Talk:Exterior algebra/Archive 3

Diagrams for n-vectors and n-forms (n = 1, 2, 3)
If it's ok I added a diagram to the Motivating examples section, so that readers get the geometric interp first thing. It's disappointing that there seem to be no diagrams for the interpretation of an n-form on WP (unless I have not looked enough). Maschen (talk) 02:02, 8 September 2012 (UTC)


 * I think that similar diagrams would be valuable at Dual space or Linear functional for geometrically interpreting linear functionals (covectors) on a vector space in terms of the annihilator of the functional and its parallel planes. The planes should be flat, semi-transparent, blatantly deviate from perpendicular, and the plane that the functional maps to 1 should stand out. References for this interpretation (e.g. Penrose's The Road to Reality) should be available. Interested? — Quondum 07:16, 8 September 2012 (UTC)


 * Definitely. Do you mean to include patterns of stacked surfaces and to run vector/s through them (hence the inner product = number of surfaces intersected)? Thanks, I'll definitely get round to this. Maschen (talk) 07:25, 8 September 2012 (UTC)
 * Also are you referring to sections 12.3-12.4 in Road to Reality? Before going any furher you might see the recent edit history of linear functional, we could make modifications from there? Maschen (talk) 07:41, 8 September 2012 (UTC)


 * Yes, and yes. I'm not too sure why 1-form and Linear functional have not been merged; AFAIK there is no distinction (other than the disciplines/contexts in which they are used). The diagram that you temporarily added gives the idea, but has detail that does not apply. It would have to be illustrated in a pure vector space without any inferred differential structure, i.e. with a clear origin from which all the vectors originate, and no "d" notation. And preferably don't use the term inner product in this context; strictly speaking it does not apply (scalar product might be permissible, and Penrose denotes it with a (rather big) dot). Nevertheless, the action of the linear functional on a vector is (in a continuous sense) equal to the number of planes intersected as you say. — Quondum 10:31, 8 September 2012 (UTC)
 * There is a discussion at Talk:one-form about this. I think the reason for the lack of merger is that a 1-form is usually thought of as a linear functional that varies from point to point.   Sławomir Biały  (talk) 12:55, 8 September 2012 (UTC)
 * PS on merging: the terms 2-form and bilinear functional are generally not equivalent, which might be a partial argument for not merging, but I find that unpersuasive. — Quondum 12:29, 8 September 2012 (UTC)

Suggestions/comments
Small gripe is that the article hasn't talked much about forms at the point of the graphic. If possible, I would split the left and right hand sides of the graphic, and include the right-hand side in the section on alternating forms. Also, the vectors do not need to be basis vectors; they can be any vectors and still this interpretation is valid. It would be better to replace the symbol ei with some other symbol, like vi or u,v,w as in the text. And, as Quondum says, the planes in all graphics should be flat, not curved (for this article). Sławomir Biały (talk) 12:57, 8 September 2012 (UTC)


 * A wealth of valuable responses. The reason for:
 * drawing curved surfaces was that for other curvilinear coordinates they are not planes (e.x. a 3-form in spherical coordinates is a mesh of concentric spherical surfaces, circular planes, and conical surfaces, coord. curves of r, θ, φ, as you both know of course). I presume that my own drawing doesn't emphasize that well enough for a reader to understand though anyway (given that the curves are nearly flat), and yes for this article they should be planes.
 * using basis n-vectors/n-forms was to emphasize the correspondence between the index notation and the geometric interpretation, and since each direction can be scalar multiplied by numbers and added to form a n-vector/n-form.
 * including basis n-vectors and n-forms side by side in the same picture was for immediate comparison.
 * It's not a problem to split and correct into new images, and will be done. Also I agree with the merge of Linear functional and 1-form. Again thanks for feedback, Maschen (talk) 17:57, 8 September 2012 (UTC)


 * Here is the diagram suggested by Quondum.
 * Any problems please say.
 * Unfortunately, although Penrose is an extremely good writer and illustrations are clear, those sections mentioned are a little confusing and I didn't see them as clearer/more direct than MTW depicts n-forms as linear functionals. This diagram is reproduced from MTW, even the letters are identical so that readers of MTW can recognize it and that readers gain a feel for common notation in the image as well as text.
 * (Right now still in the process of splitting/fixing the other image). Maschen (talk) 18:23, 8 September 2012 (UTC)

About the previous File:Gradient 1-form.svg (now redrawn) --->

Agreed with above (no "inner" product, flat level surfaces etc.), except that if Φ (now f, since this is more likely to be used for a function) is a scalar function (0-form) then surely the exterior derivative of it dΦ = α is a 1-form (surface stack) at least locally (Poincare's lemma)? To that end I fixed the image, but still include the statement in the caption.

(Another reason for reverting was my once-confusion between "circular" notation: dΦ for exterior derivative of Φ, and dΦ for gradient (MTW terminology) of Φ in place of &nabla;Φ, hence my incorrect statement "dΦ =  = <&nabla;Φ, dr>", completely wrong... since the diff form dΦ is the total differential (in a more general way, as the directional derivative, not an entry in the scalar product)...). Maschen (talk) 20:44, 8 September 2012 (UTC)


 * I agree that these diagrams are much better than Penrose's version. I would remove any arrow associated with the functional (1-form); rather use some technique to associate varying values (e.g. annotation and/or a colour variation from one plane to the next). Why three subdiagrams for σ? Only one is needed, and removal of the arrow makes them identical. It'd also be nice to have the spacing for β visibly different from that of α (and, being really picky, changing angles so that none can be interpreted to be perpendicular: arrows to planes, arrows to arrows). The captions need some tweaking, but that can wait. The annotation of f is quite not correct: f is varying from plane to plane according to the caption. Despite my quibbles, this is looking good. — Quondum 22:31, 8 September 2012 (UTC)


 *  File:Gradient 1-form.svg:  See what you mean now... How to generally annotate the planes of constant values (not just using "f = 12, 13, 14...")? Using f = c + 1, c + 2, c + 3... where c is any initial constant? Or just use "f = a" for the first and "f = b" for the last plane, where a < b are constants? Or just remove them?


