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Personally, an appealing part of GA is clarifying that the magnetic field is best viewed as an "oriented area", instead of a pseudo-vector [1], which is a conce
by sgdpk 6y ago
Personally, an appealing part of GA is clarifying that the magnetic field is best viewed as an "oriented area", instead of a pseudo-vector [1], which is a concept that is frequently ill-defined. Usually, lecturers say that the magnetic field is essentially a vector, but to be careful that it flips directions when you mirror space. But how then can you check this by just looking at the usual three coordinates of the vector?
With GA, you associate the magnetic field with an oriented plane element. Oriented planes have this flip-mirroring property. Because space is 3D, you have exactly one vector to which this plane element is perpendicular, so you can identify the field with a vector. This is called the Hodge duality, and this particular form is accidental to 3D space.
If you need more convincing, the magnitude of the magnetic field is given by the cross-product, which calculates the area spanned by two vectors.
You could argue that you don't need GA for this, only to use the exterior product instead of the cross product. But by using an algebra that contains both vectors and oriented areas, you can effectively sum the electric and magnetic fields. This will give you pretty compact and beautiful equations, which is the essence of this article.
[1] https://en.wikipedia.org/wiki/Pseudovector https://en.wikipedia.org/wiki/Pseudovector
- aesthesia 6y agoYou still don’t need geometric algebra to combine the magnetic and electric fields, if you view them as differential forms on 4-dimensional spacetime. From what I can tell, this approach is basically equivalent to what this article does. You could directly translate everything into the language of differential forms, because the only geometric products here are in fact just exterior products.
- yiyus 6y agoThis "competition" between geometric algebra and differential forms makes me uncomfortable. As far as I see (and I'm not an expert), they are just different ways to express very similar concepts. They are still not exactly the same, since the geometric product is not defined in DFs, and there is no hodge star operator in GA, for example, but everything you can do using one formalism in practice can also be easily done using the other one. What am I missing?
- DreamScatter 6y agoYou're mistaken, GA does have a Hodge star, as I've explained many times before https://grassmann.crucialflow.com/dev/algebra https://grassmann.crucialflow.com/dev/algebra The exterior product can be derived from the geometric product, so differential forms occur in geometric algebra.
- yiyus 6y agoYou can easily define it, that's what I meant saying that you can do the same things in practice, but it's not usually defined (at least in the books and articles I've read), and certainly it is not so ubiquitous as in DFs texts. And, of course, the exterior product is contained in the geometric product. I guess that, in the same way, you could define a geometric product operator when using a DFs formulation. Would you then say that geometric algebra occurs in differential forms? In any case, you did not attempt to answer my original question. Are GA and DFs just different ways to define "equivalent" concepts or is there some more fundamental difference that I am missing?
- DreamScatter 6y agoNo, I would say differential forms occur in geometric algebra, not the other way around.
- yiyus 6y agoFair enough. I have seen some comments (not in this thread, it was some time ago) that suggested that DFs allow the same as GA in practice, and everything GA does is adding an unnecessary geometric product, but exterior products should be enough (not my opinion, I can try to find the original comment if you want). I do not know enough to have an own opinion. You obviously know more than me about this, so I will ask you a slightly different question: if I learn GA well enough and totally ignore differential forms, what will I miss?
- chobytes 6y ago
- bollu 6y agoFor anyone looking for a sane way to study this stuff, what finally made sense to me was (1) study discrete differential geometry from Keenan Crane's notes: https://www.cs.cmu.edu/~kmcrane/Projects/DDG/ https://www.cs.cmu.edu/~kmcrane/Projects/DDG/ ; (2) Read these notes on geometric algebra that formally lay down what the hell a GA space is and what operations one can perform on them: https://arxiv.org/pdf/1205.5935.pdf https://arxiv.org/pdf/1205.5935.pdf (Geometric Algebra: Eric Chisolm)
- patrec 6y agoFor clicking convenience: https://www.cs.cmu.edu/~kmcrane/Projects/DDG/ https://www.cs.cmu.edu/~kmcrane/Projects/DDG/ (the link above is broken since HN includes the ]; in the URL).
- pastrami_panda 6y agoThis is also an excellent and interactive blog-post building up to the understanding of how rotors can replace quaternions in GA: https://marctenbosch.com/quaternions/ https://marctenbosch.com/quaternions/
- wwarner 6y ago+1 to this exposition!