> Geometric algebra is, as the article points out, a more powerful version of the usual vector notation
> the stuff that's already taught is better
These two statements seem contradictory.
> But it is deficient in various ways when compared to tensor notation (for calculations) and differential forms (e.g. if you want to work basis-free)
The author made no claims about tensor notation or differential forms; perhaps those might displace all vector-like notation (GA or otherwise) in 100 years, in which case the author's claim can be weakened to "in 100 years, GA will be the dominant form of vector notation".
As for "the stuff that's already taught is better", notice that such stuff includes:
- Complex numbers
- Pseudovectors
- Cross-products
- Matrix algebra
- Dirac notation
You seem to agree that GA is "more powerful" than the Gibbs-style vector algebra normally taught. The article is arguing that GA is also a simpler and more consistent approach, which I agree with (complex numbers are certainly simpler on their own, but are a little redundant if we're using GA for the rest).
From my own experience in formal education (UK high school and undergraduate physics), I never encountered tensors or differential forms. I've since learned a little about tensors (for general relativity), but that's been due to my own curiosity; I've learned a little about GA for the same reason. I never used quaternions (hence why I left them out of the above list), although I'm aware of them and that they're used e.g. in computer graphics. I used vectors, pseudovectors, and matrices a lot, and I'm certain those topics would have been easier to learn and comprehend if they'd used GA instead.
Differential forms are a particular kind of tensor and tensors can be defined in terms of multilinear maps. As spekcular says, the standard curriculum covers differential forms, tensors and vectors. This entails becoming familiar with multivectors, the wedge product and multilinear algebra, making geometric algebra a relatively small delta to pick up.
On the other hand, the standard course will also prepare you for mathematical topics like lie derivatives, differential geometry and de Rham cohomology.
Other than physics, the standard approach equips you with the mathematical machinery underlying many topics in machine learning and statistics like Hamiltonian monte carlo, automatic differentiation, information geometry and geometric deep learning.
The central advantage of geometric algebra over the standard approach isn't that it's better or more general, it's that pedagogical material for it is generally leagues and magnitudes better than those for the standard course.
> As spekcular says, the standard curriculum covers differential forms, tensors and vectors... making geometric algebra a relatively small delta to pick up.
Could you be a bit more specific about which "standard curriculum"/"standard approach" you're talking about?
For example, in my formal education (high school; masters with physics major, comp. sci. minor; 4 years of a comp. sci. PhD (abandoned)), I did not encounter differential forms, tensors, multivectors, the wedge product or multilinear algebra (or quaternions, lie derivatives, differential geometry, (co)homology, etc.).
Maybe you're talking about a "standard approach" for a pure mathematics curriculum, or perhaps physics/math grad school?
All I can say is that high school and undergraduate physics (in the UK, circa the late naughties) (a) does not standardise on those topics, (b) is filled with tricky operations which are easy to mix up or perform the wrong way around (e.g. cross products, matrix multiplication, pseudovectors), and (c) many of those annoyances would simplify-away under GA.
It's a cliche that physicists (certainly when teaching) cherry-pick the parts of mathematics they find useful. All of those concepts would certainly be useful in a physics course, but would perhaps be too much to fit in; yet there's certainly enough scope to cherry-pick GA (since we can drop Gibbs-style vector algebra[0] to make room). Perhaps something else, like differential forms, might be even better; I honestly don't know (maybe I'll do some reading about it).
[0] By "Gibbs-style" I mean the 'cross product and dot product ought to be enough for anyone' approach that permeated my undergraduate learning.
By standard approach I mean the typical material covered for someone studying vector calculus properly. This will be stuff like differential forms and the basics of tensors, manifolds and multilinear maps at the undergrad level. Differential geometry and cohomology are examples of courses which build on them.
I agree with you that pseudovectors, cross products and vector calculus are a terribly adhoc way to teach this stuff but a course covering linear algebra with differential forms elegantly unifies, corrects and generalizes them. Standard is also in contrast to the geometric algebra/calculus alternate path.
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.
From my view, it goes both ways: geometric algebra/calculus is a more transparent version of the standard approach and the translation back to it is also a relatively small delta to pick up.
Either way of going about what is in essence the same material entails becoming familiar with multivectors, the wedge product, and multilinear algebra, whether you do it through geometric algebra or the standard approach.
That makes sense but my argument is since further material (some examples which I listed) assumes and builds upon the standard approach, you'll likely be better off taking that path.
> I never encountered tensors or differential forms.
Your UK undergraduate physics must have been a bit different to mine. About a third of my physics course was taught by the maths dept, and tensors/algebras were very much a part of that.
