Talk:Tensor/Archive 1

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I'm very convinced that both treatments of tensors should be included. Why not? The modern tratement should ofcourse stay, because hey, it's the modern treatment. However the classical treatment should also remain, because 1. it's very difficult to learn the modern treatment without first learning the modern treatment. 2. most physicist rely on the classical treatment, in fact.. 3. there are some proofs which _require_ the component form of tensors that belongs to the classical treatment.

I've begun a page on the classical treatment of tensors. I think the two treatments should remain separate but equal. Maybe they can even compete for clarity. Perhaps this page should have a brief introduction and then a branch to the two treatments.

Kevin Baas 2003.03.14

Note: Much of this discussion is regarding the old version of the tensor page, which can be found on /Old. Since then a new version was written, originally on /Alternate, but which has received praise from a few people in both math and physics camps, and so was moved up. The stuff on how to best display a tensor symbol is still unresolved and very relevant.

This will probably end up the usual physics way of defining tensors, in terms of objects whose components transform according to certain rules. There's also a usual mathematics way, which involves defining certain vector spaces and not worrying about any given components until bases are introduced. The covariant vectors, for instance, are the elements of a dual space. This has the advantage of letting us know what we are really working with, and I think we should come up with a way to at least fit it into the description. Ideas?

I'm trying to motivate it from a perspective that is a little easier to grasp in a conceptual way (ie. applied math), and then I was hoping to move to a more mathematical way of dealing with them. The problems I had in learning what they were were caused by not being able to understand how they could be used in a geometric fasion. If you would like to add a section that describes them from the other perspective, I think it would be highly informative! I encourage you to do so! Oh, also, its a real pain to type set tensors, any suggestions? --Gavin E. Mendel-Gleason

I had very much the opposite problem. :) I would love to give you a hand but I'm not sure how to fit a formal construction into the above sort of presentation. If you have any ideas, I would be happy to hear them! As for type-setting, you are doomed to a life of misery. ;) But wiki supports using triple apostrophes to bold, that might be a bit easier than using the strong tags. --JG

The current page describes a very classical approach to the subject which (fortunately) is starting to be replaced by differential geometry. This approach is still very popular in engineering mathematics however.

At any rate, my opinion is that the current page is both too specific and too complicated. A tensor, really, is just a linear operator taking elements from one linear space to another linear space (or to the same space). While indicial notation is necessary for computation, it obscures the fact that tensors are invariant with respect coordinate systems, and that is one reason why they are useful. For example, some state of stress in a solid should not depend on the observer.

Also, even engineers are coming to recognize that there is a useful distinction to be made between the linear operator and some matrix associated with the linear operator. Using indicial notation from the outset obscures this very important point. DMD

Not really a map between spaces, but a sort of generalized Cartesian product. Tell you what - I'll try rewriting the page on Tensor/Alternate page tomorrow, and we'll see what turns out best. :) -JG

Superb! Tensor/Alternate should be the main page, the current work can be an example linked from it or something.

Question: why is "Tensors" a top-level heading under Mathematics? Shouldn't it be part of a Differential Geometry section, or maybe even part of Linear Algebra?

Couple of very good reasons, not the least of which is that it is going to be at least 50 years before the engineering and science community give up indicial notation. So we are stuck with, current students are stuck with it, and until the math world pulls differential geometry into the undergraduate curriculum, this situation will not change. So the top level heading makes sense, at least in this day and age. However, tensors should be wikied in from both differential geometry and linear algebra. DMD

Note to the person that has taken an interest in the Tensor page: The rest of that "bullshit" is useful for people that do things like keep your drinking water clean, design automobiles and airplanes, and, in short, get their hands dirty doing things that others may find intellectually distasteful. Since I have had enough of working with deformation tensors this morning (for a paper), I think I will go help a friend hang a sliding glass door...

As it stands, even the engineering perspective on tensors is not intelligible. "Tensors change coordinate systems?" I thought that's what matrices are for. What in the world is a tensor for an engineer? The mathematical treatment is equally poor. We still haven't given the definition of a tensors of type (m,n), covariant, contravariant tensors etc. The algebraic approach to tensor products given in the article is only a tiny part of the whole story. This is all a big mess. Ideally, I would like to see an article which both describes the index jungle used by engineers and the index free approach by differential geometry, and how the two connect. --AxelBoldt

Axel, for what its worth, my copy of Eisenhart's ``Riemannian Geometry is tensor analysis at its classical peak: a nest of indices. In engineering, one generally speaks of tensors as multilinear maps or even (occasionally) as a section of a certain bundle over a manifold. In the first sense, we associate a matrix with a (multi)linear operator. That is, this matrix may change the coordinate and group representation of some quantity. Case in point: the conductivity tensor used in groundwater flow takes the head at a point (the gradient, part of the cotangent bundle), and produces the specific discharge. If the conductivity tensor has no action in the direction of the gradient, there is no specific discharge, no matter what the magnitude of the gradient. Also, and very importantly, head is in units of length, and specific discharge is in units of length/time. Also, it is a big mess. My only opinion is that we should try to cover as many bases as possible, which includes the engineering aspects of the material, but not deprecate any one aspect at the expense of another. DMD

That's what I tried to do on /Alternate, and it does define tensors of type (m,n) and sort of explain how they relate to indices. But it's not well written, and not much use to someone who doesn't know what a tensor is, in my opinion.

With all due respect, I do not think the mathematical treatment of tensors should be geared to people who will use it to keep my water clean, certainly not at the price of dealing with the 3d case on an index by index basis, hiding the concept of a determinant 10 ft. down. The algebraic treatment is the one most convenient, and the one most people I know of (math graduate

students) think of when they hear the word "tensor". Speaking of which: does anyone know how to do the tensor sign in HTML? It really hurt me not to do it. Before we do covariant and contravariant tensors it's way more important to do anti-symmetric tensors (again from a universal-properties POV): these are actually useful when dealing linear forms, like determinants. Sorry I didn't notice the /Talk, I'm fairly new here. We should probably decide what issues we want to cover, and how. However, anything which is linked from the front page of math should have no indices, even if it means there is a /Engineering and /Math sections...

There is no problem with any mathematical treatment of anything. The problem is one of perspective. The vast majority of people that deal with tensor quantities deal with indices. Let me put it another way: Every graduate level engineer has to deal with tensors in indicial notation. There are *alot* of graduate level engineers. There is also *alot* of written material dealing with tensors written in indicial notation. I have at least two shelves of the stuff, including a few groundwater monographs. Indicial notation is not ``bullshit. It is a fact of life, and it won't go away unless mathematicians present a compelling case to the engineering community to *not* use indicial notation. It will be hard. Indicial notation lends itself to writing computer programs, which are useful for making money doing stuff like predicting the subsurface flow of TCE (a nasty dry-cleaning slovent). Oh yeah, the door. I think we will put in a 3-track sliding glass. DMD

Agreed, there isn't anything wrong with indicial notation, and it comes in handy for a lot of things. I find it really confusing, though, when tensors are presented based entirely on indices without the appropriate mathematical definitions - it turns them into collections of arbitrary definitions, and it is hard to see why for instance there is a difference between contravariant and covariant tensors. So I don't think the two approaches should be separated out, as they are referring to the same underlying object, and a decent presentation should really make it clear how. What I'd like done is what I did on /Alternate, but with someone good at math-writing having tweaked it to be more understandable. But now two people have said they like it as is, so maybe my standards are just wrong...

Yes! Indicial notation is horribly confusing! And it sucks! My only point is that all of us are stuck with it until the mathematicians and engineers can put there heads together and banish it. I am glad to see the material on the alternate page moved up, and with a little more work, discussion should segue naturally into indicial form. Then, at the end, top it off with a treatment of dyadic notation... DMD
Actually, I don't think the index notation can ever be banished; it's just a matter of emphasis. Tensors definitely shouldn't be defined using index notation, just like linear maps shouldn't be defined as matrices. But if you want to do any concrete calculations, you need coordinate systems and indices; it's the same with linear maps and matrices really. --AxelBoldt
With respect to analysis on manifolds, banishment would be good, and the sooner the better. Lots of engineering gets much easier (e.g., elasticity). Indices can always be attached once the results are clear! Or at least this is how I have convinced myself :/

I don't know what to do about this mess, but here's a couple of ways to get a tensor product sign in HTML (neither very good). You'll have to edit the page to see how they are done.

  1. ⊗ This should work in all well-behaved browsers, as long as a suitable Unicode font is available. It doesn't work in any browser I tried.
  2. Ä This is non-standard, but works in most browsers.

--Zundark, 2001 Sep 30

Halmos uses Ä, as does most of the mechanics monographs I have read.
I assume Halmos uses a circled multiplication symbol. Both of the above are circled multiplication symbols. The first one apparently doesn't display properly in your browser. It doesn't work in mine either. --Zundark, 2001 Oct 1

Please use ⊗, my browser supports only the first. And <font> is non-standard HTML.

I don't think it's really a good idea to use either -- as I said, neither is very good. Expecting ⊗ to work is unreasonable, even though it will work for some people. (I would be interested to know what font you are using that allows this to display properly.) <font> is standard HTML 4.01 Transitional, but the Symbol font isn't standard. It will work for the majority of people though, which cannot be said for the first method. The best solution is probably to use a GIF. --Zundark, 2001 Oct 1

If we are going to use GIFs, we should have both a large form for the product between spaces and a small form for the product between elements. But how will this work when people are viewing the text at different sizes?

