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Spinor

Currently the article contains a section on spinors as generalisations of tensors. This seems a bit odd. I'm not sure in what sense spinors are supposed to generalize tensors. A spinor can of course be thought of as a generalisation of a vector. The generalization of a tensor would be a half-integer object, i.e. the tensor product of a tensor with a spinor. Would it be OK, if we just drop this section?TR 14:22, 28 April 2011 (UTC)

I don't know, and I would love to know. Here is some vague speculation, that perhaps you can clear up. One can build up tensors from vectors, in that the vectors form a vector space V and the (p, q)-tensors on V are elements of $T_{q}^{p}:=V\otimes \cdots \otimes V\otimes V^{*}\otimes \cdots \otimes V^{*}$ (p and q times, respectively). If only rotational coordinate changes are desired, then we consider the action of a special orthogonal group on V. If general linear coordinate changes are desired, then we consider the action of a general linear group. That is, we have some Lie group, and V is a representation of it, and this representation extends to $T_{q}^{p}$. So if the Lie group happens to be a spin group, instead of a special orthogonal group, then you consider representations V of the spin group, and these extend to $T_{q}^{p}$ and are called spinors? Honestly, does this make any sense? Mgnbar (talk) 13:32, 29 April 2011 (UTC)

Well, you can consider it like this (I think). An ordinary vector bundle, V, can be thought of as carrying a (spin) 1 representation of SO(n). A (p, q)-tensor field on V is an section of $T_{q}^{p}:=V\otimes \cdots \otimes V\otimes V^{*}\otimes \cdots \otimes V^{*}$ (p and q times, respectively), therefore carries $1\otimes \cdots \otimes 1\otimes {\bar {1}}\otimes \cdots \otimes {\bar {1}}$ representation of SO(n), which may be viewed as a direct sum of integer spin representations. If the base space allows a spin structure, these representation may be lifted to integer spin representations of Spin(n). You can also introduce a vector bundle S, which carries a spin-1/2 representation of Spin(n). The sections of that bundle are commonly known as spinors. Obviously, you can consider arbitrary tensor product of S and V, which will decompose in various spin representations. For example, V⊗S has irreducible component with spin 3/2. But I'm not sure these are ever called spinors, that term seems to be reserved for the actual spin-1/2 fields.

(The above is written in the context of vector and tensor fields, since that is more familiar, but I think it more or less translates to the context of vector spaces as well.

I'm not sure that this really constitutes a generalization of the concept of a tensor. It seems more like the generalisation of a certain specialisation of the concept of tensor. It seems somewhat far fetched to discuss them in this article.TR 14:54, 29 April 2011 (UTC)

Thanks for your response. When I get the time and energy I will try to absorb it. Cheers. Mgnbar (talk) 17:10, 29 April 2011 (UTC)

In some areas, fields belonging to these tensor product bundles are indeed called spinors. For instance, in relativity theory the "Weyl spinor" is an element of the symmetric tensor product of four copies of an irreducible spin representation of the Lorentz group. As a module for the spin group, this is isomorphic to a certain irreducible tensor representation of the Lorentz group (the Weyl factor appearing in the Ricci decomposition), but the isomorphism is somewhat awkward, and the two objects are generally handled in completely different ways. Sławomir Biały (talk) 20:14, 29 April 2011 (UTC)

Basis and type lost

This webpage states, "When converting a tensor into a matrix one loses two pieces of information: the type of the tensor and the basis in which it was expanded." If this is true in general, and not just of TCC, it should be part of the section on multi-dimensional arrays because it clarifies the difference between the two. ᛭ LokiClock (talk) 05:13, 15 May 2011 (UTC)

I would never have said it that way, but it's a legitimate point that might help some readers, so I'm fine with your adding that. Mgnbar (talk) 14:01, 15 May 2011 (UTC)

Hmm, link doesn't work for me right now. However, the thing is; In the multi-dimensional array approach a tensor is a multi-dimension array [i]satisfying a certain transformation law[/i]. The information about the type and the basis in which the tensor is expressed is contained in this last bit. That is, the data for a tensor (in this approach) is: a) a multi-dimensional array b)a transformation law. Without b) it is just a multidimensional array.TR 17:08, 15 May 2011 (UTC)

Right, whereas a matrix is just the multi-dimensional array, without the transformation law. So in converting a tensor to a matrix, you lose the transformation law or the type and basis, however you want to encode that extra information. I think that you two agree. Mgnbar (talk) 20:18, 15 May 2011 (UTC)

But don't we already adequately say that a tensor is a multidimensional array with a transformation law? Sławomir Biały (talk) 11:09, 16 May 2011 (UTC)

