Talk:Tensor/Archive 1

message: WHy do we have:
* Tensor
* Tensor/Old
* Tensor-classical?
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 [[
 space|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.
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:
# 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 http://www.wikipedia.org/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' 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)
thread_title:  PLethora of pages? 

message: 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)
thread_title:  Disambig needed 

message: Hi,
I have a question here about the tensor notation, if anyone can help me... Thanx
Cdang 15:38, 26 Nov 2004 (UTC)
thread_title:  Generalized Hooke's law 

message: 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)
replies: text: 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)
thread_title:  Abstract index notation? 

message: 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)
thread_title:  discussion at [[Wikipedia talk:WikiProject Mathematics/related articles]] 

message: 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)
thread_title:  Minor correction 

message: 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)
replies: text: 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→龜 19:07, 25 October 2005 (UTC)
thread_title:  how do you pronounce? 

message: 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. 
http://www.cs.cornell.edu/cv/OtherPdf/SevenSpr.pdf .  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 ( http://mathworld.wolfram.com/Tensor.html ), 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 
http://www.nada.kth.se/~johanh/tensorrank.ps .
71.240.24.135 04:26, 3 January 2006 (UTC)
replies: text: 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)
thread_title:  rank 

message: 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)
thread_title:  Rewritten article 

message: 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)
thread_title:  Babangida? Arnold? 'gnem on a poominge'??! 

message: 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
<P>
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.
<P>
In other words, the tensor of a quaternion is simply its
modulus.
<P>
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).
<P>
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.
<P>
(Information for this article was provided by David Wilkins and Julio
González Cabillón.)
<P>
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).
<P>
The term TENSOR ANALYSIS was introduced by Albert Einstein in
1916 (Kline, page 1123).
<P>
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)
thread_title:  Origin of word "tensor" 

message: 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)
replies: text: 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)
thread_title:  Type 

message: 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)
thread_title:  Category:Introductory physics? 

message: 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 (talk • contribs) 09:10, 27 June 2007
As the above mentioned broken link:
* A thread discussing basic and in depth definitions as well as various examples
is still broken I have removed it. Troelspedersen 10:53, 5 November 2007 (UTC)
thread_title:  Link to thread discussing tensors is broken 

message: 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 (talk•contribs) 07:46, 3 November 2007 (UTC)
replies: text: I say merge. Kevin Baastalk 12:26, 3 November 2007 (UTC)

text: 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)

text: I'd agree w/deletion, too. Kevin Baastalk 23:17, 3 November 2007 (UTC)

text: support deletion--kiddo 01:12, 4 November 2007 (UTC)

text: In line with the above discussion, I've flagged Rank of a tensor for deletion. Rick Norwood (talk) 14:25, 12 February 2008 (UTC)
thread_title:  Tensor rank 

message: 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)
replies: text: 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)
replies: text: 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)
replies: text:  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)
thread_title:  Two usages 

message: 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 140.247.242.90 (talk) 00:15, 15 February 2008 (UTC)
replies: text: That's why there are articles on the tensor product and the tensor algebra.  Silly rabbit (talk) 00:31, 15 February 2008 (UTC)
thread_title:  This article is a joke 

message: 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 92.104.106.158 (talk) 22:50, 10 March 2008 (UTC)
replies: text: 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)
replies: text:  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)
thread_title:  Definition 

message: 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.
131.111.55.94 (talk) 15:18, 13 March 2008 (UTC)
replies: text: Sounds good to me, unless there are any objections. silly rabbit (talk) 15:20, 13 March 2008 (UTC)
thread_title:  origin of word tensor 

message: This article is so horribly bad, that it should be deleted and rewritten by a mathematician.  —Preceding unsigned comment added by 213.39.159.220 (talk) 10:05, 20 August 2008 (UTC)
thread_title:  Needs a new start 

message: 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)
replies: text: "There is no royal road to tensors". Charles Matthews (talk) 12:34, 9 May 2009 (UTC)
thread_title:  Functions from matrices to scalars? 

message: 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)
thread_title:  Deletion 

message: 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)
thread_title:  Merging? 

message: 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)
replies: text: 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)
thread_title:  'Tensor' in physics and math 

message: 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. 129.120.60.81 (talk) 17:11, 22 September 2009 (UTC)
thread_title:  Tensor Rank 

message: 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)
replies: text: Good idea. LokiClock (talk) 01:19, 12 June 2009 (UTC)
replies: text: 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)
thread_title:  [[Talk:Tensor/Rewrite|Rewrite]] 

message: 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.
*Tensor (intrinsic definition)
*Classical treatment of tensors
*Intermediate treatment of tensors
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)
thread_title:  December 2009 merger proposals 

message: 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)
replies: text: 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)
replies: text: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Definitions section