 *  File:1-form linear functional.svg:  The reason for the arrow normal to/on the surface stacks is because
 * IMO it is clumsy to have a separate arrow to one side (like MTW actually draw),
 * in this way the geometric significance of normal (to coord plane) is emphasized,
 * if the arrows are removed, how else to more directly indicate the positive sense? As you suggest, I think a colour variation (lightest from first, darkest to last) sounds like the best solution, but then the other split images need updating... also extra words are needed in the captions just to explain that... (the arrow has an automatic implication)
 * There isn't much of a problem with the arrow to the surfaces, but it is less clutter/redundancy in some important respects. I would think the reader can gain the idea of a 1-form as a linear functional far easier using the current orientations. Changing all the angles would mean changing all the inner product values... which is going to take time to make such changes (but I'll still do it)...
 * I will prepare the changes but not implement them till further comments are proposed. Maschen (talk) 23:02, 8 September 2012 (UTC)


 * In order of your bullets:
 * There is no purpose to the arrow aside from indicating which side of the stack is increasing. This could be indicated by a "+" and a "−" on the other side in the absence of annotation; with annotation it is entirely unnecessary.
 * There is no concept of perpendicularity in this diagram; we must de-emphasize any suggestion of it as much as possible.
 * Answered by my first bullet.
 * The annotation does not need f= at all. We only need -δ, 0, δ, 2δ etc. Including 0 is important. I think ot would be better to use real values (no delta): -0.5, 0, 1, 2.5 or suchlike. The reast is easily handled in the caption. — Quondum 23:52, 8 September 2012 (UTC)
 * PS: There is not much gained by changing the angles – give that a miss. — Quondum 23:54, 8 September 2012 (UTC)


 * Ok that makes much more sense (thinking too much about "tangent and normal...").
 * Although again (and I'm sorry), the +/- is exactly equivalent and actually more ambiguous than an arrowhead, which again has the automatic implication for directed increase (to a reader: "+/-" what? Explain by "The sense of the 1-form is + to -"?). Also in books like Penrose's, covectors/1-forms are actually drawn as full arrows identically to vectors. No, nothing is more simple or more intuitive than the arrowhead: we need a diagram that averages between books like MTW and Penrose. Also, it is graphically far quicker to draw an arrow than anything else (typing +/-), and colour gradients can take time... Of course, arrows are definitely redundant using numerical annotation, or using a colour gradient.


 * Sorry to continue arguing like this... I'd be happy to change to numerical annotation/a colour gradient while waiting a day or so for more comments before reloading them anyway (admittedly a bit self-centred on my own preferences...) Thanks, Maschen (talk) 00:36, 9 September 2012 (UTC)


 * My (rather emphatic) objection to the embedded arrows, notwithstanding Penrose's use thereof (note: putting it to the side as you say MTW does turns it into an annotation, which would be equivalent to my + and −; this is an important difference between Penrose and MTW), is precisely that it would immediately be assumed to be a vector, or to have some true content related to a specific vector, which it doesn't (the only content of such a vector would be to indicate which plane maps to 1). We should not "average" the two diagrams, we should select the most correct features. IMO, we should ignore Penrose, since his diagram is clearly deficient. Think about it this way: a covector is nothing other than a map from vectors to scalars. The depiction we are making is a contour map of the vector tips that map to each scalar value. This "mapping of vector tips" rather than "number of surfaces intersected" may be far easier to put across. — Quondum 07:47, 9 September 2012 (UTC)

Ok, agreed about Penrose (but please still keep the citation in the captions, for reader's interest). As an alternative or addition to the colour gradient, we could put a small arrow over/under/to one side of the symbol for the 1-form, not the surface stack itself, but the symbol, in the sense of increase, something like this: $$\overset{\nearrow}{\boldsymbol{\alpha}}$$ or $$\overset{\leftarrow}{\boldsymbol{\alpha}}$$? When applying the colour gradient to the alternating multiforms, the colour gradients can merge in places making it less clear which is the increasing sense (still visible). Just a small arrow is needed. This is a (far more) compact equivalent to MTW, so their originality is preserved. I wish I thought of this before... I'm not still "fighting for the arrow", just a possible extra fine little which may make all the difference? Thanks, Maschen (talk) 09:33, 9 September 2012 (UTC)


 * The colour gradient might introduce more problems than it's worth, as you point out. What the only thing that is necessary is a way of indicating the plane that maps to zero (which might be obvious if the origin is clear), and a sense of the scale factor as you move between planes, which could be indicated by annotation of at least one other plane with a scalar value. Where the scale factor is not of interest but the sign is, the small arrow may work (and I agree that it is preferable to + and −); it could also be included in diagrams with annotated plane(s) if desired for uniformity. I think that it's essential to get the concept of a covector (1-form) as simple and clear as possible.
 * As one goes to the higher-degree forms, the diagrams become inaccurate, a sort of short-hand for "the wedge product of these 1-forms", and so the finer detail becomes unimportant; the point then is largely to keep an echo of the diagram of the 1-form for each of the factors. — Quondum 12:08, 9 September 2012 (UTC)


 * Ok - happy that we are now settled on simply keeping it as simple as MTW. No faff; just surfaces with either:
 * numerical annotations (for scale and illustrate constant slices, I plan to use −1, −0.5, 0, +0.5, +1, which is a very simple sequence and has positive/negative fractions/integers included so readers are not restricted to think they are only nat numbers) exactly as you suggest, or
 * small arrows as I suggest,
 * that's it. 0_^ Just a couple of questions:
 * Are the senses of σ exactly as originally depicted in  File:1-form linear functional.svg: ? If so how to include? I think just annotations for that one would be better, no arrows.
 * It worries me that you say the multiforms are inaccurate depictions when Sławomir said the interpretation is valid (after his suggested corrections). Why the inaccuracy?
 * Thank you! Maschen (talk) 13:29, 9 September 2012 (UTC)

