Comment by Cybiote | Hacker News Reader

Comment by Cybiote | Hacker News Reader

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spekcular is correct. One could draw the following graph very roughly capturing how these concepts developed:

Grassman’s exterior algebra -> Differential forms

Cartan’s Exterior Derivatives -> Differential forms

Quaternions -> Clifford Algebra

exterior algebra -> Clifford Algebra -> geometric algebra

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.

> 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.

> Differential forms are a half-baked formalism.

I can't emphasize enough how wrong this is. It's the standard formalism in research-level physics and math for good reasons.

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.

?? You'll likely be better off seeing different perspectives of the same thing.

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