|
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. |
If you need to teach undergraduate mechanics, I highly recommend you at least read some of Hestenes’ New Foundations for Classical Mechanics http://geocalc.clas.asu.edu/html/NFCM.html
> without any distinctive advantages
The most basic distinctive advantage is that you can invert vectors (which is incredibly useful!!) without needing to pretend that vectors are matrices, complex numbers, or some other kind of object.
GA takes most of the advantages of complex numbers vs. R² for representing plane geometry, but extends them to arbitrary dimension, and extends them further (when using complex numbers for plane geometry you end up representing vector–vector products via the obscure z̄w product involving complex conjugation, and it is easy to get confused about the difference between a vector vs. a scalar+bivector).
But there are a wide variety of other powerful (and geometrically interpretable) algebraic identities which can be applied to vectors, blades, and multivectors, ranging from awkward to impossible to express using the language of differential forms, Gibbs-style vectors, etc. Physicists often end up resorting to tedious coordinate-by-coordinate calculations for stuff that would end up being an easy vector expression in GA. Learning these identities and how to apply them takes years and a lot of practice solving problems using GA.
My own experience for the first few years of knowing that GA existed but not being too fluent with it was that I would work some problem (mostly 2–3 dimensional geometry problems) out in coordinates, spending like 2 pages of scratch paper for the opaque intermediate calculations, with high chance for mistakes, then eventually find that most of the ugly bits along the way canceled and yielded a nice result. Then I would think a bit more about the problem, skim through a list of GA identities, and find I could have shortened that 2 pages of work to 3 lines, each of which had an obvious geometric interpretation.