Implementation
Combining the ideas into one would lead to this >
 * Maschen (talk) 13:49, 9 September 2012 (UTC)
 * This is a very nice picture. Do you think anyone might read anything into them being depicted horizontally? I've edited the caption; see what you think.
 * As you did in your other diagram, it may make sense to include say two arbitrary vectors from an origin on the "0"-surface, touching two surfaces, so that the caption can deal with them as example mappings.
 * Perhaps I should have said "non-isomorphic", not "inaccurate". The diagram of the 1-form is complete in the sense that it contains all information necessary to calculate the 1-form (assuming you know how the space is calibrated in terms of vectors), and vice versa, with no excess information. The diagram of a higher-grade form contains excess information: it depicts three 1-forms and their information. You can construct a 3-form from three 1-forms, but you cannot determine the original 1-forms from a 3-form. So that is far tricker to make an intuitive picture, and for now I cannot think of a better way than what you've done. So in summary: vectors, and 1-forms we can depict geometrically very well ("faithfully"); the rest are not "faithful" depictions but are as good as anything I can think of. — Quondum 16:13, 9 September 2012 (UTC)


 * Thanks for your clarifying response, the isomorphic concept is a good point, but those split images are to show the elements of the graded algebra in steps (what would really be fascinating is n ≠ integer - fractal !? Ok off a tangent for this thread but still). You mentioned to include vectors though the surfaces, which is the point of  File:1-form linear functional.svg:  (see above for the redraw). This one here is a stand-alone 1-form diagram.
 * About the captions, by all means do rewrite them! For this one it is more precise and nice but what is n? The dimension of the vector space? We know but the reader may not. Writing "(n − 1)-surfaces" literally sounds like "the number of surfaces is n − 1", perhaps just write "n-hypersurface"? Maschen (talk) 17:40, 9 September 2012 (UTC)
 * The "surfaces" in question are actually just (hyper-)planes. The caption should indicate this.  In general, any function, not just a linear one, can be visualized in terms of its level surfaces.  But, of course, in that case they will not be planes.   Sławomir Biały  (talk) 18:44, 9 September 2012 (UTC)


 * Caption tweaked, thanks. (For those that rewrite captions just do it, no need to cross out/underline things as previous versions can always be recovered and it's easier to read). Maschen (talk) 19:18, 9 September 2012 (UTC)
 * A quibble about "level", which here I know means "of constant value", but might be misinterpreted by most readers to mean "horizontal". I changed the caption accordingly, plus removed excess detail of dimension (hyperplane says it all). — Quondum 21:31, 9 September 2012 (UTC)
 * Phew... what a long thread I caused. Aside from fixing captions, is everything now fine? Maschen (talk) 22:10, 9 September 2012 (UTC)
 * It would be nice if it was clear that the 0-plane always contained the origin (via axes?), a fact I'd ordinarily put into the caption, but that threatens to overload it. That the planes are uniformly spaced is also a requirement, but I think that the diagram suggests that adequately. — Quondum 07:27, 10 September 2012 (UTC)
 * For this one it may be ok, for the others it may clutter. How do you want the axes oriented? I.e. if we use x, y, z then x, y in the horizontal planes and z in vertical direction cutting through the planes? Maschen (talk) 07:35, 10 September 2012 (UTC)
 * Yes, axes would be unnecessary elsewhere, especially if they have other details such as vectors that will achieve the same as a side effect. They can also be very thin, no markings: just mutually perpendicular crosshairs (obviously mutually perpendicular in 3-space, projected onto the plane, accuracy unimportant). They can be oriented any way (and yes, visibly through planes) as long as they are not parallel or perpendicular to the planes, as that might make an incorrect implicit suggestion. The 1-form representation can be rotated in space if need be. If not rotated, z could come up through the nearer corner of the top plane, and x and y can go through near the left and right near edges of the −0.5 plane, or x down towards us through the near −0.5 corner fractionally to the left and y can go horizontally right. — Quondum 13:48, 10 September 2012 (UTC)


 * I'll get to it, just busy cleaning up Polynomial greatest common divisor right now. Thank you. Maschen (talk) 13:58, 10 September 2012 (UTC)
 * ✅ Maschen (talk) 14:46, 10 September 2012 (UTC)
 * Very, very nice. — Quondum 14:59, 10 September 2012 (UTC)


 * I absolutely apologize for crossing out "thank you" just then, indeed - thanks ...... *blush* Maschen (talk) 15:19, 10 September 2012 (UTC)

Going back to where we started, shall we add the 1-form diagrams to 1-form/dual vector/linear functional (possibly differential form)? Maschen (talk) 22:15, 10 September 2012 (UTC)


 * The perfect spot for is Linear functional. We haven't clarified the status of 1-form (when does it imply a differential structure?), and at this point I'm hesitant to use these diagrams where a differential structure is implicit. Candidate places for  include one-form and dual vector. — Quondum 07:00, 11 September 2012 (UTC)


 * Ok forget the diff forms article, that was just a suggestion. We have spent much time on this so I will leave them here for just one more day for others to comment before we show them in those articles you link to (unless you or someone beats me to it)... ^_^ Maschen (talk) 07:37, 11 September 2012 (UTC)


 * Suppose we've waited long enough; let's add them. Maschen (talk) 23:08, 11 September 2012 (UTC)

New images for scalar multiplication

 * 250px|thumb|Comparison of scalar-multiplying a 1-form α (stack of surfaces) and 1-vector v (arrow). Above, the length of a vector, and the number density of surfaces of a 1-form, are doubled. The factor of two can be replaced by any real number.,
 * thumb|250px|Scalar multiplication in the exterior product of vectors constituting a 2-vector. In the above case the lengths of the vectors are doubled, consequently the area of the 2-vector quadruples.,
 * thumb|250px|Scalar multiplication in the exterior product of 1-forms constituting a 2-form. In the above case the number density of 1-form surfaces double, consequently the number density of the 2-form surfaces quadruples.