It will look a bit odd if their font size is too large or too small, but I don't think this matters too much. The first method shown above comes out as a question mark or an empty rectangle on most browsers. With Netscape 6.01, which somehow displays the first one correctly, the second one comes out as an A with an umlaut. Compared to these problems, the correct symbol shown at an inappropriate size seems a minor problem. However, image support in Wikipedia seems very primitive. It's easy to put an image inline, like this, but I can't see how to set the width, height or alt attributes for the IMG element. The alt attribute is mandatory in HTML 4. --Zundark, 2001 Oct 1

Right, and alt is important for blind people for example. The size issue of image formulas is elegantly solved by the math wiki software (which we should move towards anyway): --AxelBoldt

So are we going to change the nasty Ä things to ⊗ symbols? The latter is at least standards-conformant and should work on Netscape 6, Mozilla, Mac OS X, and other unicode-aware systems/applications. -- Taral

If we use alt, then ⊗ is the only way anyway, since it alt tags cannot contain <font> stuff. In my browser, lynx, it displays as (x), which is pretty good considering it works inside a text-console. Even seeing people like to use non-graphical browsers -- and I think I deserve being catered to.

Also, I've written sub-article about Tensor of modules over non-commutative rings. It should be fleshed out a bit, and I'm hoping I'll have the time.

I also prepared a place for a sub-article about wedge tensors, since this is an important quotient space of tensors. I'm sick of the fighting on the main Tensor page, so I'll try and flesh these out and let AxelBoldt move them around.

I think it fits best under Tensor product, which is the standard name for the universal construction I believe. I moved everything over, but it is somewhat disjoint right now and needs to be cleaned up. --AxelBoldt

The Tensor product stuff looks like a great start. I am looking forward to seeing how everything gets cleaned up. Hope to learn a bit too!

Oh, and somebody should mention useful theorems about commutation of ⊗ and *, associativity of ⊗ etc. We're also missing a general discussion of universal properties and how they help define structures.

I agree, an article about universal property is needed, and it should definitely be mentioned on the Tensor product and Category theory pages. --AxelBoldt

I just checked the EB article on tensors, and it is an incomprehensible babbling about "functions that transform according to certain rules when coordinate systems are changed". We win :-)

There's a good old usenet thread that about tensors; maybe we can incorporate some of their examples: --AxelBoldt

This is pretty good, it's the kind of material where most people first encounter (rightly or wrongly) notions of tensors. Some of us engineers can should be able to provide practical, real world examples.

I really don't think the intro with examples is that good an idea. It strongly breaks up the flow of the article, potentially confusing the readers by equating tensors and tensor fields before the two have been presented properly, and doesn't tell them anything that saying most quantities in physics and engineering are tensors, which it says anyways, wouldn't. (Btw, not all quantities are tensors - spinors aren't.)

Well, I think that the initial paragraph with examples says a lot more than that; it gives the crucial intuition behind tensors: linear machines that eat vectors. Somebody who knows vectors as "arrows" and has never seen vector spaces will have some idea about what a tensor is after reading the examples. If they want the rigorous treatment, they can just read on.
Also, it does not equate tensors and tensor fields but points out their difference.
I don't see how it breaks the flow of the article: before we jumped directly into the math, now we give some intuition beforehand. --AxelBoldt

I wonder if the notation we use in the component change formula is clear. We implicitly assume that the matrix (xk'k) is the inverse of (xii'). I think one could also argue that those two matrices are the same, just using different letters for the indices. What's the standard way of writing the component change formula? --AxelBoldt

PLethora of pages?

WHy do we have:

Does "old" mean "no longer to be used"? In which case, no need -- page history keeps old versions.

COuld we find better names, eg "Tensor (classical approach)", or even "Classical definition of tensors"? -- Tarquin 16:01 Apr 28, 2003 (UTC)

Humm, for tensors, there is two approach:
  1. The classic method, component aware, used by Einstein, called tensor analysis and prefered by physicist. It is very grossly a generalization of the concept of [[
space|vector]] and matrix and allow to write equations independently of the coordinate system.
  1. The modern, component-free appproach, a field of differential geometry where a physical property is described by a tensor field on a manifold and doesn't need to make references to coordinates at all. It's a more mathematical approach.
Both approach are equaly valid and entirely equivalent (at least for a physicist).
Tensor-classical is a relatively recent essay but the author didn't seem to understand tensors very much.
Tensor and Tensor/Old are older articles.
Some time ago I put in Talk:Tensor-classical, the following draft proposition:
What I think this article should be:
  1. Start by explaining that a tensor is a generalization of the concept of vector and matrices.
  2. Then explain that tensors allow to express physical laws in a form that apply to any coordinate systems.
  3. Say that tensors are heavily used in Continuum mechanics and Theory of relativity (beacause of the previous point)
  4. Introduce the two species: contravariant/covariant, introduce notation and ranks (~= number of indices).
  5. Define the contravariant/covariant component by showing how they transform under a change of coodinate system.
  6. Special cases:
    1. tensors of rank(0/0) => scalars,
    2. rank(1/0) => vectors in differential geometry or contravariant vectors in tensor analysis,
    3. rank(0/1) => one-forms in differential geometry or covariant vectors in tensor analysis.
  7. Give some example: Curvature tensor, Metric tensor, Stress-energy tensor
P.S. I'm not a tensor specialist myself, so ...
Hope it help. -- Looxix 21:45 Apr 28, 2003 (UTC)

In response to the above: Sometime ago I posted this in here, but i put it at the top instead of the bottom:

I'm very convinced that both treatments of tensors should be included. Why not? The modern treatement should ofcourse stay, because hey, it's the modern treatment.

However the classical treatment should also remain, because

1. it's very difficult to learn the modern treatment without first learning the classical treatment.
2. most physicist rely on the classical treatment, in fact..
3. there are some proofs which _require_ the component form of tensors that belongs to the classical treatment.

I've begun a page on the classical treatment of tensors. I think the two treatments should remain separate but equal. Maybe they can even compete for clarity. Perhaps this page should have a brief introduction and then a branch to the two treatments.

Kevin Baas 2003.03.14

-btw, I welcome the idea of changing the names. I think we should transitionally keep the old names as links to the new ones so that people don't have the floor disappear beneath them.

Whatever a tensor is, I doubt that it is a 'compound Jacobian'.

Charles Matthews 18:23 29 Jun 2003 (UTC)

It's a rough intuitive idea. I wanted to provide the reader with a basic geometrical intuition, so that he can take in the information quicker and on a more sophisticated level. If there really were such a thing as a compound Jacobian, then I wouldn't have written it. But, ofcourse, there is not (hence, it is not linked), as the reader can see immediately, and they can pick up from context that the discussion is informal. If you think you can outline the concept any more effectively, be my guest. But DON'T replace it with a grammatically over-complicated and visually incoherent sentence. I'm sick of people doing that! I'd much rather have that sentence removed. I do understand it's flaws.
User:Kevin_baas 2003.06.29

Problem is, a Jacobian is a derivative, in matrix form - and that has no connection with what a tensor is. Really. If you said 'compound matrix' that would say more, and be much less mistaken

Charles Matthews 08:56 30 Jun 2003 (UTC)

Oh really? Is that what a Jacobian is? I thought it was an ice cream flavor! My mistake. User:Kevin_baas 2003.06.30

OK. I've put in my own "sorta kinda" take on this. Providing people with intuitions is good, provided that you don't confuse the analogy with the reality. Ask any physicist! -- Karada 15:33 30 Jun 2003 (UTC)
I like that better, Karada. Thanks.
Charles, regarding the Jacobians and tensors, they are, in fact, intimately related. As you know(I assume), the metric tensor, the arcstone of the classical treatment, which converts between covariant and contravariant form, can be produced by multiplying the Jacobian with it's inverse. That's a pretty obvious and direct connection. I really hope you don't believe that what you said is really true. In fact, I'm curious what compelled you to say something like that. User:Kevin_baas
Because I know what a tensor is? I don't mean some tensor field, I mean a plain tensor. The metric tensor is of course important in Riemannian geometry; the Jacobian as derivative is important when you change variables from one chart to another. It doesn't mean that these pieces of language can be used freely. It all has to parse.

Charles Matthews 16:16 30 Jun 2003 (UTC)

OK, here's what I think we need:

  • an introduction in tensor that can be understood by a high-school grad
  • the classical treatment, for a first year physics undergrad
  • the mathematical treatment, for a final-year physics undergrad (or first/second year maths undergrad?)

Each article should have its own three steps:

  • introduction
  • motivation
  • rigorous treatment, or handoff to a rigorous treatment in the case of tensor

-- Karada 16:22 30 Jun 2003 (UTC)

Ambitious. The current page tensor does a reasonable job. Throwing the intuition back on differentials: could be worse, certainly - if people don't comprehend dxÄdy from that, they are at least no worse off and not misled. But what about tensor fields? Well, I suppose the classical approach as defined jumps straight to those.

Charles Matthews 16:34 30 Jun 2003 (UTC)

Re Karada's program:

Something like this?

