3 ms·

You'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/

You'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 ago
You 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 ago
  No, I would say differential forms occur in geometric algebra, not the other way around.
  - yiyus 6y ago
    Fair 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
      Differential forms aren't exactly comparable to GA... I would instead look at the relation between exterior algebra and GA. Differentials are a concept that the comes from doing calculus on manifolds, and exterior products of differentials are just used for tracking information about oriented volumes. To answer your question (switching differentials forms for for exterior algebras), you wont miss anything, as the wedge product is part of a GA.
- deleted 6y ago
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yiyus 6y ago

You 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 ago
No, I would say differential forms occur in geometric algebra, not the other way around.
- yiyus 6y ago
  Fair 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
    Differential forms aren't exactly comparable to GA... I would instead look at the relation between exterior algebra and GA. Differentials are a concept that the comes from doing calculus on manifolds, and exterior products of differentials are just used for tracking information about oriented volumes. To answer your question (switching differentials forms for for exterior algebras), you wont miss anything, as the wedge product is part of a GA.
deleted 6y ago
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