Maybe they'll help, especially for clarifying pictorially how scalars can "slide" out of an exterior product. M&and;Ŝc2ħεИτlk 12:11, 18 August 2013 (UTC)


 * I don't think the first illustration is really relevant for this article, and the last illustration (while interesting) might be a distraction, since most of the article is not about the exterior product of forms. One thing I would like to see is an illustration of the wedge product of three vectors that includes the orientation.  The orientation on a solid ball can be thought of as giving the ball a counterclockwise spin around any given axis.  I think something on the cube should be drawable.   Sławomir Biały  (talk) 13:25, 18 August 2013 (UTC)


 * You seem to be trying to illustrate multilinearity in the exterior product. This is not too obvious initially from the illustrations (aside from the unfortunate coincidence of multipliers chosen). I see little value in illustrating this; an algebraic treatment of multilinearity is more direct and less demanding on the reader. If this was what you were trying to illustrate, I don't think that they'll help. — Quondum 18:13, 18 August 2013 (UTC)


 * I'm just showing what scalar multiplication is like in this context nothing more. Let's forget the images if they are not helpful, but I'm not sure how "obviously" most readers will interpret the scalar multiplication of forms (vectors are easy, IMO not so much forms).
 * Agreed the first is not relevant, and the third overkill, but the third was for comparison with vectors.
 * About the exterior product of three vectors, there already is the interpretation in the first diagram in the article lead, but agreed that the shape need not be a parallelepiped, and could be anything. A diagram for a 3-vector as different shapes is in preparation and will be posted shortly hopefully. (Would a parallelepiped, sphere, and an irregular splodge, all with orientations, be ok?) M&and;Ŝc2ħεИτlk 19:34, 18 August 2013 (UTC)




 * Note: I changed the above images to links to save space.
 * For now here is a PNG, since SVG and PDF didn't work for some reason (the shading shouldn't be a problem...). An SVG/PDF version can be produced later, but for now... is this what Sławomir Biały meant? By all means criticize. M&and;Ŝc2ħεИτlk 19:55, 18 August 2013 (UTC)
 * The axis should be an oriented axis (i.e., an axis with a north pole designated).  Sławomir Biały  (talk) 20:23, 18 August 2013 (UTC)


 * Ach, yes missed that (else the orientation is ambiguous), it is fixed now with an arrow at the tip of the axis for the north pole. M&and;Ŝc2ħεИτlk 21:32, 18 August 2013 (UTC)


 * The leftmost (cube) should have the v arrow blued a bit, as though it is being seen through the cube, to give the right visual cues. The axis an equatorial arrow don't make sense to me; I would have thought that the orientation of a bounded volume is best depicted by a local circulating direction on the boundary, as with the cube; I've seen you do this on other irregular shapes. — Quondum 01:11, 19 August 2013‎ (UTC)


 * The leftmost parallelepiped does have a hint of blue, it's just too pale to see (now fixed). The "global" orientation on the sphere and irregular shape are still the same as for the local circulating arrows you're thinking of. Another version to show this is below. M&and;Ŝc2ħεИτlk 09:56, 20 August 2013 (UTC)




 * Yup, better colouring. I'm getting the hang of the "global orientation", as a spin on an axis in 3d as suggested by Sławomir. Every part of an oriented volume is oriented in this sense, so would it not make sense to shrink the axis-and-spin figures to represent a local property? This would also allow a few of of them in a diagram, which makes it clearer that the choice of axis for the spin figure is completely arbitrary. I quite like the idea, since it is independent of any boundary. — Quondum 21:51, 20 August 2013 (UTC)


 * About localizing the spin... Do you mean to add an axis at each circulation? I thought by convention the orientation was just left or right on the surface, nothing more. If there is any axis it would be the outward normal to the surface, but I don't see why that's necessary. Or have I missed your point? M&and;Ŝc2ħεИτlk 08:13, 24 August 2013 (UTC)


 * Yes, I do mean adding an oriented axis at each local circulation, but more to the point, that this circulation-around-a-vector applies at each point throughout the volume, like a bag of spinning tops (each with only one sharp point). Where a spin axis happens to penetrate a boundary (the tops whose points touch the cloth of the bag), aligned to match an outward surface normal, this picture corresponds to the local circulation on a surface.  This picture is thus independent of but consistent with the local-circulation-on-a-boundary picture.  No boundary needs to be defined at all, making it a more powerful picture.  It works on an unbounded manifold, whereas the surface circulation needs the construct of subdividing an unbounded volume into bounded regions, and a rule for transferring the outward circulation from one artificial region to the opposite circulation on the boundary of a neighbouring region.  The spinning-top analogy simply requires continuous transfer of orientation throughout the volume along every path.  The surface circulation also does not as obviously work for a disjoint boundary ("outwards" remains unambiguous, but transferring circulation direction from one portion of a boundary to another disconnected portion is less obvious).  — Quondum 11:18, 24 August 2013 (UTC)


 * I can follow most of what you're saying, but still thought the circulations on the surfaces were enough, and still think the axes are superfluous. Anyway, shall get round to modifying this in time. M&and;Ŝc2ħεИτlk 07:40, 25 August 2013 (UTC)


 * On the surfaces, we are in agreement: the axis is superfluous. I was not suggesting a change to the diagrams with the surface circulations (those are already plenty neat and complete), only to the ones already with the "spin around an axis" picture suggested by Sławomir. It is in the alternate picture where spins depicted locally at any point throughout a volume as circulation around a vector where the spinning top picture applies. One could also simply shrink the "spin on an axis" symbol (circulation around a vector) into the volume in the diagram, though I still feel a single such symbol might create the incorrect impression that the axis chosen has some significance. — Quondum 11:51, 25 August 2013 (UTC)