In tensor:

  • quick intro: point out that this is only an intro, two rigorous articles also available, classical and modern, suggest reading order (this, classical, modern).
  • physical quantities in classical physics: squeezy / squashy / twisty (think stress, shear, etc.)
  • refer back to partial dervatives, Jacobian...
  • try to develop an intuitive idea of "generalised Jacobian" with reference to real physical quantities...
  • now consider the problem of transforming these in changes of coordinate systems...
  • Elwin Christoffel's work
  • Tullio Levi-Civita's creates tensor analysis in his 1887 paper
  • Gregorio Ricci-Curbastro ("Ricci")
  • Levi-Civita and Ricci's 1900 Méthodes de calcul differential absolu et leures applications
  • discuss covariant, contravariant...
  • point to rigorous classical treatment article here
  • Einstein picks them up and runs with them, Einstein notation.
  • Application to special relativity theory...
  • Then general relativity theory...
  • Just as with vectors, coordinate-free becomes the advanced treatment...
  • Throwing away the coordinate-based scaffolding
  • Motivation for this...
  • point to rigorous modern treatment article here
  • Further reading: modern abstract differential geometry

The Anome 16:57 30 Jun 2003 (UTC)

Also see Application of tensor theory in engineering science, which is just a skeleton. The Anome

OK, I think I've spotted a major problem with the previous structure, which is trying to go to tensor fields and differential geometry much too fast, without explaining elementary tensor concepts on the way. I've imported stuff from the PlanetMath treatment into the classical treatment which should go part way to sorting this: but the intro article also needs not to rush quite so fast: "tensor" is not the same as "tensor field" in general, although it is a useful shorthand and the most common use of tensors. -- The Anome 12:36 2 Jul 2003 (UTC)

The abstract treatment has also been reformatted: could someone proofread it please? -- The Anome

After all the bashing and editing, there isn't that much difference between the "component free" article (which refers to components, after launching with tensor products) and the "classical" one (which deals with tensor products as well as components) -- does anyone now want to have a stab at integrating them into a single technical article, starting with the "modern" classical approach and then generalizing to the abstract approach? -- The Anome 13:13 2 Jul 2003 (UTC)

Consider that, with respect to co-ordinates, the classical approach is serially monogamous, and the modern one is celibate until in the right frame (of mind?). Mutual comprehension is more likely than a unified point of view.

Charles Matthews 18:56 2 Jul 2003 (UTC)

Charles: I can't parse that.

Anome: See:

  1. the above discussion
  2. the discussion in Talk:classical treatment of tensors (with me and AxelBoldt)

Kevin Baas

Well, it was a joke.

Charles Matthews 07:33 4 Jul 2003 (UTC)

I've been looking at and I think the comments about the use of is there are worth bearing in mind. If we have to live with 'X says a tensor is this, and Y says a tensor is that', well, so be it. I think I've now got my head around Kevin's 'compound Jacobians' and if a field quantity does transform according to tensor maps formed from the (Jacobian) derivative, under change of co-ordinates, then the chances are that it is a tensor field. I find this circular (if it is said that tensors are this) - as before, the circularity is bound up with trying to get to tensor fields in one conceptual leap. Not only that, there seems to me still to be a category-mistake, in identifying the 'Jacobian' with the 'tensor'. I can see that saying that the intuition about tensors is one person's way of looking at it will always be a problem.

Does seem to me that the tensor page is converging to a respectable NPOV, if only at cost: throwing technical development elsewhere. I'm now a bit concerned that multilinear algebra will have to take a tutorial strain it wasn't designed to do.

Charles Matthews 15:50 4 Jul 2003 (UTC)

I do not know the distinction between tensors and tensor fields. Also, the way I've learned tensors, that is, from the classical text (Synge's book), a tensor is defined by a coordinate-system transformation, which inextricably is defined by differentials. Indeed, it is intuitively obvious that the geometry of a continuous space can only be defined through the use of infinitesimals.

If there are, in fact, tensors which do not involve differentials, then they neccessarily belong to discrete spaces. Discrete space is a subset of continuous space. Thus the discrete case should be treated as a special case of the general space (because it is), not the other way around. I would also like to see how one can justify discrete 'squeezy-twisty things'. If none of this makes sense, then I do not understand what was intended by the implication that differentials are not intrinsic to the definition of tensors.

In any case, it seems clear that I am not the only one confused about the tensor-tensor field labels, or what sounds like a 'non-differential tensor'. There seems to be an intuitive gap somewhere that needs to be addressed. -- kevin baas

OK, I've restored your version at the old location, and moved the new one to Intermediate treatment of tensors. Please note that Synge's book and your understanding of tensors are not the only valid way of looking at things. -- The Anome 18:44 4 Jul 2003 (UTC)

You know, there really are vectors that aren't vector fields. If you want to make all vectors into translation-invariant vector fields, you can. Just the same for tensors and tensor fields. But the argument about continuous space is an intellectual vicious circle. It may be perfectly good on intuitive grounds - bootstrap arguments often are. But not good foundationally.

Charles Matthews 20:24 4 Jul 2003 (UTC)

The argument about continuous space is called logic. And yes, logic is an intellectual vicious circle. I disagree with the "not good foundationally". The argument is derived from "first principles". It also holds a posteri (or however you spell it). (I also disagree with you on the notion that bootstrap arguments are good on intuitive grounds. But that's irrelevant, because you are here meaning something different by "intuitive" than what I originally meant. I meant that the situation is straight-forward and can be seen without recourse to a technology.) However, the argument itself may be irrelevant, if I do not understand the distinction correctly, or if it is somehow mistated.

In any case, my intention was not to make an argument. Rather, it was to point out an obscurity, which still remains to be addressed. Could you please be more precise about the distinction between a tensor and a tensor field. I'm sure there is an important and canonical distinction. I would like to know what it is. -- Kevin Baas

Anome, thanks. I readily concede that Synge's book and my understanding of tensors are not the only valid way of looking at things. My purposes are purely pedagogical, and dogmatism and self-righteousness are the two worst enemies of pedagogy. -- Kevin Baas

When we stray away from Kevin's preferred approach to tensors, this seems not to add up. I think I'll confine myself to editing, when I can see some improvement to make. If someone feels they can criticise my edits for lack of NPOV or ignorance, they should go right ahead. As far as I can see a tensor field with constant entries is as good as a tensor, on flat space. Can't put it clearer than that. If you're not on flat space, you need something equivalent to what is written at tensor field, which depends on already having the tensor concept. As I wrote above, little point in arguing about intuitions not attached to groups of people who share them.

Charles Matthews 09:48 5 Jul 2003 (UTC)

This is still unclear to me. Do you mean that, by default, a tensor is simply a linear translation, or a basis? If not, how is it different? Again - I do not mean to get you upset. I simply want a straight-forward description. This means one that does not include "As far as I'm concerned..." or anything of the sort. -- Kevin Baas

The more I read about tensors, the more I realise that my previous understanding of tensors was somewhat simplistic. I'm currently reading Geometric Methods of Mathematical Physics by Bernard Schutz, and discussing this with friends.

Schutz says the following:

"Consider a point P [of a manifold] M. A tensor of type (N, N') at P is defined to be a linear function which takes as arguments N one-forms and N' vector (spatial)s and whose value is a real number."

So there you have it! What could be simpler than that? Of course, the problem is that without a proper understanding of what a manifold, vector, one-form are, this is a completely opaque statement. However, at this point you are 57 pages into his book, so you should by then know what each of those are. The problem is that we don't have a 57 page Wikipedia article to precede the tensor article.

The trouble is, there are two extremes here:

  • people who've had a quick intro to tensors in order to use them as a tool in physics or engineering, and have an elementary idea that works well for them, and generates correct results, but annoys the mathematicians
  • pure mathematicians, who probably know better than anyone else what a tensor really is, but are unable to communicate the ideas to anyone other than other pure mathematicians

and most of us are somewhere in the middle.

What we need are some mathematical physicists to help out. -- The Anome

Please keep in mind, people, that "This article attempts to provide a non-technical introduction to the idea of tensors..." - Kevin Baas

As someone with absolutely no tensor experience, I can say that the current page (thanks for all the hard work, by the way) ALMOST gets me to a place where I can understand them. I feel like a real understanding of them is in my peripheral vision, but it flees when I turn to look at it. I think something more explicit in the beginning that discussed the relationship (logical, not practical) between vectors, matrices, and tensors would be helpful. Also, maybe the linear operator page could say something about the overall relationship to functors? Of course, much of this confusion is probably arising from the fact that I have less math background than the typical target audience for this article. Defiler 01:52, 23 Oct 2003 (UTC)

As someone with tensor experience, I can say that there is lots of room for improvement on this page. There are sections wich wander randomly into nonessential details. There is way too much technical jargon used without any conceptual aids. ect. Really people, tensors are not as complex as this page makes it seem. The basic idea of a tensor is REALLY SIMPLE. That's what this page is for: The Basic Idea. "A tensor is ___". The description should not involve any jargon from either treatment. That jargon is too be introduced in the particular treatment and not before. Keep that crap off of this page, and get to the f-ing point. Please. Thank you. --Kevin Baas

Kevin, I would greatly appreciate any clarification you can bring to this article. I agree, tensors should not be this confusing. -- The Anome 09:52, 28 Oct 2003 (UTC)

I suggest, amongst other things, moving out all references to tensor densities to a page of their own. They'd be the icing on the cake here.

I do not suggest any move away from NPOV, by 'favouring' one approach over another.

Charles Matthews 09:59, 28 Oct 2003 (UTC)

Here's an approach that blends the two...

  • Motivation: local linearity of physical phenomena, linearity over many variables, need to be invariant over coordinate transforms
  • Give some examples: vectors, ship motion example
  • Show that some kinds of "vectors" transform differently from others: give old-fashioned names: "covariant vector", "contravariant vector".
  • Now explain that the covariant vectors are really one-forms.
  • Now construct tensors as multilinear maps of vectors and one-forms.
  • Show that tensors defined in this way have the properties we desire to explain physical things, such as correct transformation under coordinate transforms
  • Show that basis vectors still work..
  • ..and transform appropriately
  • derive the component-based treatment from this, as a trivial consequence of what went before
  • then go on to link to further reading:

-- Anon.

Actually, there is always a problem if you want to join up the 'bilinear' with the transformational approach. And there is a second problem with saying one-forms are covariant when actually they pull back. Now I'm convinced these things can get resolved, but I've dropped out once from the discussion after too much testosterone-laden talk, and may do again. Let's just say, the page still needs work.

