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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).

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

I agree. My best sense of a tensor is that it describes things that simply can't be described by vectors, but like a vector and a scalar they represent physical quantities. The canonical example is the stress state of a material at a point: you can pull on something in one direction and shear it in a transverse direction. You could try to describe this with several vectors, but the stress state at a point is neatly described by a single tensor. With that tensor, you can figure out how much compression that point is under (a scalar), or how much it is under tension in a particular direction, but a tensor captures all of this. (That sounds a bit circular now that I read it; let me know if that makes any sense.) Tensors are usually represented as a matrix, but don't need to be. Just as complex numbers can be represented as a matrix, tensors are considered mathematical objects in their own right.

That said, I could not give you a definition of what is a tensor and what is just a matrix. That's one thing I'd like to see on this page. —Ben FrantzDale (talk) 13:08, 11 June 2008 (UTC)

[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)

Many years ago I passed a course in tensor math but now I can't remember what tensors are--if I ever really knew. I came here to refresh my memory, but I'm also finding these articles difficult. Let me suggest a way to "light a candle."

I remember in high school I had a difficult time understanding vectors, until I learned about adding velocities. We did it by drawing arrows in different directions tail to point, the drawing the effective resultant vector. A very physical interpretation, with no mention of coordinates. But also easily understood in terms of coordinates, and in terms of independence from the particular coordinate system.

Using that analogy, I'm speculating that tensors are the "next step up" from vectors. If scalars are values that have NO direction; vectors have ONE direction, then tensors have MANY directions, or maybe ALL directions at once. And there seems to be an essential component of anisotropism--different values in different directions--but in such a fashion that the value changes smoothly ("linearly"? in the limit) as you "turn" from one direction to another. I'm imagining something like a cross between a porcupine (arrows in all directions of different lengths but points forming a smooth surface) and a squishy rubber ball or jello cube, such that if you squash down on the top they spread out at the sides, whack-a-mole style. I haven't yet touched on the covariant/contravariant concept, but I suspect that is also a crucial concept.

Of course, I'm just speculating, so I could be totally off base.

In mathematics, it's hard to understand what an object "is" apart from the operations on it. So an introductory article should include at least the most common elementary operations, e.g. the various "products," with corresponding graphical/physical interpretations.

Is it possible to give a graphical illustration of tensor quantities and tensor operations, analogous to the graphical representation of addition of velocity vectors? That might help some of us newbies.

Drj1943 06:10, 4 January 2007 (UTC)

Tensors are geometric objects, so illustrations are possible sometimes. However, most of the interesting examples involve higher rank, or are tensor fields, or both; helpful illustrations could be a real challenge.

The article can be improved, but it will take time for all the contributions to accumulate. Meanwhile, here are a few thoughts, working with the mathematicians' definitions.

Begin with a smooth surface in ordinary 3-dimensional Euclidean space, such as a torus. Fix a point on the surface; tangents to curves passing through there are vectors spraying out in different directions, but all lie in a common plane, a 2D tangent space, T_p. We now consider three tensor fields, all varying smoothly.

Rank 0: To each point p of the surface, assign a scalar value f(p).
Rank 1: To each point p of the surface, assign a vector value from T_p.
Rank 2: To each point p of the surface, assign a metric tensor that maps a pair of tangent vectors to a number.

Formally, the tensors that make up the vector field are linear functions that take a 1-form (a dual vector) and produce a number. --KSmrq^T 11:15, 4 January 2007 (UTC)

I think as an introductory article the best starting point would be a concrete illustration of a rank 2 tensor (perhaps preceeded by an ilustration of a scalar (rank 0) and vector (rank 1). The sort of thing I have in mind would be: this discussion of a stress tensor. Once the concept scalar -> vector - > rank 2 tensor is established then the rest could follow. The article also needs a better lead paragraph, the disinction between a tensor field and a tensor as used in maths is not so fundamental. I'll think a bit more about this. —The preceding unsigned comment was added by NHSavage (talk • contribs) 09:52, 31 March 2007 (UTC).

Oops sorry, I didn't sign then... just thought I'd add another link which I think is useful: Tensor from Wolfram Math World. My idea for the lead paragraph would be something along the lines of: In mathematics, a tensor can ne considered as an extension of the sequence scalar (zero indices), vector (one index), matrix (2 indices) to an arbitrary number of indices. In other words a scalar can be defined as a tensor of rank zero, a vector is a tensor of rank 1 and a metrix is a tensor of rank 2. Just as a vector can be represented by a 1D array of numbers and a matrix by a 2D array, a tensor of rank 3 can be represented by a three dimensional array of numbers. They are used in physics in general relativity, elasticity and fluid mechanics. Just a starting point but more informative than starting with the difference between a tensor and a tensor field. This may be oversimplfying things but if this is meant to be an introduction then that is probably ok IMHO.