 * OK. Trying to orientation throughout the volumes one way or another would be useful, but it would become messy... M&and;Ŝc2ħεИτlk 11:33, 27 August 2013 (UTC)

I don't see how it would become messy. Replacing the large yellow circular arrow and axis vector with say three of the same, only say 10%–15% of the size, distributed through the volume with random orientation and located to avoid collision with existing features (i.e. the vectors being wedged) should do it. Also, the leftmost figure (the parallelopiped) is inconsistent with the others, because it uses the surface circulation picture. — Quondum 12:36, 27 August 2013 (UTC)


 * Now I see what you mean clearer, thanks for clarifying that. Yes the parallelepiped will be changed in the axis figure. M&and;Ŝc2ħεИτlk 12:39, 27 August 2013 (UTC)
 * Done, better? M&and;Ŝc2ħεИτlk 13:03, 27 August 2013 (UTC)
 * You've captured the idea nicely. A minor suggestion: the direction of spin of the small figures is unclear, but this could be easily fixed by colour cues: make the nearer half of the circulating arrow visibly brighter (less blued).  Then you can decide whether retaining the "global" spin figures is useful or not. — Quondum 02:56, 28 August 2013 (UTC)

Determinant module
The treatment of determinant seems rather cursory. I would suggest a section Determinant module treating the top exterior power of a vector space and its relationship to the determinant of a linear map: this might be a good target for a redirect Determinant module, currently requested at WP:RA/M. It might mention that a module is free iff the determinant module is free, see. Deltahedron (talk) 19:39, 2 February 2014 (UTC)


 * More on the relationship with the determinant would certainly be a welcome addition. There is already a little in the section on functoriality, but it's fairly minimal at present.  Sławomir Biały  (talk) 22:51, 2 February 2014 (UTC)

The introduction is too narrow
The introduction to Exterior algebra treats wedge products as something specific to Euclidean geometry, despite the fact that Exterior Algebra is used more often in non-Euclidean contexts. The material on Euclidean Geometry should be moved into Exterior algebra. Shmuel (Seymour J.) Metz Username:Chatul (talk) 18:15, 19 February 2015 (UTC)


 * I agree with this point. One of the primary features and beauties of exterior algebra is that it captures much of the geometrical content of a vector space without reference to a metric structure, i.e. only using the multilinearity. It seems inappropriate to open the lead with a misleading specific association with Euclidean geometry. Another point: the lead opens with a mention of the exterior product, not of the algebra, which makes it badly structured. The lead is meant to open with a description of the topic, not to pedagogically lead the reader through concepts that eventually build up to the topic. —Quondum 17:37, 20 February 2015 (UTC)

No definition of i
I'm new to writing here in the talk section, so please be kind. This is just a comment on the section about the universal property of the exterior algebra. It mensions a map i:V -> /\* V, but the map i is never defined in the article.

FredrikMeyer (talk) 09:17, 5 June 2015 (UTC)


 * D.Lazard (talk) 12:58, 5 June 2015 (UTC)


 * The parenthesis "(here $i$ is the natural inclusion of $V$ in $Λ(V)$)" was added. Isn't the model at  a better construction, e.g. "together with a linear map $i : V → Λ(V)$ satisfying $i(v)^{2} = 0$ for all $v ∈ V$"? It is not clear to me that this works in this case, but "natural inclusion" feels too vague. —Quondum 13:54, 5 June 2015 (UTC)


 * I agree that "natural inclusion" was too vague. This was because, in the definition of the exterior algebra, it was not said what are the elements of degree 0 and 1. I have fixed this. Feel free to improve my formulation, if needed. D.Lazard (talk) 15:34, 5 June 2015 (UTC)

Subtlety obscured by reference to Euclidean space
The lead contains the statement: "The magnitude of the exterior product of two vectors in a Euclidean vector space gives the area of the parallelogram defined by these vectors". This seems to hide the fundamental geometric property and insight that lengths, areas, etc. of the same orientation can be meaningfully compared without a magnitude of any kind being defined. It would be nice to highlight this very general property in the lead. After all, the exterior algebra is is essentially about properties that are agnostic to any quadratic form. —Quondum 19:44, 26 November 2016 (UTC)

fee fo fo fum
✅ I'm pretty sure that fo in "The first non-obvious result obtained from this approach is the expression fo characteristic polynomial." is wrong. But I don't know what would be right.  Ϣere Spiel  Chequers  11:41, 15 January 2017 (UTC)

Abuse of term "inner product"
The term inner product has a well-defined meaning in mathematics. This article uses it in the sense that David Hestenes redefined it in the context of geometric algebras, namely as a nondegenerate bilinear form, not necessarily positive-definite. It similarly abuses the terms norm and Gramian matrix. We could replace the terms with more typical terminology for what is meant, but I would like to suggest that mentioning a bilinear form is beyond the scope of the topic, and should preferably be covered under Geometric algebra. Covering this here invites the reader to confuse the scope of the topic, and possibly even to lose sight of the generality of exterior algebra, especially with the bilinear form and the interior product essentially becoming the same thing under the identification induced.

