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I cannot recommend Alan MacDonald's two books enough (Linear and Algebra; and, Vector and Geometric Calculus). They literally replace reams of texts on complex

I cannot recommend Alan MacDonald's two books enough (Linear and Algebra; and, Vector and Geometric Calculus). They literally replace reams of texts on complex numbers, quaternions, octonions, spinors, etc. under a single unifying concept. (They don't replace those concepts; and, for sure, if you want to learn quantum you're going to need them; but they make getting your head around them so much easier.) One issue is that most of the "online" references are crap. MacDonald's presentation is a wonderful slow ramp: it starts with really basic linear algebra and then just slowly segue's into geometric algebra. I firmly believe that in 100 years we just won't be teaching 'bare' linear algebra, complex, quaternions, ... we'll just teach GA and then introduce the physical concepts separately.

yiyus 6y ago
Those books are great. They are my favorite GA books (together with Hestenes' papers). I just wanted to add that he also has wonderful youtube videos. And I also agree with that 100 years prediction. I'm even trying to do my part so that we do not have to wait so much!
topspin 6y ago
To save others the effort the two books are: Linear and Geometric Algebra (October 2020 printing) 978-1453854938 $30 new @Amazon Vector and Geometric Calculus (also Oct. 2020 prining) 978-1480132450 $35 new @Amazon Affordable, surprisingly.
zrkrlc 6y ago
I spent around two months being completely enamoured by geometric algebra until I realised it flies in the face of some pretty strong trends of mathematics in the last couple of decades. How come? Well, we’ve been moving towards more and more powerful generalisations and unifying concepts over the years. First, it was the various flavours of algebraic this-and-that, then category theory, then today, people are trying to do hot things with HoTT. (of course math is much, much bigger than this, but this is the story I particularly have been watching unfold before me) The trouble with geometric algebra is that it’s merely a special case of a much more general concept, namely tensor algebras. Tensor algebras are the most general algebra you can possibly get from things that behave like vectors; with GA you are stuck with vectors where the distances are measured by something squared as opposed to any other power (or some other crazy function like the taxicab metric). Using geometric algebra feels like typeless programming because you’re combining everything into a single quantity called the multivector whose components aren’t straightforward to keep track of. Furthermore, you have to decide the dimension in advance because IIRC these types change when you add more dimensions (the formulas however do not, surprisingly). That said, it’s really marvellous how compact things can get if you stay in 3+1 dimensions. Maxwell’s equations become literally a one-liner, and the horrible, horrible mess that is rotational dynamics becomes so much more tractable for mortals like me. One can literally do the entire physics undergrad curriculum using GA without breaking a sweat. Trouble is, you’d have to learn some basic algebraic theory and that’s a hard sell for people who are already afraid of math because “it’s too abstract”.
- adamnemecek 6y ago
  GA and tensors are closely related. I would say that tensors are more like typeless programming, the fact that you pick an algebra in advance is similar to encoding your invariants in terms of types. Tensors are cool and all but they are a pain to work with. GA is more intuitive for working with space.
- Koshkin 6y ago
  > Maxwell’s equations become literally a one-liner There still must be a limit to this. For example, I do not find "dF = 0" especially enlightening. It's almost like saying that "42" is the answer to the question about life, universe, and everything...
  - deleted 6y ago
    [deleted]
  - layoutIfNeeded 6y ago
    I guess it depends on how you define “dF”. The less arbitrary the definition is, the more likely that you’ve found something profound by stating “dF = 0”.
  - adamnemecek 6y ago
    Check the one line https://slehar.files.wordpress.com/2014/05/maxwells1.png https://slehar.files.wordpress.com/2014/05/maxwells1.png From https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction/ https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-v...
    - alquemist 6y ago
      That's cheating. One would need to also define F, c, ε0, J, assuming ∇ is known from calculus. Which would make for 5 lines :)
      - handol 6y ago
        You can't really get away with just knowing del from vector calculus, as this is the geometric-calculus del. ∇F = ∇ . F + ∇ /\ F But for defining the terms you're going to need a couple paragraphs to go along with the one equation.

yiyus 6y ago

Those books are great. They are my favorite GA books (together with Hestenes' papers). I just wanted to add that he also has wonderful youtube videos. And I also agree with that 100 years prediction. I'm even trying to do my part so that we do not have to wait so much!

topspin 6y ago

To save others the effort the two books are: Linear and Geometric Algebra (October 2020 printing) 978-1453854938 $30 new @Amazon Vector and Geometric Calculus (also Oct. 2020 prining) 978-1480132450 $35 new @Amazon Affordable, surprisingly.

zrkrlc 6y ago

I spent around two months being completely enamoured by geometric algebra until I realised it flies in the face of some pretty strong trends of mathematics in the last couple of decades. How come? Well, we’ve been moving towards more and more powerful generalisations and unifying concepts over the years. First, it was the various flavours of algebraic this-and-that, then category theory, then today, people are trying to do hot things with HoTT. (of course math is much, much bigger than this, but this is the story I particularly have been watching unfold before me) The trouble with geometric algebra is that it’s merely a special case of a much more general concept, namely tensor algebras. Tensor algebras are the most general algebra you can possibly get from things that behave like vectors; with GA you are stuck with vectors where the distances are measured by something squared as opposed to any other power (or some other crazy function like the taxicab metric). Using geometric algebra feels like typeless programming because you’re combining everything into a single quantity called the multivector whose components aren’t straightforward to keep track of. Furthermore, you have to decide the dimension in advance because IIRC these types change when you add more dimensions (the formulas however do not, surprisingly). That said, it’s really marvellous how compact things can get if you stay in 3+1 dimensions. Maxwell’s equations become literally a one-liner, and the horrible, horrible mess that is rotational dynamics becomes so much more tractable for mortals like me. One can literally do the entire physics undergrad curriculum using GA without breaking a sweat. Trouble is, you’d have to learn some basic algebraic theory and that’s a hard sell for people who are already afraid of math because “it’s too abstract”.

adamnemecek 6y ago
GA and tensors are closely related. I would say that tensors are more like typeless programming, the fact that you pick an algebra in advance is similar to encoding your invariants in terms of types. Tensors are cool and all but they are a pain to work with. GA is more intuitive for working with space.
Koshkin 6y ago
> Maxwell’s equations become literally a one-liner There still must be a limit to this. For example, I do not find "dF = 0" especially enlightening. It's almost like saying that "42" is the answer to the question about life, universe, and everything...
- deleted 6y ago
  [deleted]
- layoutIfNeeded 6y ago
  I guess it depends on how you define “dF”. The less arbitrary the definition is, the more likely that you’ve found something profound by stating “dF = 0”.
- adamnemecek 6y ago
  Check the one line https://slehar.files.wordpress.com/2014/05/maxwells1.png https://slehar.files.wordpress.com/2014/05/maxwells1.png From https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction/ https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-v...
  - alquemist 6y ago
    That's cheating. One would need to also define F, c, ε0, J, assuming ∇ is known from calculus. Which would make for 5 lines :)
    - handol 6y ago
      You can't really get away with just knowing del from vector calculus, as this is the geometric-calculus del. ∇F = ∇ . F + ∇ /\ F But for defining the terms you're going to need a couple paragraphs to go along with the one equation.