Not quite, because it isn't stated what information is contained in the transformation law. Though the transformation law is given importance, the language doesn't indicate that the basis and type are separate information from the matrix, and that the transformation law is what contains that information. As a definition, the information should be given the name "transformation law" right next to the fact that an array doesn't have the information by that name. ⁂ It almost says it here: Taking a coordinate basis or frame of reference and applying the tensor to it results in an organized multidimensional array representing the tensor in that basis, or as it looks from that frame of reference. The coordinate-independence of a tensor then takes the form of a "covariant" transformation law that relates the array computed in one coordinate system to that computed in another one. ⁂ My experience, for debugging purposes: I assumed that a transformation matrix would have its type and basis converted before being applied. I figured that the transformation law was a description of how to alter a transformation matrix to properly operate on another matrix, and saw that the basis and type would be the description required to make such a conversion. I thought basis and type converters were other matrices, so I was prevented from concluding that the transformation laws of the two tensors replaced the converter operator I was expecting. ᛭ LokiClock (talk) 13:44, 24 May 2011 (UTC)

The lead doesn't have a precise definition, and doesn't even discuss tensor type or bases: it's aimed at a much more basic level than that. If you're looking for a definition, you should look in the section prominently labelled "Definition". In that section, tensor type is defined and characterized in terms of the transformation law. Sławomir Biały (talk) 20:28, 24 May 2011 (UTC)

I wasn't talking about adding it to the lede, just showing that the lede almost says what needs to be said. Search for "transformation law" in the article, and you'll see that nowhere comes closer to explicitly stating what structure the transformation law introduces to the array. Instead, "As multidimensional arrays" concludes with, "The definition of a tensor as a multidimensional array satisfying a 'transformation law' traces back to the work of Ricci," after the specifications for the notation and without having previously indicated how what was being talked about is a description of "a multidimensional array satisfying a 'transformation law.'" I have read the article many times, and just before my last reply I payed particular attention to the section on multidimensional arrays, where this information is relevant. ᛭ LokiClock (talk) 17:03, 25 May 2011 (UTC)

At the very end of Tensor#As multidimensional arrays, it says explicitly what the transformation law is, and how this is related to the tensor valence. Sławomir Biały (talk) 17:28, 25 May 2011 (UTC)

Near the end, it says, "The "transformation law" for a rank m tensor with n contravariant indices and m-n covariant indices is thus given as..." The "thus" slips by the fact that you've just established a definition - it doesn't indicate that this location is that of the transformation law's definition, not a previously established object which is useful to derive but not necessary to understand the concept, like when demonstrating the discriminant of a polynomial form. Besides, that paragraph only describes properties of tensors, it doesn't contrast it with the array. There needs to be a constant alternation between the two if you hope to use the array analogy as a self-contained definition. ᛭ LokiClock (talk) 17:50, 25 May 2011 (UTC)

There's an entire paragraph explaining that there are two different kinds of indices and that these transform differently (one by R and the other by the inverse of R). Then it gives the explicit transformation law for a general tensor. I fail to see what is not clear about this. Sławomir Biały (talk) 18:00, 25 May 2011 (UTC)

Those statements are clear on their own, yes. They clearly describe properties of tensors, and the transformation law. The transformation law and those indices are not clearly contrasted with simple matrices. It gives the impression that the difference between a tensor and a matrix should have been clear by the time you start talking about the properties of tensors, and that from then on you should be considering properties of tensors that can be derived from the facts of this relationship. ᛭ LokiClock (talk) 18:05, 25 May 2011 (UTC)

I've added words to the effect that computing the array requires a basis, and attempted a formal definition emphasizing the basis-dependence. Sławomir Biały (talk) 18:32, 25 May 2011 (UTC)

Okay, I tried to add what I'm getting at. ᛭ LokiClock (talk) 20:53, 25 May 2011 (UTC)

Notation used in "array" section

Three comments about this "As multidimensional array..." subsection. First, in the definition, can we have the change-of-basis matrix R always on the left of vectors, rather than on the right, for consistency? Second, can we switch the notation (n, m - n) to the simpler notation (n, m)? Third, I feel that the Einstein summation convention (ESC) is a helpful time-saver for experts, but an unnecessary hindrance to beginners. Many people who encounter math have trouble getting past to the notation to the content. I've spoken with students who were under the mistaken impression that the ESC is somehow an essential aspect of tensors. So I vote to avoid ESC wherever possible. Mgnbar (talk) 12:55, 26 May 2011 (UTC)

On the last part. You will need to employ some sort of notation for the sums over indices. The Sigma notation for sums is also a notation convention, and I don't see it as helpful for most readers too use a more cumbersome notational convention. Put differently, how exactly were you planning on writting something like:

${\hat {T}}_{i_{n+1},\ldots ,i_{m}}^{i_{1},\ldots ,i_{n}}=(R^{-1})_{j_{1}}^{i_{1}}\cdots (R^{-1})_{j_{n}}^{i_{n}}R_{i_{n+1}}^{j_{n+1}}\cdots R_{i_{m}}^{j_{m}}T_{j_{n+1},\ldots ,j_{m}}^{j_{1},\ldots ,j_{n}}.$

in another notion? TR 13:07, 26 May 2011 (UTC)

The change of basis should be on the right. A basis is naturally identified with an invertible map from R^N to a vector space. Elements of GL(N) compose on the right with these. Put another way, if you arrange the elements of the basis into a row vector f=(X₁,...,X_N), then multiplication is naturally on the right f R. Sławomir Biały (talk) 13:28, 26 May 2011 (UTC)

Note that since the elements of the basis are indeed vectors, it is conventional to have linear transformations of them act on the left. This convention is used in the rest of the section. A bigger, issue at the moment though is that this "formal definition" cannot be sourced, and hence should not appear at all, IMHO.TR 13:58, 26 May 2011 (UTC)

A basis is not a vector. A change of basis is a numerical matrix (in GL(N)) that acts on the right. A physicist would call this a passive transformation. If you want to act on the vectors in the basis, then you can act on the left with an element of GL(V). A physicist would call this an active transformation; this is not what is meant by "change of basis". The definition should be easy to source. Off the top of my head I can recomment, for instance, Kobayashi and Nomizu or Richard Sharpe's "Differential geometry". I'll try to add references at some point. I don't see that the definition is in any way controversial (except possibly that I need to convince folks that the matrix multiplication really is in the correct order). Sławomir Biały (talk) 14:13, 26 May 2011 (UTC)

You are right. However, only very few of the readers of this article will be comfortable with things acting on the right. Note that for the part where R is written in components the actual order does not matter. Only a few lines above, the R components and basis vectors appear in the opposite order. (On a related note, it would be more consistent to use e_i for the basis vectors as in the text.) Notation wise this should all be made more consistent.TR 14:36, 26 May 2011 (UTC)

About the type (n,m-n), this is a lot more convenient in labelling the indices in the transformation law. If you say the type is (n,m) then the labels in the transformation law become even more complicated.TR 14:57, 26 May 2011 (UTC)

Most of these comments come down to how basic we're willing to get, to help the reader actually learn the thing. Wikipedia is not a textbook, but surely we can put in a little more detail, to make the notation less daunting.

Regarding ESC: I would put in one or two more examples of a change-of-basis formula, such as the one for (1, 1)-tensors, with the summation signs intact. For the fully general change-of-basis formula, I would put

${\hat {T}}_{j_{1}\ldots j_{q}}^{i_{1}\ldots i_{p}}=\sum _{k_{1}=1}^{N}\cdots \sum _{k_{p}=1}^{N}\sum _{\ell _{1}=1}^{N}\cdots \sum _{\ell _{q}=1}^{N}Q_{k_{1}}^{i_{1}}\cdots Q_{k_{p}}^{i_{p}}T_{\ell _{1}\ldots \ell _{q}}^{k_{1}\ldots k_{p}}S_{j_{1}}^{\ell _{1}}\cdots S_{j_{q}}^{\ell _{q}}.$

(for appropriate $Q=S^{-1}$). Yes, this is a monstrosity, but that's the point. It takes a lot of work to change the basis, and this formula shows that. Also, the summation signs are tedious; this motivates the ESC. I would show the ESC version of this formula immediately after it:

${\hat {T}}_{j_{1}\ldots j_{q}}^{i_{1}\ldots i_{p}}=Q_{k_{1}}^{i_{1}}\cdots Q_{k_{p}}^{i_{p}}T_{\ell _{1}\ldots \ell _{q}}^{k_{1}\ldots k_{p}}S_{j_{1}}^{\ell _{1}}\cdots S_{j_{q}}^{\ell _{q}}.$

Regarding the left/right action of the change of basis matrix: I understand Slawomir's math, but I disagree with it as pedagogy. At least on the first pass, it's easier for the uninitiated reader to see something like

$\{v_{1},\ldots v_{N}\}\mapsto \{Rv_{1},\ldots ,Rv_{N}\},$

which shows each vector getting transformed as you would expect. Mgnbar (talk) 15:18, 26 May 2011 (UTC)

Yes, but the change-of-basis transformation is not of the form $(Rv_{1},\dots ,Rv_{N})$. That's clearly the sort of thinking that needs to be discouraged. Sławomir Biały (talk) 21:35, 26 May 2011 (UTC)