Charles Matthews 18:02, 13 Nov 2003 (UTC)

How about this: one could, for example, give a definition in the first paragraph, involving abstract concepts like "coordinate systems" and "measures" and the like, and then one could go on about other stuff, and continue to throw more ideas into the mix in every paragraph...

When, ofcourse, the original intention is just to make that first paragraph clear: what exactly does a tensor do? what is it? (if these terms or approach is too classical, then perhaps, instead of trying to make the paragraph inbetween classical and modern, and thus jumping erratically between the two and never acutally hitting either of them or anything distinct and coherent enough to paint a clear picture, we may be better off having two interpretations of a tensor; two "definitions" per se, at least tentatively, to see how it fleshes out, and then wwe could find the shared ideas in those pareagraphs and work from there, still, perhaps, using those paragraphs in whole within the body.)

Perhaps, instead of throwing out more ideas, we just make those abstract ideas clear with basic and straightforward physical examples. Take them through a problem, minus the mathematical specifics.

The main point i'm trying to make is, take a concise definition and flesh it out a piece at a time, rather than throwing more jargon into the mix.

--Kevin Baas

Well, I think that could re-open debates that may have resolved themselves by now. My preferred option at present is to develop a glossary of tensor theory, from roughly where we are.

Charles Matthews 08:42, 17 Nov 2003 (UTC)

Wow! I did not understand a single sentence! Very useful article! :))

Please follow the idea of "Encyclopedia". I feel that the authors are nice guys and actually mean good, but I dont understand what they mean.

New To Advanced Math

Hi; I'm trying desperately to understand many of these advanced principals of mathematics, such as lattice groups, but no matter how many times I review the material, it doesn't sink in. Could someone please provide examples, problems to solve (with their solutions) and/or ways to visualize this? beno 26 Jan 2006


This page needs serious trimming. Taral 17:09, 19 Jun 2004 (UTC)

Would someone edit this page so that it is at least in a readable style? As I do not yet understand tensors, I cannot perform this edit (for fear of losing information).

I agree - this article is incomprehensible to me, and is very badly written. Sorry to be negative, but can someone _Please_ re-write this. I can't believe that this is the best we can do. I would re-write it myself if I understood what a tensor was. :) Williamstown33 (talk) 11:12, 11 June 2008 (UTC)

I agree. My best sense of a tensor is that it describes things that simply can't be described by vectors, but like a vector and a scalar they represent physical quantities. The canonical example is the stress state of a material at a point: you can pull on something in one direction and shear it in a transverse direction. You could try to describe this with several vectors, but the stress state at a point is neatly described by a single tensor. With that tensor, you can figure out how much compression that point is under (a scalar), or how much it is under tension in a particular direction, but a tensor captures all of this. (That sounds a bit circular now that I read it; let me know if that makes any sense.) Tensors are usually represented as a matrix, but don't need to be. Just as complex numbers can be represented as a matrix, tensors are considered mathematical objects in their own right.
That said, I could not give you a definition of what is a tensor and what is just a matrix. That's one thing I'd like to see on this page. —Ben FrantzDale (talk) 13:08, 11 June 2008 (UTC)
I agree. I cannot understand a word in this article, and without learning about tensors I can't learn General Relativity which I long to learn about!The Successor of Physics 12:03, 15 March 2009 (UTC)

Disambig needed

The article says:

The word tensor was introduced by William Rowan Hamilton in 1846, but he used the word for what is now called modulus.

which is fine except that modulus is a disambiguation page. So which of the 6 possible meanings did he use? I can discount 3 of them straight off, but that leaves another 3 still. --Phil | Talk 15:03, Sep 21, 2004 (UTC)

I believe he used the word "modulus" itself and not the word for what is now known as modulus. --MarSch 10:50, 13 April 2006 (UTC)

Disambiguation needed!! There are large well established groups of users (which has been recognized but not very well dealt with up front) of tensors. Tensorial properties differ substantially depending on what the "active" definition is. Starting from the most abstract definition fails to be useful except to the cognocenti. Divide and conquer! Start by saying that most used tensors are special cases of the more abstract general case and then specifically describe the (easier to understand) special cases (remember "compare and contrast" ?). There seems to be some confusion as to whether a tensor is a set of numbers, a set of functions, a set of operations, or something else. {I am unqualified to say too much more (or this much)]. Maybe start with applications rather than the math. Tensors are used in equations which describe certain real world phenomena such as fluid flow, structural loads on buildings, electromagnetic fields, gravitational fields, etc. They are also used in more abstract mathematical areas such as topology (?). They are needed because scalars, vectors and matrices are inadequate (why?) and also when their notation is more compact or easier to work with. Please be careful in not confusing things like fields, functors, operators, operations, functions, transformations, vectors, etc. Please define terms like "invarient". Obviously conservation or invariance are key concepts for this and should probably be tackled up front.

Generalized Hooke's law


I have a question here about the tensor notation, if anyone can help me... Thanx

Cdang 15:38, 26 Nov 2004 (UTC)

Abstract index notation?

As a physicist with a mathematical bent, I have for some time had a strong preference for the "abstract index notation" for tensors introduced by Roger Penrose. (This is the notation used by Robert Wald in his textbook General Relativity, for instance.) It applies to the modern component-free approach to tensors, but it looks like a component formalism. For instance, raised and lowered indices represent whether each "slot" of a tensor acts on elements of the vector space or its dual. And contraction between vector and dual vector slots is represented by a repeated index (which looks like the Einstein summation convention for components, even though no component sum is implied).

I feel like using that notation could help to simplify the article: it's a fully modern mathematical approach which should satisfy mathematicians, but it is in general very easy to convert it to a component formalism (just substitute component labels for the abstract indices) which makes it directly useful to physicists and engineers who haven't studied the mathematics in depth. (In fact, those who don't care about the mathematical details could probably get through the less modern parts of the article without even realizing that they weren't looking at component expressions.)

Do other people out there have experience with this notation? Are there pitfalls in using it that I haven't considered?--Steuard 22:30, Apr 18, 2005 (UTC)

Well, we are duty-bound to include all major points of view in this, the central tensor article; favouring just one approach goes against our charter. This does cause problems, which are particularly obvious here, in this case. But we can't address them by assuming that the right way to teach tensors is ...; there just are a number of aspects. Charles Matthews 07:04, 19 Apr 2005 (UTC)

Most physicists who have done 2 years maths methods can (and need to) handle the expansion of (curl (curl u)) (and the equivalent expansions for eg u. del (u) needed in fluid mechanics, eg for 'Crocco's relation') - can someone write down/derive these expansions for me in modern notation? After a few weeks (admittedly casual) acquaintance with Penrose, I still wouldn't wish to, (and even less, would I expect to be able to teach it to a fellow physicist). Linuxlad 09:15, 19 Apr 2005 (UTC)

Curl is supposed to be done with Hodge duals. Curl of a curl would be like *d*d with the exterior derivative, so related to a Laplacian like *d*d + d*d*. It is perfectly true that some mutual incomprehension results from divergent (sorry) ideas about how to get to the needed bits of vector calculus. A mere encyclopedia article is unlikely to sort out schisms, such as existed between J. W. S. Cassels and George Batchelor in Cambridge. Being a mathematician, I am always going to stick up for a sensible answer to 'what a tensor is', coming near the beginning. One thing I feel is needed is to get tensor density off this page, and treated properly on its own. There is a good reason for that, namely that it 'breaks' the Bourbaki approach to tensors (comes back when one moves onto tensor fields). Charles Matthews 09:36, 19 Apr 2005 (UTC)

It would be fun to have your take on the Cassels/Batchelor interaction. (I remember going to a joint CEGB/DAMTP 'Problems Drive' where GKB was paricularly heavy on our lead mathematician, who spent too long going through his new FE fluids code - GKB wanted only to hear the essential science of it...)

discussion at Wikipedia talk:WikiProject Mathematics/related articles

This article is part of a series of closely related articles for which I would like to clarify the interrelations. Please contribute your ideas at Wikipedia talk:WikiProject Mathematics/related articles. --MarSch 14:10, 12 Jun 2005 (UTC)

Article is not approchable

This article attempts to provide a non-technical introduction to the idea of tensors, but fails, because it seems rather caught up on the philisophical/pedagogical nature of tensors, rather than a concrete description of what they are. While it may be true that While tensors can be represented by multi-dimensional arrays of components, the point of having a tensor theory is to explain further implications of saying that a quantity is a tensor, beyond that specifying it requires a number of indexed components, an article that doesn't actually list what the properties of tensors are, does neither.

I'm sorry to sound a little negative, but although I have a passable understanding of scalars and vectors, I can't even begin to understand tensors from this article - to me it appears you have to know what a tensor is before you can learn what a tensor is. I'd try to fix it myself, but I don't know what tensors are. (Hence my issue.)

Although tensors can be treated as an abstract quantity, it might be beneftial to talk about the concrete aspect of tensors first, so that a rudimentary understanding can be gained by non-experts. Then you can move on into the abstract generalization. E.g. with vectors: in a vector space a vector is much more general than the conventional "list of numbers" view, but that view is usually presented first, and once that is understood, generalization proceeds from there.

