and on manifolds, and they can access a significant amount of standard physics and mathematics literature.

Having proceeded in this way, what does introducing GA do except suck up a lot of class time? To me, it seems clunky and without any distinctive advantages.

Another question to think about: if this notation system is so good, why don't working mathematicians or physicists actually use it? For example, people thought Feynman diagrams were strange at first, but they proved their value and consequently caught on.

Again, my argument is that this is not some revolutionary esoteric knowledge, it's well-understood stuff that people don't teach for good reasons.

The author went to MIT, worked for NASA, and now lives in San Francisco https://twitter.com/mferraro89

It is unexceptional (indeed, expected) to get through an American undergraduate science or engineering degree without ever taking an abstract algebra course (much less the 2+ apparently expected of German pure math students).

But in any event, the top post here by Garlef is barking up the wrong tree. Division algebras, field extensions, and galois theory (per se) are not the tools to use for studying arbitrary-dimensional geometry. What you want is Clifford algebra (which Clifford himself, and later Hestenes, call “geometric algebra”) and then geometric calculus, which can be used on arbitrary manifolds, in non-metrical contexts, etc.

Basic geometric algebra should be taught to advanced high school students and all undergraduates studying any technical subject.

Math students looking for a math-style introduction to geometric algebra should try Chisolm (2012) https://arxiv.org/abs/1205.5935

Well, I think the point is that in rigid body dynamics, the configuration and phase spaces naturally form a manifold and then the equations of motion are in terms of differential forms on the cotangent bundle of the these manifolds. This is commonly expressed in terms of the language of exterior algebras, hodge duals, etc. That's what is driving all of this, and is usually covered in a good class on mathematical physics. Again, there is nothing new here except marketing, but marketing plays an important and useful role.

I remember for a long time, people coming from the math end of things would look down a bit on physicists laboriously working everything out in complex tensor notation when there are these elegant canonical descriptions arising from differential geometry that look very simple and beautiful and are completely coordinate-invariant.

But then when you want to actually calculate something, you end up doing all the painful tensor contractions anyway, so the physicists would likewise often lookdown on the mathematicians for writing these simple one liners that described all of mechanics but not really understanding how to calculate stuff.

So if repackaging some of the basic facts of differential geometry as "Geometric Algebra" gets physicists to be excited about it, then that's a good thing. Just like repackaging some of the laborious tensor calculus computations into differential geometry has gotten a lot of mathematicians excited about physics. It really is much more pleasant to work in a coordinate-free manner using differential structures associated to the natural manifold suggested by the problem, rather than being stuck in euclidean space and needing to deal with lots of fictional forces and complex change of basis formulas.

For example, look at this text: https://depositonce.tu-berlin.de/bitstream/11303/2482/1/Doku...

> for someone studying vector calculus properly

If you can’t invert vectors, you aren’t studying vector calculus properly. ;-)

Differential forms are a half-baked formalism.

Unfortunately I don’t know of any great undergraduate level geometric calculus textbooks. Ideally there would be something like Hubbard & Hubbard’s book (http://matrixeditions.com/5thUnifiedApproach.html) written using GA as a formalism.

Hestenes & Sobczyk’s book (http://geocalc.clas.asu.edu/html/CA_to_GC.html) is a hard slog, and not appropriate for an undergraduate audience.