Talk:Tensor

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[edit] 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

/Archive1

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).

[edit] 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.

[edit] Generalized Hooke's law

Hi,

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

Cdang 15:38, 26 Nov 2004 (UTC)

[edit] 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...)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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→龜 19:07, 25 October 2005 (UTC)

[edit] rank

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)

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)

[edit] 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)


[edit] 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

    (a_1,a_2) \otimes (b_1,b_2) = (a_1b_1,a_1b_2,a_2b_1,a_2b_2)

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? 128.255.85.4 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
Tijk
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. 128.255.85.111 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)


[edit] 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)

[edit] 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)

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