Hodge duality is similarly not relevant without a bilinear form, although there are similar maps Λ → Λ∗ and Λ∗ → Λ induced by the interior product and an n-form that might be worth mentioning (a source: Jayme Vaz, Jr.; Roldão da Rocha, Jr. (2016). An Introduction to Clifford Algebras). This seems more appropriate, since an exterior algebra already induces an n-form up to a scalar multiple. —Quondum 21:21, 11 February 2017 (UTC)


 * It is quite common in Mathematics for different authors to use different nomenclature. In particular, some authors refer to a Minkowsky inner product as simply an inner product, due to its importance in Relativity. Either usage is proper, as long as the article is consistent and mentions the alternative usage. Shmuel (Seymour J.) Metz Username:Chatul (talk) 20:48, 14 February 2017 (UTC)


 * Sure, but this article does not do as you say it should. It does not define the terms, does not mention the alternative usage, and worst, it links to the article Inner product, which is the incorrect but confusingly similar meaning.  Since this article is broader in scope than simply GR, most particularly I would say that it is primarily a mathematics rather than physics article, so deferring to physics terminology seems inappropriate or at the least confusing.  —Quondum 03:48, 17 February 2017 (UTC)

The ideal I
In the section: 'Formal definitions and algebraic properties", why is I an ideal? — Preceding unsigned comment added by Yliu0128 (talk • contribs) 20:57, 10 September 2017 (UTC)

Restored old version of the lead
I see that the new lead has moved dramatically away from readability for the typical reader. The wedge product is widely used in computer graphics applications, so the article needs to be readable to a college-level undergraduate. The lead that I replaced started with the universal property, the mere mention of which in the more accessible lead met with resistance back when the lead was discussed in a community-wide discussion here. I have restored the original, widely discussed, lead. Sławomir Biały (talk) 11:19, 22 March 2018 (UTC)

Cayley & Sylvester
The § History says Cayley and Sylvester had a theory of multivectors. There is no reference to them. Does someone know a source for this assertion? — Rgdboer (talk) 03:06, 31 January 2019 (UTC)

/* Applications */ Entire section is uncited.
The only citations in the applications section are in the subsections on Leverrier's Algorithm and Superspace. I don't know how to flag that appropriately. It is very difficult to verify that the text is accordant with accepted definitions. 184.17.203.207 (talk) 14:29, 17 June 2018 (UTC)


 * I count three references. It seems like much of the rest is just summarizing other articles.  It should still be referenced, but is not alarming.   Sławomir Biały  (talk) 20:28, 17 June 2018 (UTC)


 * I would love to have a reference for the intriguing paragraph in Applications / Physics regarding the EM field.Kbk (talk) 16:01, 31 January 2020 (UTC)


 * I linked to Electromagnetic tensor for that section. This is covered in "most"(?) textbooks on electromagnetism. Some are easier than others, some are more comprehensive. 67.198.37.16 (talk) 04:53, 16 November 2020 (UTC)

Orientations shown on the last figure
Dear Maxchen,

Greetings. I have read the wikipedia article, "Exterior algebra" and enjoyed it very much. Thank you so much for the last figure (in the article). It is very nice. However, I am confused by the orientations shown as arrows in the figure. It seems to me that the orientation is not right-handed. It may be left-handed. For example, εΛη should have shown a counterclockwise direction as shown in the (left-side) of the first figure. Thank you in advance. You can reach me at sangdhong@gmail.com, in addition to ShdSiriCan in the Wikipedia.

Best regards, Sang — Preceding unsigned comment added by ShdSiriCan (talk • contribs) 01:35, 16 June 2020 (UTC)


 * , the author of the diagram,, does not appear to be very active on Wikipedia any more. I agree that there are concerns if the detail of orientation in the diagram are considered, both for the illustration of the 2-form and and of the 3-form.  For the time being (until someone can update the diagrams), perhaps we should disregard the orientation detail.  —Quondum 11:20, 16 June 2020 (UTC)


 * It looks right-handed to me. epsilon=thumb, eta=forefinger, omega = middlefinger. 67.198.37.16 (talk) 04:59, 16 November 2020 (UTC)

Some problems with this article
I find that the introductory section is much longer than any introductory section should be.

It contains much useful information that ought to be in later paragraphs but not in the introduction. The reason for this is that any article ought to begin with a brief description of its subject so that it is easy to understand and not full of so much information that the reader can (as with this article) find it difficult to focus on the most important information.

And then: The section titled Formal definition and basic properties is essential. But before the formal definition ought to come an informal definition that is nonetheless accurate.

The informal definition must come before (or at the beginning of) the Motivating examples section, because otherwise the reader has little idea of what the examples are examples of.

This Motivating examples section has good information. But it is misplaced because just as with the introductory section, there is way too much information and the information is way too specialized for the lower dimensional cases. All or almost all of this information belongs in a later section perhaps titled something like Two- and three-dimensional cases.

I hope someone knowledgeable on this subject will re-organize this article to make it much more digestible to a typical reader.2600:1700:E1C0:F340:A839:92DA:B936:7FC (talk) 17:13, 31 July 2019 (UTC)


 * Regarding the Introduction, I just read through it for the first time, and I feel that (at the current revision) its level and length are quite appropriate. It surveys Exterior Algebra in plain language, and the level of sophistication increases smoothly from paragraph to paragraph, so it's possible to stop at any point without missing something essential.  It's a good introduction, and I wouldn't want it to be any terser. I did move an introductory definition of differential forms to the Applications / Differential Geometry section.Kbk (talk) 17:21, 31 January 2020 (UTC)


 * On the surface, the above critique sounds reasonable. But looking at the current article, it is hard to imagine what kind of "informal definition" could be given, beyond the one that already is there (of area/volume). This informal definition is only a few sentences long, it is hard to imagine how to turn it into something longer than that. So I don't know how to actually accomplish the suggestions. 67.198.37.16 (talk) 20:06, 16 November 2020 (UTC)

Hodge duality
Is it sure that the definition of hodge duality given here is equivalent to the standard definition? — Preceding unsigned comment added by 89.135.87.8 (talk) 20:32, 18 October 2017 (UTC)


 * I see that this problem persists in , where two incompatible definitions are given. The Hodge dual of an element of the exterior algebra conventionally matches the second definition.  The first definition is, in a sense (in that it does not need a volume form to be chosen), a more natural operator.  This is defined by Vaz & da Rocha (2016) p. 43 and there called a "quasi-Hodge isomorphism".  There are two of them, one being the inverse of the other up to a grade-dependent sign.  It seems natural enough to include these, but not using the name "Hodge dual".  —Quondum 15:56, 10 February 2021 (UTC)

Motivating Examples
In the sentence on the geometrical interpretation of wedge product, it reads, "... signed lines, areas, volumes, etc. ...". Should "lines" not be "lengths"?

