Talk:Exterior algebra

Structure of the first sections.

The first section after the lead is § Motivating examples. This section is a mess, as it supposes already known what is the wedge product, and it uses many other concepts that are not supposed to be known by most readers, such as "a metric tensor field and an orientation", "the orientation $\mathbf {e} _{1}\times \mathbf {e} _{2}$ and with the metric ${\begin{bmatrix}1&0\\0&1\end{bmatrix}}$", etc. So, this section must be moved after a definition section, shortened, and rewritten for showing that use of exterior algebra simplifies these examples. Such a renewed section could also be renamed "Motivating applications".

The next section is § Formal definitions and algebraic properties. In particular, it uses "is defined" for one of several definitions, and is uses a definition that involves too much algebraic knowledge. So this section must be moved later in the article, renamed "Other definitions", include the definition with the universal property, and show that the definition as a quotient of the tensor algebra is, in fact, a proof of the existence of a solution for the universal property.

This implies to create a first section "Definitions" after the lead that could begin with Let V be a vector space over a field F, and $(e_{i}\mid i\in I)$ be an ordered basis of it. For every natural number, the kth exterior power of V is the vector space ${\textstyle \bigwedge ^{k}V}$ that has a basis formed by the "formal wedge products" $e_{i_{1}}\wedge \cdots \wedge e_{i_{k}}$ such that $i_{1}<\cdots <i_{k}.$ Using the convention that the empty product is 1, one has ${\textstyle \bigwedge ^{0}V=F}$ and ${\textstyle \bigwedge ^{1}V=V.}$ The exterior algebra $\bigwedge V$ is the direct sum ${\textstyle \bigoplus _{k=0}^{\infty }\bigwedge ^{k}V,}$ that is, the vector space with all above formal wedge products as a basis. Then, the section may continue with the definition of the (wedge) product in this vector space, the fact that this makes the exterior algebra an alternating algebra, the fact that a change of basis for V induces a change of basis on the exterior algebra, and thus that the definition does not really depend on the base choice, etc. This in this section that the definitions of blades, multivectors, etc., given in the lead could be moved.

I'll probably not have the time for restructuring the article myself. So, if there is a consensus for the above project, it would be great if someone else do it. D.Lazard (talk) 11:14, 6 October 2023 (UTC)

I believe that it would be better not to define in terms of a specific basis. Perhaps have a section called Informal definition at the beginning of Motivation and say that it's an algebraic abstraction from oriented areas. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:46, 6 October 2023 (UTC).

I also don't think you should invoke a basis as a fundamental definition, if possible. –jacobolus (t) 14:57, 6 October 2023 (UTC)

I agree that, conceptually, a coordinate free definition is much better. But, as it is generally the case conceptual simplicity involves more abstractness and more technical background. A typical example is the definition of a real affine space as $\mathbb {R} ^{n}$ with the zero and the standard basis forgotten. This is because of the technicality needed to define "forget", that many Wikipedia articles talk of $\mathbb {R} ^{n}$ as "the" affine space of dimension n. Here, the conceptually simplest definition is as a solution of a universal problem. But this needs some technicality to show the existence of a solution and to establish the properties that are needed for manipulating multivectors. Moreover, this very abstract definition requires some expertise to be really understood. So, it is difficult to introduce this definition in the first section of the body.

The definition given in the article as a "formal definition" (I never understood what could be an informal definition) is not convenient either for the first section, since it involves advanced concepts of algebra such as "tensor algebra", "generated ideal" and "quotient by an ideal".

So, we are left with the definition that prevailed before Bourbaki, which it the definition that I have sketched above. Also, this definition is especially important since it is used in most reasoning and computations with the exterior algebra.

