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The Cambridge University GA Research Group’s website along with the “Geometric Calculus R & D Home Page” should serve as a good introductions to geometric algebra, along with the Wikipedia reference – “Geometric algebra”. A somewhat more advanced exploration in the context of differential geometry can be found in the book titled “A New Approach to Differential Geometry using Clifford's Geometric Algebra”. A development with more of a 'universalist' flavour (universal as in universal algebra) is presented at the geocalc.clas.asu.edu page “Universal Geometric Calculus ”.

Should geometric algebra be presented to undergraduates in engineering, physics and mathematics early in their curriculum instead of, or alongside, linear algebra?

Good answers should try to analyze:

  • which subject is more fundamental, in an attempt to identify where they should fit in in a curriculum

  • what changes would be required in the general curriculum in order to meaningfully support an early introduction of geometric algebra, and whether such changes would be feasible given the current infrastructure

  • why such curriculum changes have not been made thus far

  • examples of personal or institutional experiences resulting from an attempt to make such a curriculum change (if any exist)

  • whether a problematic or beneficial slippery slope exists -- could an (escalated, seemingly more "extreme") argument be made for the introduction of "applied abstract algebra" early in high school as a consequence of serious consideration given to the main question?

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    $\begingroup$ In terms of experiences: I've gotten some recommendations about Macdonald's textbook doing exactly what you propose. $\endgroup$ – Willie Wong Mar 14 '14 at 13:00
  • $\begingroup$ @WillieWong Hey, that's neat. Where can I read about this? $\endgroup$ – user89 Mar 16 '14 at 5:30
  • $\begingroup$ Eh, read about what? The recommendations given to me were word-of-mouth, so I doubt you can read about it anywhere. If you want to know more about the texbook, follow the link in my previous comment. $\endgroup$ – Willie Wong Mar 17 '14 at 8:16
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    $\begingroup$ Macdonald also has a follow-up textbook: Vector and Geometric Calculus. The two books could be used to replace traditional courses on linear algebra and multivariable/vector calculus. $\endgroup$ – J W Apr 1 '14 at 16:50
  • $\begingroup$ Emil Artin wrote a very nifty book called Geometric Algebra but it covers material rather different from what was meant in the original question. Here is a reference to Artin's book: books.google.com/… $\endgroup$ – Joseph Malkevitch Apr 8 '16 at 21:25
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As Willie Wong suggests in the comments to your question, it could be possible to use Alan MacDonald's textbook Linear and Geometric Algebra as an alternative to a more conventional linear algebra course. See also MacDonald's webpage on the book at http://faculty.luther.edu/~macdonal/laga/.

If you have the freedom to teach linear algebra MacDonald-style before multivariable/vector calculus then you can follow up with Vector and Geometric Calculus. Once more, see also MacDonald's webpage on the book at http://faculty.luther.edu/~macdonal/vagc/.

The OP mentions undergraduates in engineering, physics and mathematics. There's also computer science. Geometric algebra has applications to computer graphics, computer vision and robotics. (Admittedly, the latter two topics, especially robotics, are also studied in engineering.) See, for instance, Geometric Algebra for Computer Science by Dorst, Fontijne & Mann. The webpage for the book is: http://www.geometricalgebra.net/. It would probably go particularly well after MacDonald's aforementioned Linear and Geometric Algebra.

I own all three of the above books, but I have not used them yet in a course. One of the challenges is overcoming the inertia of standard approaches, especially when they work well for most practical purposes. A new approach may need to be demonstrably superior before having a chance of ousting the incumbent.

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    $\begingroup$ ... and it won't show superiority into tried $\endgroup$ – vonbrand May 25 '14 at 16:24
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    $\begingroup$ @vonbrand: Are you referring to a Catch-22 situation? Was "into" meant to be "until" in your comment above? $\endgroup$ – J W May 25 '14 at 17:02
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    $\begingroup$ Yes, stupid soft keyboard $\endgroup$ – vonbrand May 25 '14 at 20:10
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My own introduction to this topic was Alan MacDonald's early paper,

A Survey of Geometric Algebra and Geometric Calculus

Capable Advanced level Mathematics and Physics pupils could be directed towards this freely available paper and in particular to Section 3.1.6. Electromagnetism. When they see the simplification (unification) offered by the multivector approach some may realize that Geometric Calculus could, in fact, be a new paradigm in mathematics. They might then seek out undergraduate courses offering the subject.

Personally, I believe that the two books, abbreviated LAGA and VAGC, will eventually become standard texts and that the first sight of Geometry (and Algebra), for even the youngest students, will be Geometric Algebra (GA).

Returning to the main question though... The entry point in the curriculum could be whenever the concept of complex numbers is usually introduced in mathematics. I would avoid any usage of the word "abstract" as in the phrase "applied abstract algebra". The whole ethos of GA is its power in the unification of physics.

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