Thanks,

Aliotra (talk) 17:33, 4 September 2021 (UTC)

Edits by user Darcourse
User made 39 edits of the article in a row, without any edit summary for explaining them. It is possible that some of these edits are improvements, but retrieving them is not the role of other editors. In the whole, these edits make the article worse than it was. For example: It is possible that some changes are minor improvements, but many fall under MOS:VAR (including those that do not change the rendering). Again, this is not the role of other editors to check separately so many changes.
 * A section is normally devoted to explanations. Here, a subsection  has been added that does not contain any prose. If explanations would be added, this would not adds nothing useful for explaining the motivations of the subject of the article.
 * The nonsensical sentence "that is, the co-ideal residues left when all tensor square elements are removed" has been added, that is not only not understandable, bud introduces a terminology (co-ideal residues) that is not defined and seems an invention of the editor.
 * Using "⋯" (MOS:DOTDOTDOT)
 * Moving punctuation outside $$$$

Thus, I'll revert all these changes. D.Lazard (talk) 12:47, 9 September 2021 (UTC)

Rank of a k-vector -> Decomposable?
I'm wondering how commonly the word decomposable is used for a k-blade, the way it is used in the section Rank_of_a_k-vector. Coming from a geometric (Clifford) algebra background, this is the first time I have seen a blade u^v be called decomposable. Instead, all the literature I've seen would call this factorizable. My issue with the word decomposable is: a 2-vector u^v+x^y is called "not decomposable". I find this term misleading, since it is decomposable into the terms u^v and x^y, as is also indicated in this section, but it is indeed not factorizable.

I therefore wonder if that term is broadly used apart from in the reference given to Shlomo Sternberg? If it isn't, I would propose switching to the term factorizable, which is broadly in use.Tbuli (talk) 16:16, 12 December 2021 (UTC)

Dependence on metric and orientation
The lead as currently written presupposes a metric and an orientation. I was going to add some text explaining that some properties were independent of the choice of metric, but it struck me that the lead was already long enough that part of it belonged in a separate section. Any thoughts on what, if anything, should be split off and how best to mention the dependencies? --Shmuel (Seymour J.) Metz Username:Chatul (talk) 12:17, 28 June 2022 (UTC)