So, I suggest to keep the above definition from a basis, and to precede it by something like There are several equivalent definitions of the exterior algebra (see below). The one that is presented here depends formally on the choice of a basis of the vector space, but this choice is not meaningful, since the resulting exterior algebra is essentially invariant under a change of basis. D.Lazard (talk) 15:58, 6 October 2023 (UTC)

I'll try to think about what the most accessible basis-free version would be. Grassmann's 1862 Ausdehnungslehre starts with a definition of a "domain" (n-dimensional real vector space) in terms of a basis but almost immediately (§24) proves that any $n$ "magnitudes of first order" which "stand in no numerical relation to one another" (linearly independent) serve as a basis. This happens before the definition of the exterior product. –jacobolus (t) 16:59, 6 October 2023 (UTC)

Grassmann's 1844 Ausdehnungslehre doesn't lead with a basis (doesn't introduce the concept of a basis or coordinates until §87, halfway through part 1), but the presentation is geometrically motivated and not nearly concise enough for us here. It would be worth citing or possibly even substantially quoting from in footnote(s) though. –jacobolus (t) 17:27, 6 October 2023 (UTC)

For what it's worth, I strongly dislike Berger's way of introducing affine spaces in his book (not unique to Berger, but Wikipedia cites him). It is done for his own convenience in the context of a logical structure built around coordinates as a fundamental abstraction, and this is relatively more concise to make rigorous, but I think it does a disservice to his students and Berger doesn't spend enough effort on emphasizing that this is just one arbitrary definition and that coordinates are best thought of as a tool of convenience rather than an essential foundation. YMMV. –jacobolus (t) 17:15, 6 October 2023 (UTC)

I strongly propose including div/grad/curl, in an informal way, in the motivating section (maybe not the first part), c.f. my first talk post about editing the introduction. MeowMathematics (talk) 12:14, 7 October 2023 (UTC)

You might be looking for geometric calculus. These are all dependent on a metrical structure. (Curl not essentially so, if you use the wedge product rather than cross product). –jacobolus (t) 05:24, 9 October 2023 (UTC)

I agree, and I would be tempted to put curl and grad as the first examples in motivations. Divergence relies on a metric, so I would put it a bit later. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:33, 11 October 2023 (UTC)

I've made a ham-fisted first attempt at the reorganisation, and rewriting the definition section. Others feel free to improve. In particular, someone please write a basis-dependent definition in front of the current definition. MeowMathematics (talk) 22:17, 8 October 2023 (UTC)

I D.Lazard's take on the lead is a clearer basic summary. YMMV. –jacobolus (t) 05:27, 9 October 2023 (UTC)

This is not the talk thread about the lead, but in any case feel free to make concrete suggestions. MeowMathematics (talk) 11:24, 9 October 2023 (UTC)

I consider Recall normal in mathematical literature, but I believe that it is problematical in Wiki. Also, I suggest that you start the formal definition section with There are several equivalent definitions of the exterior product. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:33, 11 October 2023 (UTC)

Why not use the simpler quotient?

The article currently constructs the exterior algebra as the quotient of the tensor algebra by the two-sided ideal generated by elements of the form $u\otimes v+v\otimes u$. That seems unnecessarily complicated: With $u=a+b$ and $v=a-b$ we have

$u\otimes v+v\otimes u=(a+b)\otimes (a-b)+(a-b)\otimes (a+b)=2a\otimes a-2b\otimes b\;,$

so this ideal is also generated by elements of the form $v\otimes v$. Is there a reason not to use this simpler construction?

Joriki (talk) 20:03, 21 November 2023 (UTC)

Joriki, your observation is valid. There is no reason to use the more complicated construction, which is also incorrect in characteristic 2. This is the result of a recent substantial revision. —Quondum 18:26, 10 December 2023 (UTC)

 Done —Quondum 20:52, 11 December 2023 (UTC)

Exterior products

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":

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

There is a discussion of these two "conventions" in John M. Lee, "Introduction to Smooth Manifolds" (2nd Edition) p.358.

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 $\wedge$. 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)

Summations

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 <math>\Sigma</math> for $\Sigma$; 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)

On edits by Timo Omburo

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)