At the vary least, the first paragraph or two should precisely define what a tensor is, instead of weasling around it. (Consider: "In mathematics, a rectangle is a certain kind of geometrical entity. The rectangle concept includes the idea of a square. Rectangles may be written down in terms of coordinate systems, or as a set of points, but are defined so as to be independent of any chosen representation. Rectangles are of importance in physics and engineering. In the field of surveying, for instance, .... While rectangles can be represented by coordinates, the point of having a rectangle theory is to explain further implications of saying that a quantity is a rectangle, beyond that specifying it requires a number of points. In particular, rectangles behave in specific ways under geometric operations. The abstract theory of rectangles is a branch of Euclidlean geometry." All true, but if you didn't know it before, there is no way from that that you'd discern that a rectangle is a planar figure with four sides and four right angles - you certainly wouldn't understand any difference between a rectangle and a rhombus.) Specifically, I might reccommend moving the approaches in detail toward the top of the article, certainly before the examples, and probably before the history (Which, by the way, is woefully lacking in discussion of *why* tensors were developed.

Great comment. Did you also look at the other tensor articles: Classical treatment of tensors, Tensor (intrinsic definition), Intermediate treatment of tensors and hopefully that's all. I really hate that we have such a mess of articles and I would try to clean it up, if the consensus wasn't against me. Maybe you can help change that, see also the above section. --MarSch 10:25, 22 Jun 2005 (UTC)
I've added some text under examples to attempt to address this issue. These examples attempts to build on the imagery of what is currently the second paragraph of this article (...multi-dimensional arrays...) and on contrasts with vectors and scalars. If this is inadequate, perhaps the "for computer programmers" section of tensor product is concrete enough? RaulMiller 13:49, 8 October 2005 (UTC)

I don't find your comment negative at all. In fact, this seems to be not only a problem with this particular article, but virutally every introduction to tensors that I have ever come across. I am still wondering what a tensor is but all I have been able to figure out so far is that it is like a vecotr, but with the components more mixed up. Arundhati Bakshi 11:40, 28 January 2006 (UTC)
Many years ago I passed a course in tensor math but now I can't remember what tensors are--if I ever really knew. I came here to refresh my memory, but I'm also finding these articles difficult. Let me suggest a way to "light a candle."
I remember in high school I had a difficult time understanding vectors, until I learned about adding velocities. We did it by drawing arrows in different directions tail to point, the drawing the effective resultant vector. A very physical interpretation, with no mention of coordinates. But also easily understood in terms of coordinates, and in terms of independence from the particular coordinate system.
Using that analogy, I'm speculating that tensors are the "next step up" from vectors. If scalars are values that have NO direction; vectors have ONE direction, then tensors have MANY directions, or maybe ALL directions at once. And there seems to be an essential component of anisotropism--different values in different directions--but in such a fashion that the value changes smoothly ("linearly"? in the limit) as you "turn" from one direction to another. I'm imagining something like a cross between a porcupine (arrows in all directions of different lengths but points forming a smooth surface) and a squishy rubber ball or jello cube, such that if you squash down on the top they spread out at the sides, whack-a-mole style. I haven't yet touched on the covariant/contravariant concept, but I suspect that is also a crucial concept.
Of course, I'm just speculating, so I could be totally off base.
In mathematics, it's hard to understand what an object "is" apart from the operations on it. So an introductory article should include at least the most common elementary operations, e.g. the various "products," with corresponding graphical/physical interpretations.
Is it possible to give a graphical illustration of tensor quantities and tensor operations, analogous to the graphical representation of addition of velocity vectors? That might help some of us newbies.
Drj1943 06:10, 4 January 2007 (UTC)
Tensors are geometric objects, so illustrations are possible sometimes. However, most of the interesting examples involve higher rank, or are tensor fields, or both; helpful illustrations could be a real challenge.
The article can be improved, but it will take time for all the contributions to accumulate. Meanwhile, here are a few thoughts, working with the mathematicians' definitions.
Begin with a smooth surface in ordinary 3-dimensional Euclidean space, such as a torus. Fix a point on the surface; tangents to curves passing through there are vectors spraying out in different directions, but all lie in a common plane, a 2D tangent space, Tp. We now consider three tensor fields, all varying smoothly.
  • Rank 0: To each point p of the surface, assign a scalar value f(p).
  • Rank 1: To each point p of the surface, assign a vector value from Tp.
  • Rank 2: To each point p of the surface, assign a metric tensor that maps a pair of tangent vectors to a number.
Formally, the tensors that make up the vector field are linear functions that take a 1-form (a dual vector) and produce a number. --KSmrqT 11:15, 4 January 2007 (UTC)
I think as an introductory article the best starting point would be a concrete illustration of a rank 2 tensor (perhaps preceeded by an ilustration of a scalar (rank 0) and vector (rank 1). The sort of thing I have in mind would be: this discussion of a stress tensor. Once the concept scalar -> vector - > rank 2 tensor is established then the rest could follow. The article also needs a better lead paragraph, the disinction between a tensor field and a tensor as used in maths is not so fundamental. I'll think a bit more about this. —The preceding unsigned comment was added by NHSavage (talkcontribs) 09:52, 31 March 2007 (UTC).
Oops sorry, I didn't sign then... just thought I'd add another link which I think is useful: Tensor from Wolfram Math World. My idea for the lead paragraph would be something along the lines of: In mathematics, a tensor can ne considered as an extension of the sequence scalar (zero indices), vector (one index), matrix (2 indices) to an arbitrary number of indices. In other words a scalar can be defined as a tensor of rank zero, a vector is a tensor of rank 1 and a metrix is a tensor of rank 2. Just as a vector can be represented by a 1D array of numbers and a matrix by a 2D array, a tensor of rank 3 can be represented by a three dimensional array of numbers. They are used in physics in general relativity, elasticity and fluid mechanics. Just a starting point but more informative than starting with the difference between a tensor and a tensor field. This may be oversimplfying things but if this is meant to be an introduction then that is probably ok IMHO.
The illustration of the scalar->vector->matrix->hierarchy needs someone good at that sort of thing to participate but I would concieve of a picture of the left and a mathematical represention of the right. So a short line with a number next to it, a 3D vector with a standard vector as a column matrix, and a stress tensor illustration with a matrix next to it.--NHSavage 10:35, 31 March 2007 (UTC)
At this point perhaps the following question will prove of use. What is a 'vector'?
Here are some possible answers:
1) A vector is a list of numbers
2) A vector is a displacement, or to be more specific a geometric quantity possessing both magnitude and direction
3) A vector is an element of a vector space
Next question: what is a 'tensor'? Rmilson 17:48, 31 March 2007 (UTC)
I can't answer the last question, although I think probably any of the 3 definitions could be valid depending on the context and the degree of rigour/perspective you want to use. However, speaking personally, I found the explanation of a tensor as the next step in the progression so to speak to be one which gave me a quick idea of what a tensor is. Surely that is what this article ("a non-technical introduction to the idea of tensors") should be aiming for, rather than a rigourous mathematical definition? --NHSavage 19:44, 31 March 2007 (UTC)
Well, if you take that approach, you might as well just look at array. There is no definition of tensor which is not equivalent to the 'rigorous mathematical definition'. (Well, qualify that for 'tensor density'.) We have had this discussion for three years, and I don't think we've been offered an expository option that is at all 'non-technical'. Charles Matthews 19:58, 31 March 2007 (UTC)
Fine, if that is correct IMHO there should not be an article which pretends to be "a non-technical introduction to the idea of tensors". On the other hand it is the starting of the Wolfram Math World article and provides a more comprehensible starting point to someone who has never come across them before. I am sure that even an introductory article has to go a lot further than this but if you can't offer a non-technical article you migth as well delete this one. I am not a mathematician, just someone who has recent had to become aquaited with tensors. (which is why I decided not to follow the usual policy of being bold).--NHSavage 19:18, 2 April 2007 (UTC)
So the lead section has had an incorrect definition introduced. It is not true that mathematical tensors are functions (Halmos strikes again, it seems). It is also misleading to say that mathematicians and physicists are talking about different things (beyond the valid point about tensor fields). Charles Matthews 20:02, 31 March 2007 (UTC)

Minor correction

I think an error slipped into the article:

The scalar quantities are those that can be represented by a single number --- speed, mass, temperature, for example.

I'm pretty sure that speed is a 1st order tensor (vector) and not a 0th order tensor (scalar)?

In technical physics-speak, "velocity" is a vector and "speed" is the magnitude of that vector. But your point is still valid: only someone who knows physics terminology well would distinguish between the two words, and the vector version is certainly the more fundamental concept. So perhaps this example should be removed or replaced for the sake of clarity.--Steuard 15:57, 13 October 2005 (UTC)

how do you pronounce?

is it ten-ser or ten-saw?

That depends where you live :) Most usually just ten-sir, I think. Karol 12:22, 25 October 2005 (UTC)
And it depends on how you pronounce saw, I suppose. The US Midwest or Southern US accent often has saw sounding like sore, which would sort of make sense for tensor. Anyway, I pronounce it TEN-sir. —HorsePunchKid 19:07, 25 October 2005 (UTC)


The definition of the word rank in the article is possibly confusing: for example, the rank (as usually defined) of the matrix A=[1 2; 1 2] is 1, while according to the definition in the article, A has rank 2 because it has 2 indices. For general tensors, there is a definition of rank under which A would have rank 1; see e.g. . The definition is as follows: an order-n tensor A has rank 1 if it can be written as the outer product of n vectors. It has rank k if it can be written as the sum of k rank-1 tensors.

For what it's worth, MathWorld has the same confusing definition of rank ( ), but I still think it would be better to use the one which aligns with the usual usage for matrices.

