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Talk:Tensor

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.

⊗ 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.
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): http://www.mathcircle.org/cgi-bin/mathwiki.pl? --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: http://groups.google.com/groups?hl=en&threadm=D73us0.GsA%40Corp.Megatest.COM --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 (x^k'_k) is the inverse of (xⁱ_i'). 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:

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 vector and matrix and allow to write equations independently of the coordinate system.
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:
Start by explaining that a tensor is a generalization of the concept of vector and matrices.
Then explain that tensors allow to express physical laws in a form that apply to any coordinate systems.
Say that tensors are heavily used in Continuum mechanics and Theory of relativity (beacause of the previous point)
Introduce the two species: contravariant/covariant, introduce notation and ranks (~= number of indices).
Define the contravariant/covariant component by showing how they transform under a change of coodinate system.
Special cases:
tensors of rank(0/0) => scalars,
rank(1/0) => vectors in differential geometry or contravariant vectors in tensor analysis,
rank(0/1) => one-forms in differential geometry or covariant vectors in tensor analysis.
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.

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:

the above discussion
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 /wiki/Wikipedia:Words_that_should_not_be_used_in_wikipedia_articles 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' vectors 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

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