Talk:Bivector

Exterior Product

The exterior product section states "To calculate a ∧ b consider the sum [...]". However, what follows is how to calculate (a ∧ b)², which is not quite the same thing.

According to exterior algebra, the calculation for two dimensions should be a ∧ b = (det [a b])e₁ ∧ e₂ = (a₁b₂ − a₂b₁)e₁ ∧ e₂.

84.202.54.247 (talk) 02:56, 9 November 2014 (UTC)

perhaps 'calculate' is not the best term. What follows is an argument that it's not a number as it has a negative square, hence the calculation of (a ∧ b)², so must be something different. It doesn't directly calculate it as it's not needed and as it's an argument not restricted to two dimensions. Personally I find starting in two dimensions harder as geometric algebra and the bivector as so restricted - there is in effect only one bivector.--JohnBlackburne^words_deeds 15:22, 9 November 2014 (UTC)

I've reworded it a bit to reduce the potential confusion. It is still not great, though. —Quondum 15:42, 9 November 2014 (UTC)

Biradials

For the true bivector see bivector (complex) where the classical concept can be understood as the Lie algebra elements of the Lorentz group in its representation with biquaternions. Dismissal of the original language gave rise to some authors using this term for a multivector, in particular a 2-vector. A discussion of the English vs Continental usage was held on this Talk venue, but (too hastily) archived (see #2). Proponents of geometric algebra may appreciate what happened with Jules Hoüel and his Theorie elemetaire des quantites complexes, Part 4:

• J. Hoüel (1874) Éléments de la Théorie des Quaternions, Gauthier-Villars publisher, link from Google Books.

See page 386. Addition of biradials. Biradiales rectangles. He says two biradiales are equal when they have the same modulus and the same versor.

Now versors have some meaning, they are elements of elliptic geometry#Elliptic space and fascinated W. K. Clifford. But the bivectors of this article are a sham and have no support outside of GA.Rgdboer (talk) 02:51, 10 March 2015 (UTC)

You may think that the first traceable use of a word is the "true" meaning of the word. I hope not: language moves forwards, and an encyclopaedia should not be written using archaic terminology. The modern use if a term can quite legitimately differ from that at earlier points in history. —Quondum 04:29, 10 March 2015 (UTC)

Progress

An editor in Switzerland made changes on May 31 attempting to salvage this article. The reference to Encyclopedia of Mathematics was removed since now the EoM article expands on Plucker coordinates under the title "Bivector", and thus EoM does not any longer support the 2-vector meaning of this article. The History section was modified, yielding even more evidence of the true bivector (complex). As there is no arguing with William Rowan Hamilton (1853) who originated both vector and bivector, the defenders this article must eventually yield to the principles on which this Project is founded and the general understanding that facts have consequences. Rgdboer (talk) 22:10, 15 June 2015 (UTC)

I think you misunderstand the EoM article entirely. The meaning it gives is exactly equivalent to 2-vector up to a fixed factor, although it expresses it in a rather archaic fashion (just the sort of language that you seems to appeal to you). In this light, your assertion seems to me to lack coherency. —Quondum 02:58, 16 June 2015 (UTC)

Why the focus on geometric algebra?

In its current form, this article is almost entirely about bivectors in the context of geometric algebra. They are certainly important there! But I think it is strange and inappropriate for a general article on bivectors to be formulated in a language that is sufficiently unfamiliar to typical readers to require considerable time defining the geometric product and its other basic concepts from the ground up.

Bivectors have plenty of applications outside of geometric algebra: they can be used to describe rotations and related physical quantities or magnetic fields (without any reference to the geometric product, etc.), they are important in the Classification of electromagnetic fields and in Petrov classification of spacetimes, they can describe areas with orientation (with applications to the effect of curvature on parallel transport: the Riemann tensor maps area bivectors to rotation bivectors), and undoubtedly many more phenomena.

None of this requires geometric algebra, and this article as written today is close to useless for someone trying to learn the basics of bivectors for those applications. Old comments on the EM field classification Talk page suggest that there's a significant mathematical literature on bivector properties and classification that could be very interesting to discuss here, for example. I expect that a substantial section discussing the role of bivectors in geometric algebra would be interesting and important to include. But it should draw on general results about them: it shouldn't be the context where those results are derived. Steuard (talk) 02:26, 11 July 2023 (UTC)

Area element

This article misinterprets the differential geometry concept of an area element. The misunderstanding is shown by the expression "vectors can be thought of as directed line segments". See Equipollence (geometry) for the proper notion as an equivalence class of such segments. The equivalence relation for area elements depends on transformations that preserve area elements. A precise expression was given by Mikhail Postnikov in his Analytic Geometry (1957) where he uses shear mapping and squeeze mapping to relate equivalent area elements. The mathematics of this relation is described at Signed area#Postnikov equivalence. In short, this article is misleading and can be considered as supplement to Comparison of vector algebra and geometric algebra. Readers should observe that "geometric algebra" is not a branch of mathematics, but algebraic geometry is. — Rgdboer (talk) 00:25, 8 January 2024 (UTC)

Feedback Request

@Quondum: My edits to this article were intended much as in the GA article. I have this feeling that this article should be even more elementary than the main GA article, as physics students often use bivectors without the rigorous details over their construction. Thus I invite you to look over my recent edits to this article as well. Jasper Deng (talk) 23:18, 6 March 2024 (UTC)

I may take a week or two to get to this – I have too much happening for a while. The definition of a bivector does not depend on the quadratic form, so perhaps this should be pointed out or not stressed as much. —Quondum 16:33, 7 March 2024 (UTC)

Query of edit comment

Quantling, on this edit, your edit comment (Use {{math}} rather than {{nowrap}} to prevent line wrapping within math expressions) puzzles me: {{math}} and {{nowrap}} have exactly the same wrapping behaviour. Can you clarify? —Quondum 05:50, 9 March 2024 (UTC)

Yes, they are the same with regards to wrapping. My thinking is that {{math}} also does other mathy things, such as using the same font as <math> does, so I use {{math}} for no-wrapping math and {{nowrap}} for no-wrapping non-math. If my comment doesn't say that to you, I'm open to other wording, which I can use in the future.—Quantling (talk | contribs) 22:59, 10 March 2024 (UTC)

All good. Your next edit comment says this. I should have put it down to an unfortunate wording in this instance. —Quondum 04:20, 11 March 2024 (UTC)