 * Regarding "presupposes a metric", could the article indicate, perhaps at the end of Section 1.1, and at an elementary level, precisely how (if true) a vector space can have no metric, no orthogonality or concept of angle?
 * In one of the maths sections of Penrose's Road to Reality (11.6, p. 211, 2004) he discusses Exterior Algebras (calling them Grassmann Algebras) as having no metric "Grassmann algebras are more primitive and universal than Clifford algebras, as they depend only upon a minimal amount of local structure... the Clifford algebra needs to ‘know’ what ‘perpendicular’ means... the ordinary notions of ‘Clifford algebra’ ... require that there be a metric on the space, whereas this is not necessary for a Grassmann algebra."
 * [snip - PGE 5-Sept-22]
 * As with some other maths and physics Wiki articles, this one indicates what something is, but, IMO, doesn't clearly contrast it with familiar counter-examples that would be helpful for the student (for whom else are encyclopedias written?) Thank you. PaulGEllis (talk) 10:45, 10 August 2022 (UTC)
 * My first thought was citing gauge fields as examples of physical entities in vector spaces with no metric, but that may not be simple enough for the general reader. IAC, you don't need a metric to multiply basis vectors with a scalar. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:17, 10 August 2022 (UTC)
 * Section 1.1 is Areas in the plane, which only applies to the two dimensional case. Inner product in Duality gives the general case. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:47, 14 August 2022 (UTC)
 * Chatul: [snip - PGE 5-Sept 22] Meanwhile, I agree that citing gauge fields could be too advanced for the general reader. Maybe something more familiar from electromagnetism? But for the level I would aim at, Inner product in Duality would be too far down the Article; maybe a Section 1.3 would be more appropriate? PaulGEllis (talk) 19:14, 27 August 2022 (UTC)
 * Is EM in 4-space too advanced? The field $$F=dA$$ is given as a 2-form or, equivalently, as an antisymmetric rank 2 tensor. Of course, $A$ is a gauge field, but at least it's an abelian gauge field. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 16:59, 28 August 2022 (UTC)
 * [snip - PGE 5-Sept-22]
 * IMO, the ME section creates an initial impression that seems to be at odds with my understanding of Penrose's description of EA. The illustration of the bivector/plane area used to motivate the idea of using a wedge product appears to need to rely on an orthogonal basis (is that what you meant by “presupposes a metric”?), right down to Axiom 5, at which point the axiom (itself providing no sense of orientation, though the main text of 1.1 and above does do so) is dismissed without explanation.
 * [snip - PGE 5-Sept-22]
 * PaulGEllis (talk) 15:16, 4 September 2022 (UTC)
 * To clarify, a Clifford algebra has an associated quadratic form but that form need not be a metric; it may be degenerate. There is a natural vector space isomorphism between a Clifford Algebra on a vector space and the exterior algebra of the vector space.
 * I would suggest explicitly mentioning the metrics in #Motivating examples.
 * Orientation is an additional structure beyond the choice of metric.
 * I would suggest explicitly mentioning the metrics in #Motivating examples.
 * A vector space has no preferred basis, e.g., if e1, e2 is a basis then so is e1+e2 , e1-e2; there are no "fixed coordinates".
 * I would suggest explicitly mentioning the metrics in #Motivating examples.
 * I'm not sure what you mean by 'Axiom 5'; the only list of axioms in the text is in #Axiomatic characterization and properties, which lists 3 axioms for inner products. If you're suggesting that the text refer to axioms in one of the references then the text should quote those axioms. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 19:49, 4 September 2022 (UTC)
 * @Chatul Thank you for the clarifications. Clearly, I don't know enough to be editing this page and will try to delete as much as possible of what might be misleading/erroneous, without damaging your contributions or my basic concerns with the Article.
 * I still find the ME section unconvincing as regards how the determinant can be fixed with coefficients in an exterior algebra that's supposed to have no concept of angle, unless they're provided by something with more structure than the exterior algebra may have.
 * Incidentally, when I referred to Axiom5 it is the 5th item in the numbered list in the ME:
 * 5. A(e1, e2) = 1, since the area of the unit square is one.
 * Regards - Paul PaulGEllis (talk) 10:49, 5 September 2022 (UTC)
 * Thanks. That is a list of properties but not a list of axioms.
 * Unfortunately, the ME section only gives two examples, both of which assume a metric. I;ve explicitly given the metric for the first example; the second example already has the text "For vectors in a 3-dimensional oriented vector space with a bilinear scalar product,". Would a third example, giving the electromagnetic field as a 2-form in 4-space instead of an antisymmetric electromagnetic tensor help? --Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:26, 5 September 2022 (UTC)
 * @Chatul: I see your inclusion of the metric for the 1st part of the ME section. (I guess that no metric could, rather unnecessarily, even be represented by a bilinear product in which the mediating matrix is all zeroes.)
 * How does the 2-form illustrate what's missing from, or inadequately explained in, the Article as it stands? Presumably, you want to use the 2-form to illustrate the usual lack of a metric in EA. Will you include some text that explains the difference to the lay-reader?
 * Incidentally, I've rethought my understanding of a vector space (VS) without a norm, so I can see now how the basis occurs and my original confusion is resolved. The reason I called item 5 in ME an axiom was that I read the phrase "if one tries to axiomatize this area" and assumed the A of A was A for Axiom. Thanks for putting me right) PaulGEllis (talk) 16:50, 8 September 2022 (UTC)
 * Yes, an inner product with all zero coefficients cannot be positive definite, and thus cannot be a metric; in fact, it cannot even be a pseudometric.
 * The text of Exterior algebra already assumed the metric that I made explicit.
 * Both existing examples illustrate forms with a metric and an orientation. There is a footnote that mentions the general case, but neither of the examples illustrate it, hence my suggestion of adding the EM field as a third example. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 12:40, 9 September 2022 (UTC)
 * @Chatul: I see your inclusion of the metric for the 1st part of the ME section. (I guess that no metric could, rather unnecessarily, even be represented by a bilinear product in which the mediating matrix is all zeroes.)
 * How does the 2-form illustrate what's missing from, or inadequately explained in, the Article as it stands? Presumably, you want to use the 2-form to illustrate the usual lack of a metric in EA. Will you include some text that explains the difference to the lay-reader?
 * Incidentally, I've rethought my understanding of a vector space (VS) without a norm, so I can see now how the basis occurs and my original confusion is resolved. The reason I called item 5 in ME an axiom was that I read the phrase "if one tries to axiomatize this area" and assumed the A of A was A for Axiom. Thanks for putting me right) PaulGEllis (talk) 16:50, 8 September 2022 (UTC)
 * Yes, an inner product with all zero coefficients cannot be positive definite, and thus cannot be a metric; in fact, it cannot even be a pseudometric.
 * The text of Exterior algebra already assumed the metric that I made explicit.
 * Both existing examples illustrate forms with a metric and an orientation. There is a footnote that mentions the general case, but neither of the examples illustrate it, hence my suggestion of adding the EM field as a third example. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 12:40, 9 September 2022 (UTC)
 * Both existing examples illustrate forms with a metric and an orientation. There is a footnote that mentions the general case, but neither of the examples illustrate it, hence my suggestion of adding the EM field as a third example. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 12:40, 9 September 2022 (UTC)


 * Since you raised the issue of Dependence on metric and orientation, and expressed the intention to add some text, may I encourage you to do so. Not feeling qualified to try to improve the article myself, my role has been to illustrate the ignorance of one general reader in the hope that further explanation helps to clarify. Regards - PaulGEllis (talk) 16:08, 10 September 2022 (UTC)
 * Is the current text in Exterior Algebra clear enough? --Shmuel (Seymour J.) Metz Username:Chatul (talk) 18:07, 28 September 2022 (UTC)
 * @Chatul:: Thanks for adding in the further points about orientation and metric. If I pursue my particular concerns any further it will probably lower the tone of the article too far - better that I go learn some more myself. Regards - PaulGEllis (talk) 19:35, 28 September 2022 (UTC)
 * @Chatul:: Thanks for adding in the further points about orientation and metric. If I pursue my particular concerns any further it will probably lower the tone of the article too far - better that I go learn some more myself. Regards - PaulGEllis (talk) 19:35, 28 September 2022 (UTC)

Style changes
In https://en.wikipedia.org/w/index.php?title=Exterior_algebra&curid=221537&diff=1116584177&oldid=1116422957#Formal_definitions_and_algebraic_properties, Kavuldra cleaned up the presentation of some mathematical text. While reviewing his improvements I noticed and wonder whether it would read better inline as So, ${\bigwedge}(V) = T(V)/I $ is an associative algebra. Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:25, 19 October 2022 (UTC)
 * Presumably the goal is to call this out dramatically, since this is being used as a definition. –jacobolus (t) 20:34, 19 October 2022 (UTC)

Clarifying the relation between the two ways to see exterior algebra: first, as a quotient algebra and second, as the space of antisymmetric multivectors
I have added details the better understand the relation between the two approaches, this being mostly important for positive characteristic of the field. This is pretty much connected to the two possible products given below in the text. 0ctavte0 (talk) 16:04, 12 June 2023 (UTC)