Talk:Exterior algebra#Dependence on metric and orientation

In elementary differential geometry and mathematical physics texts the exterior product seemed to more often be defined as the "determinant convention":

: $$

\omega \wedge \eta

= \frac{(k+m)!}{k!\,m!}\operatorname{Alt}(\omega \otimes \eta),

as opposed to the "Alt convention":

: $\backslash omega\; \backslash wedge\; \backslash eta\; =\; \backslash operatorname\{Alt\}(\backslash omega\; \backslash otimes\; \backslash eta)$

There is a discussion of these two "conventions" in John M. Lee, "Introduction to Smooth Manifolds" (2nd Edition) p.358.{{cite book | last=Lee | first=John | title=Introduction to Smooth Manifolds | publisher=Springer | publication-place=New York Heidelberg Dordrecht London | date=2012-08-26 | isbn=978-1-4419-9981-8 | page=358}}

{{blockquote|The choice of which definition to use is largely a matter of taste. Although the definition of the Alt convention is perhaps a bit more natural, the computational advantages of the determinant convention make it preferable for most applications, and we use it exclusively in this book. (But see Problem 14-3 for an argument in favor of the Alt convention.) The determinant convention is most common in introductory differential geometry texts, and is used, for example, in [Boo86, Cha06, dC92, LeeJeff09, Pet06, Spi99]. The Alt convention is used in [KN69] and is more common in complex differential geometry.}}

While the current article does have a section Alternating multilinear forms it is deep in the article and its practical relevance is not pointed out. I suspect a majority of people consulting Wikipedia about exterior products are taking courses or reviewing material in elementary differential geometry or physics and would appreciate an additional section, located near the beginning of the article, along the lines of Lee's textbook or other introductory presentations. Pmokeefe (talk) 15:26, 28 December 2023 (UTC)

: This is a fair enough observation, though it is based on an essentially a distinct definition (making it more than just a convention), where we delineate the alternating subset of the tensor algebra over a field of characteristic zero, on which define a new operation $\backslash wedge$. This is not directly equivalent to the approach using the quotient by the ideal, and requires characteristic 0. So this veers into pedagogy, where we would be addressing the embedding of the exterior algebra into the tensor algebra. Given the nonequivalence and the need to explain the inherent difference, it seems difficult to do more than put a mention of the section near the beginning. —Quondum 21:41, 28 December 2023 (UTC)

:The distinction is more or less just a matter of whether you consider a wedge product to represent a simplex or a parallelotope / what you consider the basic unit for hypervolume to be. If you want to relate the simplex and the parallelotope with the same specific corner, you have to multiply or divide by this scaling factor.

:But if we are defining the wedge product of vectors to "be" its own new kind of object (a "blade") without explicitly basing it on previously defined concepts, then it doesn't really have any inherent unit, and whether you consider it to represent a simplex or a parallelotope doesn't change the algebra in any meaningful way (if you make up a basis and start trying to do concrete computations for solving some practical problem, you may need to pick an interpretation). –jacobolus (t) 20:25, 29 December 2023 (UTC)

{{reflist-talk|title=}}

This is an excellent article.

But it would be improved if not just some, but all, summations were denoted by the uppercase 𝚺 notation.

Currently there are several places where the Einstein summation convention is used instead, with virtually no explanation.

It would be much better if all summations are denoted by 𝚺.

The 𝚺 summation notation is understood by all disciplines that use mathematics.

The Einstein summation convention is not.

— Preceding unsigned comment added by 2601:200:c082:2ea0:2df9:7a03:f281:1107 (talk) 17:34, 18 March 2024 (UTC)

:Please use {{tag|math|content=\Sigma}} for $\backslash Sigma$; not everybody has the correct Unicode fonts to handle Sigma.