I recall, after freshers week, the dean getting everyone together. He said two things:
- Hopefully you all had a great fresher's week, now it's down to business, and
- Make sure you have fun at college.
He also had a projection on the overhead saying "If you can't blind them with science, baffle them with bullshit". I'm reasonably certain the second statement above was the latter, because...
He then casually mentioned a "maths refresher" 2 week course that all freshers had to take before "the real stuff" started. That "maths refresher" was the entire Further Maths 'A' level syllabus. In two weeks. Those of us who had done Further Maths at school were fine. Those that hadn't were shell-shocked.
> That "maths refresher" was the entire Further Maths 'A' level syllabus. In two weeks. Those of us who had done Further Maths at school were fine. Those that hadn't were shell-shocked.
Heh, that reminds me of my first physics course in an under-graduate computing degree (in Romania). The curriculum was so well designed overall that this Physics course needed linear algebra concepts that would be taught halfway-through the semester in Algebra, integration along a surface and similar that would be taught at a similar in the Calculus course, and some Statistics I don't remember that would be only be taught in the second semester.
The prof's solution? He taught a 3-hour course covering all of the above, and considered that good enough for all future courses. This particular Physics course later went on to cover analytical mechanics (generalized coordinates, Lagrangians, Hamiltonians), electricity, general relativity, statistical thermodynamics, and quantum mechanics, all in a single semester.
Needless to say, 99% were happy they passed and couldn't tell you a single thing about any of these subjects a few minutes after the final exam.
> That "maths refresher" was the entire Further Maths 'A' level syllabus. In two weeks. Those of us who had done Further Maths at school were fine. Those that hadn't were shell-shocked.
Heh, I recall managing to coast for a short time thanks to having done AS Further Maths.
The Further Maths syllabus was quite modular, and the modules our teachers picked had some discrete math (sorting algorithms, Dijkstra's algorithm, bridges of königsberg, etc. which was useful for comp. sci.), and some which complemented the regular maths course (complex numbers and more calculus, which was certainly useful for physics).
Our Further Maths was a lot less modular. Preparation for it started in the 2nd year (so 12/13 years old), when we were streamed for maths - if you were in set-1, you studied to take 'O' level (showing my age here) in 4th year (so a year earlier than most) on an accelerated schedule.
That meant you could take AO (a halfway house between O and A) when everyone else was taking their normal O levels. The thing is that the extra stuff in AO was all Pure Maths, and formed a fair amount of the easier "P1" maths syllabus for the normal A level maths exam, which had P1 and Me1 (Maths with mechanics 1, basically statics).
Because you'd done that work already prior to the A level years, you could take "A level maths" after only 1 year (which looked really good on UCCA applications :), so you've now done an exam consisting of the two 'P1' and 'Me1' papers in the first year of your A levels.
Which meant that in the second year of your 'A' levels, you could do 'Pure Maths' (P1, P2) and 'Further Maths' (Me1, Me2) for a total of 3 maths A levels.
On top of that, you had your other two subjects (mine were Physics and Chemistry), and because it was the JMB board, everyone got to do "General Studies".
Getting all of them gave you 6 A levels, even though some of the work was duplicated in the maths arena (over different years of course :)
S levels were a bonus on top - there was no fudging for those, though, you just took what you thought would be useful to study. They gave me maths and physics because I'd said I was going to do physics at college... :)
> the stuff that's already taught is better
These two statements seem contradictory.
> But it is deficient in various ways when compared to tensor notation (for calculations) and differential forms (e.g. if you want to work basis-free)
The author made no claims about tensor notation or differential forms; perhaps those might displace all vector-like notation (GA or otherwise) in 100 years, in which case the author's claim can be weakened to "in 100 years, GA will be the dominant form of vector notation".
As for "the stuff that's already taught is better", notice that such stuff includes:
- Complex numbers
- Pseudovectors
- Cross-products
- Matrix algebra
- Dirac notation
You seem to agree that GA is "more powerful" than the Gibbs-style vector algebra normally taught. The article is arguing that GA is also a simpler and more consistent approach, which I agree with (complex numbers are certainly simpler on their own, but are a little redundant if we're using GA for the rest).
From my own experience in formal education (UK high school and undergraduate physics), I never encountered tensors or differential forms. I've since learned a little about tensors (for general relativity), but that's been due to my own curiosity; I've learned a little about GA for the same reason. I never used quaternions (hence why I left them out of the above list), although I'm aware of them and that they're used e.g. in computer graphics. I used vectors, pseudovectors, and matrices a lot, and I'm certain those topics would have been easier to learn and comprehend if they'd used GA instead.