The illustration of the scalar->vector->matrix->hierarchy needs someone good at that sort of thing to participate but I would concieve of a picture of the left and a mathematical represention of the right. So a short line with a number next to it, a 3D vector with a standard vector as a column matrix, and a stress tensor illustration with a matrix next to it.--NHSavage 10:35, 31 March 2007 (UTC)

At this point perhaps the following question will prove of use. What is a 'vector'?

Here are some possible answers:

1) A vector is a list of numbers

2) A vector is a displacement, or to be more specific a geometric quantity possessing both magnitude and direction

3) A vector is an element of a vector space

Next question: what is a 'tensor'? Rmilson 17:48, 31 March 2007 (UTC)

I can't answer the last question, although I think probably any of the 3 definitions could be valid depending on the context and the degree of rigour/perspective you want to use. However, speaking personally, I found the explanation of a tensor as the next step in the progression so to speak to be one which gave me a quick idea of what a tensor is. Surely that is what this article ("a non-technical introduction to the idea of tensors") should be aiming for, rather than a rigourous mathematical definition? --NHSavage 19:44, 31 March 2007 (UTC)

Well, if you take that approach, you might as well just look at array. There is no definition of tensor which is not equivalent to the 'rigorous mathematical definition'. (Well, qualify that for 'tensor density'.) We have had this discussion for three years, and I don't think we've been offered an expository option that is at all 'non-technical'. Charles Matthews 19:58, 31 March 2007 (UTC)

Fine, if that is correct IMHO there should not be an article which pretends to be "a non-technical introduction to the idea of tensors". On the other hand it is the starting of the Wolfram Math World article and provides a more comprehensible starting point to someone who has never come across them before. I am sure that even an introductory article has to go a lot further than this but if you can't offer a non-technical article you migth as well delete this one. I am not a mathematician, just someone who has recent had to become aquaited with tensors. (which is why I decided not to follow the usual policy of being bold).--NHSavage 19:18, 2 April 2007 (UTC)

So the lead section has had an incorrect definition introduced. It is not true that mathematical tensors are functions (Halmos strikes again, it seems). It is also misleading to say that mathematicians and physicists are talking about different things (beyond the valid point about tensor fields). Charles Matthews 20:02, 31 March 2007 (UTC)

[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

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

T_ijk

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] When is a matrix not a tensor

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

$\begin{bmatrix} \cos \theta & -\sin\theta \\ \sin\theta & \cos\theta\end{bmatrix}$
$\begin{bmatrix} \partial y_1/\partial x_1 & \partial y_2/\partial x_2 \\ \partial y_1/\partial x_2 & \partial y_2/\partial x_2 \end{bmatrix}$

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

a matrix is a mixed tensor of type (1,1). not quite sure why the divergence in terminology. i would guess it is called a tensor when it represents some physical quantity that is invariant under change of basis, a geometric object. if it is viewed as a linear map, it is called a matrix. this distinction is, of course, very artificial. maybe other folks can shed some light on this. Mct mht 00:06, 7 May 2007 (UTC)

Hua. So a matrix, M, would be $M_i^j$ in indicial notation? —Ben FrantzDale 00:40, 7 May 2007 (UTC)

Yep. A matrix is a like a number, in that it can "mean" many different things, just as a number can be a temperature or pressure or distance. The most common thing a square matrix means (represents) is a linear transformation, or linear operator, mapping vectors to vectors. This is so common that some folks think only in terms of matrices, where a mathematician would not. Now consider the thing we call an inertia tensor; we can arrange the nine moments as a symmetric matrix, but it might be confusing to try to identify that matrix with a linear transformation. If we can write a tensor using only two indices, we can put its components in a matrix; however, a common tensor like that for Riemannian curvature has too many indices to fit into a matrix. --KSmrq^T 00:55, 7 May 2007 (UTC)

It appears one part of the answer is, given just a matrix, you can't know for sure because being a tensor rather than a matrix is a transformation property, and a matrix doesn't have that idea. In other words, tensors are commonly implemented in terms of matrices.

As for the examples I gave, at least the rotation matrix is not a tensor. Consider ninety-degree rotation. That matrix looks like [[0 -1][1 0]] in good old Euclidean coordinates. But if we want a ninety-degree rotation with the basis {(2,0), (0,1)}, it'll look like [[0 -1/2][2 0]].

Similarly, if we have the same rotation and we want to use the basis {(0,1),(-1,0)}, the rotation matrix remains unchanged. If I have a tensor [[0 -1][1 0]] in the usual basis, then in {(0,1),(-1,0)}, I think it winds up looking like the identity(?). If that's about right, then I understand. —Ben FrantzDale 03:15, 7 May 2007 (UTC)

Not exactly.

Suppose we are given a fixed finite-dimensional real vector space, V, with standard inner product; R³ will do fine. We know what it means for a real-valued function f on V to be linear: f preserves sums and scales. We can collect all such functions (or "forms") into another real vector space, V^∗, which has the same dimension as V. In fact, given an ordered basis (e₁,…,e_n) for V, we can choose an ordered basis (ω₁,…,ω_n) for V^∗ such that basis function ω_i applied to basis vector e_j yields 1 if i = j and 0 otherwise. We call V^∗ the dual of V. In some literature, especially outside mathematics, the distinction between these two spaces is blurred, but we want to be more careful.