Also, unfortunately, it appears to be hard to compute the rank of a general tensor: see . 04:26, 3 January 2006 (UTC)

You have a valid point on this, but if usage is not consistent, we sometimes have to resort to a choice of conventions. Charles Matthews 09:17, 3 January 2006 (UTC)

Rewritten article

Putting the 'Brief overview' so far up the article is very unfortunate. While it may provide reference material for those who have already had a course on tensors, it is going to look like a slew of symbols to those who have not. This is not how we should approach this admittedly-difficult topic. The sentence It can be deduced from the above that a rank 3 tensor is the same as a 3 dimensional matrix is just about everything that should be avoided here. Charles Matthews 09:17, 3 January 2006 (UTC)

Rank 3 tensor same as 3d matrix?

It can be deduced from the above that a rank 3 tensor is the same as a 3 dimensional matrix.

I don't think that's correct, anyway. The other tensor related articles are quite emphatic about saying that a tensor isn't the same as its representation, since the representation is dependent on the particular basis one chooses. The other problem is that the matrix dimensions represent covariance and contravariance, therefore by my reading, a tensor of rank (0,3) should be denoted as a vector, while still having total rank 3, for example by the Kronecker tensor product


which represents a rank (0,2) tensor notationally as a vector. By extension the same idea applies to representations of rank 3 tensors, which can be represented as vectors or matrices depending on covarient and contravariant ranks. In any case, I'm pretty sure the quoted statement is wrong and should be deleted. It would be nice if someone who knows the subject could give a clear description of the relationship between the rank of a tensor, the notation used to represent it and covariance and contravariance. Does it, in fact, ever make sense to represent a tensor as a three dimensional matrix? 18:55, 21 January 2006 (UTC)

The 'rank' thing is unfortunate; we may have been landed with it by someone's insistence on terminology from a very old textbook. Certainly tensors can be written as
meaningfully. Charles Matthews 12:19, 28 January 2006 (UTC)
Well, I see rank is used this way very often. Charles Matthews 12:21, 28 January 2006 (UTC)
I realize the rank of a tensor and the rank of a matrix are two different things. My point has to do with the fact that saying a 3d matrix is the same as a rank 3 tensor is misleading. The example I gave, I think anyway, shows that a notational vector can represent a rank 2 tensor, and if you were to repeat kronecker product with a third vector you would have a (notational) vector or matrix representing rank 3 tensor. Am I wrong about that? Also in introductions to tensors, the notions of covariance and contravariance are treated as though they are related to column and row representations of vectors, which would seem to imply that the duality of covariance and contravariance maps onto the row and column representations in a matrix, so does the third dimension of a 3d matrix represent a covariant or contravariant direction?
Basically, what I'm getting at, I think, is that, as far as I can tell, there isn't a one to one correspondence between vectors and matrices as devices for representing information, and tensors which are abstract things defined by the type of operations they participate in. So saying that a rank 3 tensor is a 3d matrix is misleading, especially since there is no such thing as 3d matrix algebra (at least none I was taught in any of my linear algebra classes). I think in introducing tensors it's probably wise to make this distinction. 23:32, 28 January 2006 (UTC)
I agree and will remove that sentence. --MarSch 11:00, 13 April 2006 (UTC)

Since that whole section was a big mess I've replaced it with:

A tensor of rank 0 is just a scalar. A tensor of rank 1 is either a tangent vector or a tangent covector. Higher rank tensors are formed by sums of tensor products of rank 1 tensors.

which is correct and unambiguous and I think states the essential points made by the previous version.--MarSch 11:19, 13 April 2006 (UTC)

When is a matrix not a tensor

I've been using tensors for years now in engineering, but still don't have a satisfactory sense of when something is a tensor and when it is a matrix. I understand that a tensor is independent of the basis it is written in, just like a vector. A vector is better thought of (to me at least) as an arrow rather than as a tuple, since the arrow does not depend on basis. The same is true of a tensor, although it isn't as easy to draw. In engineering applications, it seems like the things that get called "tensors" are usually physical quantities such as stress or a strain gradient tensor whereas other linear operations such as rotation are more often called "operators". For example, are either of these tensors (or matrix representations of tensors in a particular basis)?

When is a matrix not a tensor? —Ben FrantzDale 22:05, 6 May 2007 (UTC)

a matrix is a mixed tensor of type (1,1). not quite sure why the divergence in terminology. i would guess it is called a tensor when it represents some physical quantity that is invariant under change of basis, a geometric object. if it is viewed as a linear map, it is called a matrix. this distinction is, of course, very artificial. maybe other folks can shed some light on this. Mct mht 00:06, 7 May 2007 (UTC)
Hua. So a matrix, M, would be in indicial notation? —Ben FrantzDale 00:40, 7 May 2007 (UTC)
Yep. A matrix is a like a number, in that it can "mean" many different things, just as a number can be a temperature or pressure or distance. The most common thing a square matrix means (represents) is a linear transformation, or linear operator, mapping vectors to vectors. This is so common that some folks think only in terms of matrices, where a mathematician would not. Now consider the thing we call an inertia tensor; we can arrange the nine moments as a symmetric matrix, but it might be confusing to try to identify that matrix with a linear transformation. If we can write a tensor using only two indices, we can put its components in a matrix; however, a common tensor like that for Riemannian curvature has too many indices to fit into a matrix. --KSmrqT 00:55, 7 May 2007 (UTC)
It appears one part of the answer is, given just a matrix, you can't know for sure because being a tensor rather than a matrix is a transformation property, and a matrix doesn't have that idea. In other words, tensors are commonly implemented in terms of matrices.
As for the examples I gave, at least the rotation matrix is not a tensor. Consider ninety-degree rotation. That matrix looks like [[0 -1][1 0]] in good old Euclidean coordinates. But if we want a ninety-degree rotation with the basis {(2,0), (0,1)}, it'll look like [[0 -1/2][2 0]].
Similarly, if we have the same rotation and we want to use the basis {(0,1),(-1,0)}, the rotation matrix remains unchanged. If I have a tensor [[0 -1][1 0]] in the usual basis, then in {(0,1),(-1,0)}, I think it winds up looking like the identity(?). If that's about right, then I understand. —Ben FrantzDale 03:15, 7 May 2007 (UTC)
Not exactly.
Suppose we are given a fixed finite-dimensional real vector space, V, with standard inner product; R3 will do fine. We know what it means for a real-valued function f on V to be linear: f preserves sums and scales. We can collect all such functions (or "forms") into another real vector space, V, which has the same dimension as V. In fact, given an ordered basis (e1,…,en) for V, we can choose an ordered basis (ω1,…,ωn) for V such that basis function ωi applied to basis vector ej yields 1 if i = j and 0 otherwise. We call V the dual of V. In some literature, especially outside mathematics, the distinction between these two spaces is blurred, but we want to be more careful.
Formally, a tensor is a real-valued function that takes a mixed list of vectors and dual vectors as arguments, and is a linear function of each. Our first two favorite examples are the dot product and the determinant (considered as a function of column vectors).
Our next favorite example is the obvious application of the dual.
Obviously, a linear operator, g:VV, does not fit the pattern of a tensor; but we can use the dual pairing to get around that. Think of dual vectors as row vectors (using the dual basis); then we can write F as a 3×3 matrix — here an identity matrix. Thus every linear operator can be written a square matrix, and also can be interpreted as a tensor of mixed rank.
So, although a rotation is not a tensor, we have a convenient way to treat it as one.
Using Einstein summation notation, this mutation is so natural and subtle it could go unremarked. Similarly, we can easily turn the dot product into a matrix, or raise and lower indices at will using the dual pairing. But we can never convert a rank three tensor like the determinant into a matrix; the former requires three indices, whereas the latter provides only two. --KSmrqT 21:21, 7 May 2007 (UTC)

Babangida? Arnold? 'gnem on a poominge'??!

Under the approaches in detail section, there is the following text:

  • The modern approach
The modern (component-free) approach views babangida initially as an abstract object, expressing some definite type of multi-linear concept. Arnold's well-known properties can be derived from his definition, as linear maps or more generally; and the rules for manipulations of tensors arise as an extension of linear algebra to multilinear algebra. This treatment has attempted to replace the component-based treatment for advanced study, in the way that the more modern component-free treatment of gaston replaces the traditional component-based treatment after the component-based treatment has been used to provide an elementary motivation for the concept of sage. You could say that the slogan is 'gnem on-a-poominge'. Nevertheless, a component-free approach has not become fully popular, owing to the difficulties involved with giving a geometrical interpretation to higher-rank tensors.

Is this actually meaninful or some prank? !jim 19:25, 6 October 2006 (UTC)

Origin of word "tensor"

The history at the web site Earliest Known Uses of Some of the Words of Mathematics disagrees with what is currently stated in this article:

TENSOR (in quaternions) was used by William Rowan Hamilton (1805-1865) in 1846 in The London, Edinburgh, and Dublin Philosophical Magazine XXIX. 27:

Since the square of a scalar is always positive, while the

square of a vector is always negative, the algebraical excess of the former over the latter square is always a positive number; if then we make (TQ)2 = (SQ)2 - (VQ)2, and if we suppose TQ to be always a real and positive or absolute number, which we may call the tensor of the quaternion Q, we shall not thereby diminish the generality of that quaternion. This tensor is what was

called in former articles the modulus.

The earliest use of tensor in the Proceedings of the Royal Irish Academy is on p. 282 of Volume 3, and is in the proceedings of the meeting held on July 20, 1846. The volume appeared in 1847. Hamilton writes:

Q = SQ + VQ = TQ [times] UQ

The factor TQ is always a positive, or rather an absolute (or signless) number; it is what was called by the author, in his first communication on this subject to the Academy, the modulus, but which he has since come to prefer to call it the TENSOR of the quaternion Q: and he calls the other factor UQ the VERSOR of the same quaternion. As the scalar of a sum is the sum of the scalars and the vector of the sum is the sum of the vectors, so that tensor of a product is the product of the tensors and the versor of a product is the product of the versors.

