Talk:Geometric algebra

Inconsistency in Electromagnetism section

The definition of the electromagnetic field tensor is inconsistent with the formulation of Maxwell's laws. The source that gives $F=(E+icB)\gamma _{0}$ also gives Maxwell's laws as $DF=J/c\varepsilon _{0}$, not $DF=\mu _{0}J$ as stated in the article. We should either change the definition to $F=(E/c+iB)\gamma _{0}$ or change our statement of Maxwell's laws. I recommend changing $F$ to be consistent with other pages. Ghartshaw (talk) 03:25, 25 September 2020 (UTC)

Add geographic applications

No mention to "geo" in the article. Easy to cite, see for example https://doi.org/10.1080/19475683.2019.1612945

See also the old (first issue) Map algebra and "modern map algebras" in GIS applications.

See also geo-objects and geo-fields conceptualized by GIS theory in https://doi.org/10.1080/13658810600965271 (also transformations from objects to fields and from fields to objects).

— Preceding unsigned comment added by 2804:431:C7C0:76DC:CFAF:85AE:A3B2:C1A6 (talk) 11:35, 21 February 2021 (UTC)

This is an article of mathematics (I have clarified this by editing the first line of the article). The concept referred to in your links is not clear, but it is certainly not related with the one of this article. D.Lazard (talk) 12:35, 21 February 2021 (UTC)

Intro sentence is a barrage of jargon

I appreciate the effort that's gone into the article, and the desire for precision, but this intro paragraph is just not suitable for the first paragraph in a Wikipedia article as per MOS:INTRO. It's completely impenetrable to a non-expert.

In mathematics, the geometric algebra (GA) of a vector space with a quadratic form (usually the Euclidean metric or the Lorentz metric) is an algebra over a field, the Clifford algebra of a vector space with a quadratic form with its multiplication operation called the geometric product. The algebra elements are called multivectors, which contains both the scalars F and the vector space V.

I think this info could even make an appearance in paragraph 2, but paragraph 1 needs to be especially accessible to readers with no advanced mathematical training. It needs to be at a level more like the following, which could perhaps serve as a starting point for us to develop a better intro paragraph here:

In mathematics, a geometric algebra is a framework for describing properties of geometric objects in space, based on the central notions of the multivector and the geometric product. It provides an alternative to more widely-used approaches based on linear algebra or quaternions.

(Please excuse me if this isn't right; take it as a starting point for further refinement. I'm not an expert—only an amateur who has been learning about geometric algebra for the last couple of days.) --Doradus (talk) 13:43, 12 August 2021 (UTC)

Looking at the page's history, I notice that a year ago the intro was much more accessible. We seem to have injected additional jargon over the last few months.

The geometric algebra (GA) of a vector space is an algebra over a field, noted for its multiplication operation called the geometric product on a space of elements called multivectors, which contains both the scalars F and the vector space V.

This is still a bit dense, but if the non-expert glosses over such jargon as "an algebra over a field", you immediately find out that GA is noted for is "geometric product" that operates on multivectors. This was a much better intro for the non-expert. --Doradus (talk) 13:53, 12 August 2021 (UTC)

Go ahead and move stuff around, just try not to lose any info in the process. You are right in principle, the "experts" do have a tendency to "jargonize" the articles. With some justification they will say that it is "math" and it is up to the casual user to learn it. The Clifford page (a GA is a CliffordA viewed more geometrically and less algebraically) is even worse.Selfstudier (talk) 14:13, 12 August 2021 (UTC)

The problem is not jargon, but vagueness and inaccuracy: there is no usable definition in the whole article. @Doradus: your suggestion is better than the current first paragraph, but not correct since an algebraic structure is not a framework. I would suggest the following first paragraph, but I am unable to certify its correctness. If it is correct, be free to use it in the article.

In mathematics, the geometric algebra (GA) of a quadratic space (a vector space with a non-degenerate quadratic form) is an associative algebra, more precisely a Clifford algebra, that is built from the quadratic space, and solves the universal problem of maps from vector spaces to Clifford algebras. Its operation is called the geometric product, and the GA contains the vector space and its exterior algebra as subspaces.
The geometric algebra allows carrying all operations of linear and multilinear algebra in a single algebraic structure (scalar product, exterior product, tensor product, determinant, tensor calculus, etc.).

D.Lazard (talk) 17:08, 12 August 2021 (UTC)

I've decided to be bold and craft an intro sentence much like the one from a year ago. I tried to retain all the info in the intro section, as per Selfstudier, but took a little of it out of that first sentence so it would be less daunting. I'm open to suggestions if anyone would like to make further changes. --Doradus (talk) 19:37, 14 August 2021 (UTC)

Reversion notation

Hi HongGong, in https://en.wikipedia.org/w/index.php?title=Geometric_algebra&diff=1016190290&oldid=1016055619 (reverted) and then in https://en.wikipedia.org/w/index.php?title=Geometric_algebra&diff=1030284754&oldid=1018780264 you changed the notation for reversion from dagger to tilde. I searched the talk page for an explanation, but could not find any. Your second edit comment merely says "The notation is correct." Notation is a convention. Since the dagger notation was defined in the article itself (just as the tilde notation now is), it was just as correct as the tilde notation (as long as it does not clash with other notation). I noticed that you've also changed a different article in a similar way (https://en.wikipedia.org/w/index.php?title=Rotor_(mathematics)&diff=1016188882&oldid=1001315838), with an edit comment of similar quality ("General cleanup"). Can you please explain what makes tilde better than dagger here? --RainerBlome (talk) 21:49, 3 December 2021 (UTC)

Meet

In the article the Grassmann exterior product is - be it between quotes - referred to as the "meet". Allthough both concepts use the same symbol, in my opinion there is no relation between the two. Madyno (talk) 16:46, 2 December 2022 (UTC)

It says that the meet is the dual of the Grassmann exterior product. -Bryan Rutherford (talk) 17:44, 2 December 2022 (UTC)