Formally, a tensor is a real-valued function that takes a mixed list of vectors and dual vectors as arguments, and is a linear function of each. Our first two favorite examples are the dot product and the determinant (considered as a function of column vectors).

$\begin{align} \operatorname{dot} \colon V \times V &\to \R \\ (\bold{v}_1,\bold{v}_2) &\mapsto x_1 x_2 + y_1 y_2 + z_1 z_2 , \\ \det \colon V \times V \times V &\to \R \\ (\bold{v}_1,\bold{v}_2,\bold{v}_3) &\mapsto \begin{vmatrix} x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3 \\ z_1 & z_2 & z_3 \end{vmatrix} . \end{align}$

Our next favorite example is the obvious application of the dual.

$\begin{align} F \colon V^{\ast} \times V &\to \R \\ (\omega,\bold{v}) &\mapsto \omega(\bold{v}) . \end{align}$

Obviously, a linear operator, g:V→V, does not fit the pattern of a tensor; but we can use the dual pairing to get around that. Think of dual vectors as row vectors (using the dual basis); then we can write F as a 3×3 matrix — here an identity matrix. Thus every linear operator can be written a square matrix, and also can be interpreted as a tensor of mixed rank.

So, although a rotation is not a tensor, we have a convenient way to treat it as one.

$\begin{align} R \colon V &\to V, \qquad V = \R^3 \\ \begin{bmatrix}x\\y\\z\end{bmatrix} &\mapsto \begin{bmatrix}-y\\x\\z\end{bmatrix} \\ R &\cong \begin{bmatrix}0&-1&0\\1&0&0\\0&0&1\end{bmatrix} \\ T \colon V^{\ast} \times V &\to \R \\ (\begin{bmatrix}a&b&c\end{bmatrix},\begin{bmatrix}x\\y\\z\end{bmatrix}) &\mapsto \begin{bmatrix}a&b&c\end{bmatrix} \begin{bmatrix}0&-1&0\\1&0&0\\0&0&1\end{bmatrix} \begin{bmatrix}x\\y\\z\end{bmatrix} \end{align}$

Using Einstein summation notation, this mutation is so natural and subtle it could go unremarked. Similarly, we can easily turn the dot product into a matrix, or raise and lower indices at will using the dual pairing. But we can never convert a rank three tensor like the determinant into a matrix; the former requires three indices, whereas the latter provides only two. --KSmrq^T 21:21, 7 May 2007 (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. --KSmrq^T 00:21, 12 November 2006 (UTC)

[edit] Type

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

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

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

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

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

Cloudswrest 19:40, 24 January 2007 (UTC)

[edit] Category:Introductory physics?

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

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

[edit] Link to thread discussing tensors is broken

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

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

[edit] Tensor rank

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

I say merge. Kevin Baas^talk 12:26, 3 November 2007 (UTC)

Actually, I believe the Rank of a tensor article states a prescription not supported by practice. It is true that tensor rank and matrix rank mean different things, and that a rank two tensor can be represented as a matrix, but that's not a true conflict. I would delete the article, not merge it. --KSmrq^T 16:29, 3 November 2007 (UTC)

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

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

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

[edit] Two usages

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

I agree. I'll attempt a rewrite unless someone else objects or steps forward to undertake the task themselves. Rick Norwood (talk) 14:27, 12 February 2008 (UTC)

It does deserve some mention, though. For instance, a "scalar" to a physicist is a scalar-valued function to a mathematician. Likewise, a "tensor" to a physicist actually means a "tensor field." It is an important, if trivial, distinction. Silly rabbit (talk) 16:09, 12 February 2008 (UTC)

Wouldn't that be the same for real numbers and just about everything else as well, then? —Bromskloss (talk) 13:39, 8 April 2008 (UTC)

[edit] This article is a joke

I'm sorry, I really am, but I have to be blunt. There is no such thing as a tensor. Please stop referring to everything from scalars to vectors to matrices as such. Tensor means ONE thing: the tensor algebra and its product. I suppose it's not wikipedia's fault that there are 17 articles that do little except to make it abundantly clear that very few physicists actually know math. —Preceding unsigned comment added by 140.247.242.90 (talk) 00:15, 15 February 2008 (UTC)

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

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

[edit] Definition

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)

A modern abstract treatment of tensors is given in the article Tensor (intrinsic definition). You may also find the article Intermediate treatment of tensors useful. These articles are mentioned in a disambiguation box at the beginning of this article. There is also a closely related notion of tensor field. Michael Slone (talk) 01:43, 11 March 2008 (UTC)

I find it strange that we have several different articles which treat the same subject on different levels of stringency. One would think the important parts could be made to fit in a single article. —Bromskloss (talk) 13:44, 11 March 2008 (UTC)

[edit] origin of word tensor

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

Maybe this could be added in the article. 131.111.55.94 (talk) 15:18, 13 March 2008 (UTC)

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