In other words, the tensor of a quaternion is simply its modulus.

In his paper "Researches respecting quaternions" (Transactions of the Royal Irish Academy, vol. 21 (1848) pp. 199-296), Hamilton uses the term "modulus," not "tensor." This paper purports to have been read on 13 November 1843 (i.e., at the same meeting as the short paper, or abstract, in the Proceedings of the RIA).

The terms vector, scalar, tensor and versor appear in the series of papers "On Quaternions" that appeared in the Philosophical Magazine (see pages 236-7 in vol III of "The Mathematical Papers of Sir William Rowan Hamilton," edited by H. Halberstam and R.E. Ingram). The editors have taken 18 short papers published in the Philosophical Magazine between 1844 and 1850, and concatenated them in the "Mathematical Papers" to form a seamless whole, with no indication as to how the material was distributed into the individual papers.

(Information for this article was provided by David Wilkins and Julio González Cabillón.)

TENSOR in its modern sense is due to the famous Goettingen Professor Woldemar Voigt (1850-1919), who in 1887 anticipated Lorentz transform to derive Doppler shift, in Die fundamentalen physikalischen Eigenschaften der Krystalle in elementarer Darstellung, Leipzig: von Veit, 1898 (OED2 and Julio González Cabillón).

The term TENSOR ANALYSIS was introduced by Albert Einstein in 1916 (Kline, page 1123).

According to the University of St. Andrews website, Einstein is reported to have commented to the chairman at the lecture he gave in a large hall at Princeton which was overflowing with people:

I never realised that so many Americans were interested in tensor analysis.

Tensor analysis is found in English in 1922 in H. L. Brose's translation of Weyl's Space-Time-Matter: "Tensor analysis tells us how, by differentiating with respect to the space co-ordinates, a new tensor can be derived from the old one in a manner entirely independent of the co-ordinate system. This method, like tensor algebra, is of extreme simplicity" (OED2).

Hamilton did give us vector and scalar used in much the same way as at present, but clearly his use of "tensor" has nothing to do with our modern sense of the term. Therefore, I do not think he should be cited. Instead we should add a reference for Woldemar Voigt. --KSmrqT 00:21, 12 November 2006 (UTC)


The term "type" e.g. type(1,1) or type(2,0), is used in the Physical Examples section but is not defined there. Also, I also didn't find it in the Glossary.

After much looking, I did find a definition of "type" (also called "valence") in the "Intermedidate" treatment, referring to the number of "covariant" and "contravariant" indices, but that introduces additional new terms into the introductory discussion.

The earlier discussion comment "Rank 3 tensor same as 3d matrix" uses the notation: rank (p,q). Is this another variation of "type"?

In any event, it seems that the distinction between "covariant" and "contravariant" is one of the significant concepts of tensors, so perhaps that should be explained also. Drj1943 05:07, 4 January 2007 (UTC)

One possible way to explain "covariant" and "contravariant" is to use a topographic map as an example. Altitude on this map can be expressed in two ways, viz, as "contour lines" with steepness represented as contour line density (this is the traditional method), or as little arrows pointing up (or down) hill, with the arrow length representing steepness. Both are a representation of the same "gradient" tensor field. Both describe the same object! The contour line method is the "covariant" representation, and the arrow method is the "contravariant" representation.

Cloudswrest 19:40, 24 January 2007 (UTC)

Category:Introductory physics?

This article is contained in Category:Introductory physics, which according to itself includes "topics in physics that are commonly taught in middle school or high school, or may be in the curriculum for college freshman." But do college freshmen (not to say middle and high schoolers) learn, or even hear about tensors? --Acepectif 12:04, 11 June 2007 (UTC)

I would say it definately does not belong there. As a current math and physics major in college, I only came across this later on and in research. Tensors were never mentioned when I took AP Physics. Nor where they mentioned in either my 200 level linear algebra class nor introductory physics class freshman year. I have only come across them in upper level (i.e. 300 level) work and research work. By all means, I support removing it from Category:Introductory physics and moving it to a list of more advanced topics --Aohara 2:35, 28 August 2007 (EST)

Link to thread discussing tensors is broken

I tried to follow the link to that thread, and it isn't going to the right place. I don't want to remove the link because the discussion on that thread might be valuable. Maybe somebody can fix it. By the way, I LOVE how this article points out that mathematicians and physicists use the term "tensor" in different ways. That's something that had confused me for a long time. —Preceding unsigned comment added by Singularitarian (talkcontribs) 09:10, 27 June 2007

As the above mentioned broken link:

is still broken I have removed it. Troelspedersen 10:53, 5 November 2007 (UTC)

I removed the comment that the acceleration is not in the same direction as the force. Note that F= ma where F and a are vecotrs. M is a scalar. Thus, F has the same direction as a. The statment is incorrect. I removed it.Mangogirl2 02:25, 15 September 2007 (UTC)

Tensor rank

There exists a very short article - Rank of a tensor - which I believe should be merged into the rank section of tensor. Comments ? Thanks. MP (talkcontribs) 07:46, 3 November 2007 (UTC)

I say merge. Kevin Baastalk 12:26, 3 November 2007 (UTC)
Actually, I believe the Rank of a tensor article states a prescription not supported by practice. It is true that tensor rank and matrix rank mean different things, and that a rank two tensor can be represented as a matrix, but that's not a true conflict. I would delete the article, not merge it. --KSmrqT 16:29, 3 November 2007 (UTC)
I'd agree w/deletion, too. Kevin Baastalk 23:17, 3 November 2007 (UTC)
support deletion--kiddo 01:12, 4 November 2007 (UTC)
In line with the above discussion, I've flagged Rank of a tensor for deletion. Rick Norwood (talk) 14:25, 12 February 2008 (UTC)

Two usages

Do we really need to make such a big fuzz about the "two usages" of the word tensor? I mean, one is a tensor and the other is a tensor valued function. Nothing difficult about that! (Unless I missed something…) —Bromskloss (talk) 19:34, 14 December 2007 (UTC)

I agree. I'll attempt a rewrite unless someone else objects or steps forward to undertake the task themselves. Rick Norwood (talk) 14:27, 12 February 2008 (UTC)
It does deserve some mention, though. For instance, a "scalar" to a physicist is a scalar-valued function to a mathematician. Likewise, a "tensor" to a physicist actually means a "tensor field." It is an important, if trivial, distinction. Silly rabbit (talk) 16:09, 12 February 2008 (UTC)
Wouldn't that be the same for real numbers and just about everything else as well, then? —Bromskloss (talk) 13:39, 8 April 2008 (UTC)

I like very much the article's clarification of the two usages of the word tensor. I was very confused by that point in the past, because it seemed to me that mathematicians and physicists were using the word differently, and I didn't get it. This Wikipedia article has been very enlightening for me for that reason.--Singularitarian (talk) 23:15, 24 June 2008 (UTC)

This article is a joke

I'm sorry, I really am, but I have to be blunt. There is no such thing as a tensor. Please stop referring to everything from scalars to vectors to matrices as such. Tensor means ONE thing: the tensor algebra and its product. I suppose it's not wikipedia's fault that there are 17 articles that do little except to make it abundantly clear that very few physicists actually know math. —Preceding unsigned comment added by (talk) 00:15, 15 February 2008 (UTC)

That's why there are articles on the tensor product and the tensor algebra. Silly rabbit (talk) 00:31, 15 February 2008 (UTC)

The anonymous user is simply mistaken. Of course there are things called tensors. They are the elements of the tensor algebra. And they are, in the case of a real base field, the same things that physicists have been calling tensors for nearly a century. Perturbationist (talk) 02:22, 21 February 2008 (UTC)


This article is about a mathematical object, yet there no precise definition of a what tensor is. I came here in the hope of finding out exactly what a tensor is and was disappointed. —Preceding unsigned comment added by (talk) 22:50, 10 March 2008 (UTC)

A modern abstract treatment of tensors is given in the article Tensor (intrinsic definition). You may also find the article Intermediate treatment of tensors useful. These articles are mentioned in a disambiguation box at the beginning of this article. There is also a closely related notion of tensor field. Michael Slone (talk) 01:43, 11 March 2008 (UTC)
I find it strange that we have several different articles which treat the same subject on different levels of stringency. One would think the important parts could be made to fit in a single article. —Bromskloss (talk) 13:44, 11 March 2008 (UTC)

Reading all this blabber, it seems to me that the only person who knows what a"tensor" is the person who invented it. I studied mathematics through differential and partial differential equations, yet I can not find a way to learn and correctly use "tensors, tensor algebra, tensor analysis" This makes in impossible for me to undertstand Einstein's 1916 Article "General Relativity" in which he doesn't explain what his "tensors" are, even with his usual vagueness. Having many times rigidly examined Einstein's "On the Electrodynamics of Moving Bodies" and found his analysis dubious, I must say I doubt his ability to understand "tensors".

origin of word tensor

In 'A Brief on Tensor Analysis', James G. Simmonds writes: "The name tensor comes from elasticity theory where in a loaded elactic body the stress tensor acting on a unit vector normal to a plane through a point delivers the tension acting across the plane at the point."

Maybe this could be added in the article. (talk) 15:18, 13 March 2008 (UTC)

Sounds good to me, unless there are any objections. silly rabbit (talk) 15:20, 13 March 2008 (UTC)

Needs a new start

This article is so horribly bad, that it should be deleted and rewritten by a mathematician. —Preceding unsigned comment added by (talk) 10:05, 20 August 2008 (UTC)

Tensors, vectors, and earthquakes.