:The Einstein Summation Convention is omnipresent in the relevant fields, even if there are fields where it is less common. I would suggest just adding a brief explanation. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:07, 19 March 2024 (UTC)

:: This article only uses the summation convention in a few places. One in the section on index notation, where it is well-justified, but should be explained and linked, and as far as I can tell only in one other place where it is not very essential. Tito Omburo (talk) 15:35, 19 March 2024 (UTC)

:::I added a link for index notation. What is the other section? Thanks. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 16:53, 19 March 2024 (UTC)

:::: Very briefly in applications where the electromagnetic field is discussed. Tito Omburo (talk) 09:12, 22 March 2024 (UTC)

::::: Thanks. #Electromagnetic field comes after #Index notation, which now links to the Einstein summation convention; is that good enough or should the reference be in both sections? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:14, 22 March 2024 (UTC)

It seems to me that this user rolled back all the changes that were a result of a careful discussion late last year among many editors.

The introduction to this article is, again, horrendously long. We have already discussed this. I do not want to have to rehash the same discussion every time - if they want to do extremely large edits, let them argue their case out in this talk page.

Otherwise, I am rolling back their changes in the lede, because of all the problems already listed below which they did not engage with, in a week or so.

MeowMathematics (talk) 06:54, 26 March 2024 (UTC)

:Personally I think the lead in Special:PermanentLink/1178847481 (which I worked on some after D.Lazard) was a better general approach than MeowMathematics's replacement which ultimately settled at Special:PermanentLink/1195382941, or the current version Special:PermanentLink/1215275338, though I'm sure it would be possible to do better than any of these. I don't think MeowMathematics's version can really be characterized as resulting from consensus, and I intended to (someday) get around to reworking it again, but didn't have the energy to wade into. YMMV. –jacobolus (t) 09:20, 26 March 2024 (UTC)

:: I'm fine with that version of the lede. The MeowMathematics version is, to me, unacceptable. I've gone ahead and put that version in. Tito Omburo (talk) 12:47, 26 March 2024 (UTC)

: I think this puts WP:BRD the wrong way around. There was no consensus for the "MeowMathematics" edits. The evidence of this lack of consensus is my rollback to earlier versions of things.

: However, on substance, the "MeowMathematics" version was clearly inferior to what had been there before. Firstly, the lede should provide an accessible overview of the article, which the MeowMathematics version did not. Secondly, the purpose of a motivation section is to motivate the definition, so it makes sense to have it be first, before a formal definition. Thirdly, the definition in terms of formal symbols was not technically correct, and also lacked a reference. Various other issues with this article were as follows. Sources had been removed from various places, which I restored. Plucker embeddings and differential forms are discussed much later in the article, and they are out of scope for a section on motivating examples. A lot of the linear algebra section was referenced to a self-published work, and seemed out of scope for this article.

: I do not see any consensus on the discussion page for MeowMathematics's edits. In fact, mostly people seemed to be at best neutral to these edits (advising them, for instance, to work on a draft before working here, or else to be bold). My changes to the lede were reverting it to an earlier consensus version based on years of discussions. My changes to the article itself mostly restored the old consensus ordering of the sections, and various consolidations of content to other parts (e.g., differential forms and Plucker embeddings to much later).

: Incidentally, my version of the article is about 10% shorter than your version, and your primary complaint seems to be that the article was too long. If you wish to change back to your version from the prior consensus version, please discuss why you think yours is better. Tito Omburo (talk) 12:31, 26 March 2024 (UTC)

A recent edit had the summary {{tqq|Motivating examples: article later uses "anticommutative" for "skew-symmetric", changed for consistency}}. The article still uses both terms. I believe that both terms should be mentioned as equivalent at first use and that all further uses stick to one or the other. Which should be preferred? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:32, 18 June 2025 (UTC)

The lead is much too long. The second and third paragraphs, dealing with blades, should be in a subsequent section. I'm not sure about the last two paragraphs, dealing with universality and generality. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:43, 18 June 2025 (UTC)

: Unfortunately the various shorter leads which have been proposed have been terrible. Tito Omburo (talk) 17:46, 18 June 2025 (UTC)

:I think the lead is actually in an okay place at the moment. Some of the historical versions of the lead have been some combination of inscrutable, incomplete, or very wordy. Compare a few selected versions in reverse chrono order: 1192296245, 1179645309, 1164784535, 1036892821, 906722806 –jacobolus (t) 22:21, 18 June 2025 (UTC)

::I actually prefer the 906722806 and 1036892821 leads. Tito Omburo (talk) 12:16, 19 June 2025 (UTC)