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What is the inverse of a vector?

mattferraro.dev

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Re: What is the inverse of a vector?

#11

Post by Grustaf »

BlackFly wrote: post #9
Grustaf wrote: post #4

Ah, another geometric algebra evangelist? I can't figure out if GA actually adds anything substantial, or if it merely lets us write some equations in a more succinct fashion. But it certainly looks cool. As to vectors, obviously they have inverses, additive inverses. Since vectors don't have multiplication, there is no multiplicative inverse, but if you define new operations on them, well then that operation can hav…

If you are working in ordinary Euclidean space with orthonormal bases, it doesn't do much for you. When you start doing calculus on embedded surfaces (the beginning of differential manifolds) it begins to be more helpful. Since Lie algebras are special types of differential manifolds, you can learn quite a bit by studying the geometry and this quickly leads into gauge theory and modern physics. The Geometry of Physic…

That books seems very interesting, but is it really about Geometric Algebra? I'm not talking about geometry in general, or differential geometry, which I'm a bit familiar with. And not Algebraic Geometry either, for that matter, which is also a fascinating subject.

I mean specifically Geometric Algebra.

It sort of seems like a notation, but it has almost a cult like following and perhaps it's more than a notation, is it a theory, a branch of mathematics?

Re: What is the inverse of a vector?

#12

Post by yuppiemephisto »

Grustaf wrote: post #11
BlackFly wrote: post #9

Earlier quoted context omitted.

If you are working in ordinary Euclidean space with orthonormal bases, it doesn't do much for you. When you start doing calculus on embedded surfaces (the beginning of differential manifolds) it begins to be more helpful. Since Lie algebras are special types of differential manifolds, you can learn quite a bit by studying the geometry and this quickly leads into gauge theory and modern physics. The Geometry of Physic…

That books seems very interesting, but is it really about Geometric Algebra? I'm not talking about geometry in general, or differential geometry, which I'm a bit familiar with. And not Algebraic Geometry either, for that matter, which is also a fascinating subject. I mean specifically Geometric Algebra. It sort of seems like a notation, but it has almost a cult like following and perhaps it's more than a notation, is…

It really is useful even at the advanced level. Following it far enough leads to the Atiyah-Singer Index Theorem and Hodge Theory. The advantage over the exterior algebra is that you have that and an interior algebra, which leads to many formulas in differential geometry becoming very natural (like Cartan's magic formula).

Re: What is the inverse of a vector?

#13

Post by musgravepeter »

Grustaf wrote: post #11
BlackFly wrote: post #9

Earlier quoted context omitted.

If you are working in ordinary Euclidean space with orthonormal bases, it doesn't do much for you. When you start doing calculus on embedded surfaces (the beginning of differential manifolds) it begins to be more helpful. Since Lie algebras are special types of differential manifolds, you can learn quite a bit by studying the geometry and this quickly leads into gauge theory and modern physics. The Geometry of Physic…

That books seems very interesting, but is it really about Geometric Algebra? I'm not talking about geometry in general, or differential geometry, which I'm a bit familiar with. And not Algebraic Geometry either, for that matter, which is also a fascinating subject. I mean specifically Geometric Algebra. It sort of seems like a notation, but it has almost a cult like following and perhaps it's more than a notation, is…

There is Geometric Algebra for Physicists by Doran and Lasenby (2003). It recasts mechanics, E&M up to gauge theories and GR into geometric algebra. I stalled out at mechanics but I've now taken it off the Tsundoku pile and may give it another chance.

Re: What is the inverse of a vector?

#14

Post by csdvrx »

spekcular wrote: post #7

The article begins: >In this post we will re-invent a form of math that is far superior to the one you learned in school. The ideas herein are nothing short of revolutionary. and concludes: > I firmly believe that in 100 years, Geometric Algebra will be the dominant way of introducing students to mathematical physics. In the same way that Newton's notation for Calculus is no longer the dominant one, or that Maxwell's…

> Geometric algebra is, as the article points out, a more powerful version of the usual vector notation. But it is deficient in various ways when compared to tensor notation (for calculations) and differential forms (e.g. if you want to work basis-free). [I'm oversimplifying a bit, but a full discussion is too long for a comment here.]

I have no opinion about the claims, but I loved the article, as I quickly saw the gains from this algebra for my very basic needs.

After checking out your 2 suggested alternatives, I'm not so convinced they are easier to understand.

Re: What is the inverse of a vector?

#15

Post by ur-whale »

Geometric algebra (Clifford Algebra) unfortunately came late historically.

It's a shame, because the whole theory is a very useful (eg for engineering / applied math) superset of linear algebra.

I really wish I had learned this first in my undergrad years, would have made a whole bunch of things way clearer from the get go:

    differential forms

    tensor calculus

    linear algebra

    etc
From zero to geo is a very good video introduction to the topic:

https://www.youtube.com/watch?v=2hBWCCAiCzQ&list=PLVuwZXwFua...

Re: What is the inverse of a vector?

#19

Post by contravariant »

BlackFly wrote: post #9
Grustaf wrote: post #4

Ah, another geometric algebra evangelist? I can't figure out if GA actually adds anything substantial, or if it merely lets us write some equations in a more succinct fashion. But it certainly looks cool. As to vectors, obviously they have inverses, additive inverses. Since vectors don't have multiplication, there is no multiplicative inverse, but if you define new operations on them, well then that operation can hav…

If you are working in ordinary Euclidean space with orthonormal bases, it doesn't do much for you. When you start doing calculus on embedded surfaces (the beginning of differential manifolds) it begins to be more helpful. Since Lie algebras are special types of differential manifolds, you can learn quite a bit by studying the geometry and this quickly leads into gauge theory and modern physics. The Geometry of Physic…

Is there some canonical metric on Lie algebras that makes this possible? Otherwise geometric algebra won't get you far.

Re: What is the inverse of a vector?

#20

Post by vanderZwan »

Garlef wrote: post #8

While the article is written very nicely, It seems that this is written out of a perspective of some missing knowledge. The basic object that the author seems to be interested in is that of an "algebra over a field" ( https://en.wikipedia.org/wiki/Algebra_over_a_field ). Specifically: Invertability of all elements with respect to the multiplication leads to the notion of division algebra and these have been studied f…

> It seems that this is written out of a perspective of some missing knowledge.

Well, the author talks about what "we learned in school", not university, so that checks out but only because you two have different audiences in mind.

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