I am not a mathematician, and I thought my difficulty in reading the article was my fault. I am glad to see more qualified people are unhappy with the presentation. Here is my level of confusion: I don't understand the qualitative difference between a vector and a tensor. I have a suspicion that a tensor is something like a space filled with vectors. I also presume I am wrong. I got into this today because I was looking up Earthquakes and they are described with tensors. But what they call a tensor looks like a vector to me at my level of understanding. What about the description of an earthquake requires a tensor rather that a vector?

JFistere (talk) 19:33, 1 September 2008 (UTC)

a vector is an element of a vector space, where two vectors can be added, which has a distinguished "zero vector", etc. a tensor is something "higher dimensional", for lack of better words. if you perform the tensor product construction on two vector spaces, the elements of the resulting spaces are called tensors.
for example, the tensor product of Rn and Rm is the n × m matrices with real entries. the velocity of a point object is a vector. the Inertia Tensor of a rigid body is, well, a tensor (in fact it's a 3 × 3 matrix); it's the rotational analog of mass.
if you have a surface, then to each point x of the surface is attached the vector space Vx consisting of vectors tangent to the surface at x. for example, if the surface is the sphere and x is the north pole, Vx is then a copy of the horizontal vectors.
for each x, picking a vector from Vx (in a "smooth" way) gives a vector field. in other words, a vector field is a (smooth) assignment of a vector to each point of the surface. similarly, one can speak of tensor fields.
don't know nothing about earthquakes. can't help you there. Mct mht (talk) 01:32, 2 September 2008 (UTC)

Functions from matrices to scalars?

In Spivak, tensors are (at first, anyway) defined as functions from V^k to R. I'm supposing right now that this could be generalized to functions mapping to any field. In any case, this seems like a very easy way to describe what a tensor is, but I haven't seen it in any of the tensor articles. Is there any reason in particular for the absence of such a treatment, or did I miss it? Capefeather (talk) 23:46, 10 January 2009 (UTC)

"There is no royal road to tensors". Charles Matthews (talk) 12:34, 9 May 2009 (UTC)

stop zigzagging

If mathematicians and physicists have different definitions of what tensors are, then they should be embodied in different articles. Instead this article keeps zigzagging at every step between one definition and the other, making it very difficult to follow what is already a difficult subject. I have a BA in math and I would like to know what tensors are; this article does not help. All I can tell is that it is like what computer programmers call a multidimentional array, with the added rule that dimensions can either be "sub" or "super". If there are further constraints on what makes a tensor, I could not follow it. CharlesTheBold (talk) 03:55, 22 April 2009 (UTC)

It's not a different definition, it is a slightly different convention on the use of language: the underlying objects are the same. I have attempted to shape up the article. Inevitably, given that this is really a big subject, details are going to have to be in articles hanging off this one.


I'm taking out the F = ma discussion, since it seems unilluminating and to be making a confused point. Charles Matthews (talk) 12:33, 9 May 2009 (UTC)


it seems that almost nobody is care about discussing. There are a lot of reasons to keep separated and still growing strong this tensors ideas. Anyway i vote keep --kmath (talk) 01:41, 11 May 2009 (UTC)

'Tensor' in physics and math

This article states that in physics, the word 'tensor' is used to mean a tensor field, but that is simply not true. In physics, for example, the radius of gyration quantity as well as the moment of inertia quantity are known to be second-order tensors, these two tensor quantities are never considered as tensor fields.

Most Deadly (talk) 22:44, 31 May 2009 (UTC)

Most Deadly (talk) 22:59, 31 May 2009 (UTC)

Most Deadly (talk) 23:13, 31 May 2009 (UTC)

OK, but the real point is that the "mathematical physics" language of tensors traditionally does not make the distinction clear. Because the component notation doesn't make clear whether the components are simply numbers, or numerical functions of position. No doubt the context usually provides the necessary prompts; but the comment arose from insistence that the "continuum mechanics" version of tensor, which certainly is a tensor field, be considered in the article on the same footing as the "pure mathematical". So one way to remain neutral on what "tensor" means is to make this comment at the start. Charles Matthews (talk) 09:53, 2 June 2009 (UTC)

Tensor Rank

There are links left in the article that point to Tensor#Tensor_Rank, which doesn't exist. Either the links need to be removed, or the section added. (talk) 17:11, 22 September 2009 (UTC)


I'm going to start a draft of the rewrite. Add to this until it has all the necessary information from the original, then someone replace it. LokiClock (talk) 22:31, 28 May 2009 (UTC)

I'm all for a rewrite. I'm moving the draft so it doesn't get conflated with the talk page. —Ben FrantzDale (talk) 20:41, 11 June 2009 (UTC) Talk:Tensor/Rewrite

Good idea. LokiClock (talk) 01:19, 12 June 2009 (UTC)
This project is stagnant. If anyone can pick out the useful bits to salvage and move to the existing article, do so. LokiClock (talk) 11:43, 24 November 2009 (UTC)

December 2009 merger proposals

It has been suggested that as a start to the clean up of the mess that is the tensor articles on wikipedia some of not all of the following articles should be merged into this article.

The above articles are basically content forks from this article. Although it is not unusual to have an "intro to ..." article for complicated subject matter on wikipedia, have four is just evidence of poor cooperation between editors. Personally, I don't think it is necessary at all to have even two articles. Tensors are not that complicated. Problems with these articles seem to trace back to language differences between physicists and mathematicians, but since the article needs to explain what tensors are in plain english and not in physicistian or mathmatician, that should not pose a real problem to writing the article. (In fact having input from the different languages might actually help to get something that is actually plain English. TimothyRias (talk) 15:20, 29 December 2009 (UTC)

  • Support. I would like to get this down to one or two articles as well (refer also to WT:WPM#Tensors on Wikipedia). In addition to a plain English discussion of tensors, that the present article already does to some extent, I don't think there is any obstruction to sketching the definition. I am comfortable with either the mathematicians' or the physicists' way of doing things—preferably we can do both. Sławomir Biały (talk) 15:29, 29 December 2009 (UTC)
  • Support. The forking plays 52-card pick-up with the basic information on and explanation of the subject. A "plain English version" is an excuse for not writing in plain English in the first place. I'm not sure what one should think the non-plain-English version would be, either, because the symbolic definitions don't constitute much more than a stub's worth of content, so you have to wonder what the rest of the page is speaking in if not English. Surely the article needs to explain conventions and terminology of the trades in which the concept is deployed, but those should be given subordinately to the uncolored concept. That is, don't start the article with, "Well it's confusing because mathematicians call one thing tensors, but physicists sometimes call other things tensors, so I'm a little confused about what I'm writing about and where the nurse went, and if you read the whole article you may find out which is which." Start with, "A tensor is such and such. Also there are tensor fields, which some physicists may simply refer to as tensors, when the intention is clear." LokiClock (talk) 20:33, 29 December 2009 (UTC)

Definitions section

Tensors should be defined through various means in the article. There are many such definitions throughout these articles already, so they should be collected in a section.

One thing that comes up with the different treatment articles, the intrinsic definition article, the "as opposed to matrix" section, etc. is that there are many approaches by which to define the concept. To ensure that all these aren't left to compete in a thencluttered intro, there should be a section for them. This isn't just a means of avoiding conflict. These perspectives each provide their own insights that the others will not, and everyone who has some knowledge of education theory, let alone common sense, should know that not everyone can learn the same way. In fact, it's a long tail with any one method, and no matter how good, one perspective on the subject will only make sense to so many people.

I'm not attached to the name or anything. If one would prefer "treatments," "formulations," or whatever, go for it. LokiClock (talk) 22:14, 29 December 2009 (UTC)

Disagree on two grounds. First, so-called definitions occur in the form "a tensor is a type of quantity that transforms in a certain way", and this is not a definition as such, but a criterion for recognition (not answering the question "what is an X?", but the question "how do I recognise whether I have an X?"). This is certainly one bone of contention. Secondly, as I have been arguing on the maths WikiProject talk page, arguments from pedagogy also lead into contentious territory; but for an encyclopedia article the basic problem is not "how is topic T to be taught?", but "what are the facts about topic T that are fundamental and verifiable, and how do we arrange them?". Lack of clarity on the second point has led us astray in the past. It is certainly true that there are different perspectives on how to teach tensors, but that means that any discussion of "how to teach tensors" must be kept neutral, not that there really are different views on tensors and the facts about them. Charles Matthews (talk) 08:56, 30 December 2009 (UTC)
On the first, that's what I mean about not being attached to the name. I picked "Definitions" just because you'd normally put that kind of thing in a "Definition" section. As for not being different views on tensors, I don't mean opinions, I mean ways of deriving the same thing, e.g. Pythagorean theorem#Proofs, which if I read you correctly is what you mean by criteria for recognition. On the second, then how do we go about such a section, without such clarity? LokiClock (talk) 19:46, 30 December 2009 (UTC)
I doubt that "derivations" is the real issue for this article, at least. The major issue for the main article "Tensor" is to convey the idea that there is a single concept of tensor (constructed on a vector space) that is studied in various ways and has various applications. This is actually an issue of "geometry", in my view, not of notation or indeed of algebra. It is still troublesome to express. Charles Matthews (talk) 22:02, 31 December 2009 (UTC)
I feel like this discussion has veered from its original course. I believe that the article can easily accommodate a section in which the notion of a tensor is made more precise by introducing the covariant transformation law a little more explicitly. This addresses to some extent the issue of how one "recognizes" a tensor, and indeed is how most physicists appear to think about them. There is also room to accommodate the definition of a tensor in terms of tensor products, although perhaps that should be further on in the article, not in a consolidated "Definitions" (or whatever) section. The particulars are, as you say, troublesome to express. Sławomir Biały (talk) 22:39, 31 December 2009 (UTC)