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Thanks for this! Now I understand what geometric algebra refers to. When people talk about it, it always sounds like a simple generalization of exterior algeb

Thanks for this! Now I understand what geometric algebra refers to. When people talk about it, it always sounds like a simple generalization of exterior algebra. And so it is. You allow a formal sum of a 0-form, a 1-form, and so on up to an n-form; instead of x∧x being zero, it's |x|². I assume the textbooks show how to do all this in a basis-independent manner. Can you do the same thing with chains, and come up with an even-more-generalised Stokes theorem? There are two things I don't get. Why start calling things 2-vectors, when you could call them 2-forms and people would know what you were talking about? Also, when discussing 3D rotations, the article decomposes a vector u as the sum of two vectors a + b, where a·i = b∧i = 0, and i is a 2-vector. I'm fine with b∧a, and I know that there is an inner product for forms of the same degree, though I'd have to look up how it works. But how do you take the inner product a·i of a vector and a 2-form? I'd guess this is an abuse of notation whose meaning is obvious if you're used to it.

creata 6y ago
> When people talk about it, it always sounds like a simple generalization of exterior algebra. And so it is. Sure, both are quotients of tensor algebras, but if anything, it's the other way around. Geometric algebra is to exterior algebra as inner product spaces are to vector spaces, or as Riemannian manifolds are to smooth manifolds. I'd say that vector spaces are more general than inner product spaces. > Why start calling things 2-vectors, when you could call them 2-forms and people would know what you were talking about? Because they're trying to appeal to the kind of physicist that doesn't know what a 2-form is. Besides, bivector is already established terminology, too. > But how do you take the inner product a·i of a vector and a 2-form? Yeah, this isn't great, since there are way too many notions of "inner product" in geometric algebra (see The Inner Products of Geometric Algebra: 10.1007/978-1-4612-0089-5_2). But I guess the intent is clear enough.
- thisrod 6y ago
  > Geometric algebra is to exterior algebra as inner product spaces are to vector spaces To get from x∧x to |x|², you need a norm. And I guess linearity must break down somewhere unless that norm is compatible with an inner product. Thanks, I missed that point. > But I guess the intent is clear enough. Once you have u and b, you can define a = u - b.
JoeCamel 6y ago
> Why start calling things 2-vectors, when you could call them 2-forms and people would know what you were talking about? There is more to 2-forms. Also, to be precise, they are functions from a vector space to the scalar field (usually reals). Sure, they form a (dual) vector space but I think it would be more confusing to call GA elements k-forms.

creata 6y ago

> When people talk about it, it always sounds like a simple generalization of exterior algebra. And so it is. Sure, both are quotients of tensor algebras, but if anything, it's the other way around. Geometric algebra is to exterior algebra as inner product spaces are to vector spaces, or as Riemannian manifolds are to smooth manifolds. I'd say that vector spaces are more general than inner product spaces. > Why start calling things 2-vectors, when you could call them 2-forms and people would know what you were talking about? Because they're trying to appeal to the kind of physicist that doesn't know what a 2-form is. Besides, bivector is already established terminology, too. > But how do you take the inner product a·i of a vector and a 2-form? Yeah, this isn't great, since there are way too many notions of "inner product" in geometric algebra (see The Inner Products of Geometric Algebra: 10.1007/978-1-4612-0089-5_2). But I guess the intent is clear enough.

thisrod 6y ago
> Geometric algebra is to exterior algebra as inner product spaces are to vector spaces To get from x∧x to |x|², you need a norm. And I guess linearity must break down somewhere unless that norm is compatible with an inner product. Thanks, I missed that point. > But I guess the intent is clear enough. Once you have u and b, you can define a = u - b.

JoeCamel 6y ago

> Why start calling things 2-vectors, when you could call them 2-forms and people would know what you were talking about? There is more to 2-forms. Also, to be precise, they are functions from a vector space to the scalar field (usually reals). Sure, they form a (dual) vector space but I think it would be more confusing to call GA elements k-forms.