Regarding type (n, m - n): The transformation law I just posted doesn't seem any more complicated than the one using (n, m - n). Does it seem more complicated to you? Mgnbar (talk) 15:20, 26 May 2011 (UTC)

Sorry, Slawomir, I now realize that you were talking about taking linear sums of vectors, whereas I was working with their coordinate arrays (and sloppily writing R where the article would have R^-1). In other words, you're using a formula like

$\mathbf {\hat {e}} _{i}=\sum _{j}R_{i}^{j}\mathbf {e} _{j}=R_{i}^{j}\mathbf {e} _{j},$

whereas I'm talking about a formula like

${\hat {v}}^{i}=(R^{-1})_{j}^{i}v^{j}$

where v is the coordinate array for any $\mathbf {e} _{i}$. Or do I still misunderstand you? If we're going to do tensors as multidimensional arrays, it makes sense to me to treat the basis vectors in the same way as any other vector. Maybe this is what TR was getting at above. Mgnbar (talk) 23:07, 26 May 2011 (UTC)

Yes, this is what I mean. No, I don't think the basis vectors should be treated like any other vector. One reason has to do with the physical motivation of this approach to tensors. A basis is a system of reference rods in V. The physicist should be free to change his measurement apparatus by reconfiguring the reference rods, but is not free to apply a change to the underlying space V. The difference here is between active and passive transformations. Secondly, one can probably develop a mathematical theory where the change of basis is expressed by the left-multiplication of $R\in GL(V)$ with a basis vector as in $R\mathbf {e} _{i}$. But on a manifold this will only be meaningful locally, and the definition of a tensor (as a global object) becomes much less satisfactory. By contrast, a tensor field on a manifold (in the usual way of defining things—i.e., with the right GL(N) action) is a GL(N)-equivariant map of the frame bundle into a representation of GL(N), and this is naturally a global notion. Sławomir Biały (talk) 01:53, 27 May 2011 (UTC)

Doesn't bra-ket notation operate on the right? The physics audience should in that case be familiar with it. I'm splitting this discussion. ᛭ LokiClock (talk) 05:58, 27 May 2011 (UTC)

I've never heard a mathematician use this "active vs. passive" terminology. If I understand correctly, an active transformation of vectors is a function $f:V\to V$. Any basis B of V induces an isomorphism of V with R^N, through which f can be viewed as a function $f_{B}:\mathbf {R} ^{N}\to \mathbf {R} ^{N}$. In contrast, a passive transformation is any invertible function $g:\mathbf {R} ^{N}\to \mathbf {R} ^{N}$. That is, f operates on vectors, whereas g operates on arrays. Similar remarks hold for higher-order tensors. Do physicists use this terminology because they spend so much time working with f_B that they (or rather novices among them) may confuse a g with an f_B?

Your other comments I don't understand, but experience tells me that I should work harder, because you'll probably end up being right. :) I don't understand how $R\in GL(V)$ --- a function, not an array --- should operate on the right or the left, rather than just operate. A tensor field on a manifold M is a section of a product $TM\otimes \cdots \otimes TM\otimes T^{*}M\otimes \cdots \otimes T^{*}M$ of tangent and cotangent bundles. If you like you can tensor-product with another vector bundle E to make tensors with coefficients in E. The transition maps of the manifold (and E if present) tell you how to change coordinates for tensors. Would you please restate your left/right assertion in this notation? Thank you. Mgnbar (talk) 13:26, 27 May 2011 (UTC)

As I see it, there are two bones of contention here that are different but related: one is the issue of whether we apply functions on the left or right of their arguments, and the other is whether we represent a change of basis by a transformation $R\mathbf {e} _{i}$ where R is an invertible endomorphism of the abstract vector space V, or by a transformation of the form $R_{i}^{j}\mathbf {e} _{j}$ where $R_{i}^{j}$ is an invertible numerical matrix. For the first issue, thinking of $R\in GL(V)$ as a function from V to V, usually we would write $Rv$ for the application of that function to an element of V. That's "operating on the left". Some people use postfix notation for function application and composition, but it's unusual enough that we should avoid it. A basis, on the other hand, is naturally identified with an invertible linear map $\mathbf {f} :\mathbb {R} ^{N}\to V$. So to compose with an endomorphism of V, we should (using the usual convention) write $R\mathbf {f}$, whereas to compose with a numerical matrix, we should write $\mathbf {f} R$. For the other issue, on a manifold when you apply a change of coordinates, you aren't doing anything to the tangent space $T_{p}M$. Rather you are applying a map $\mathbb {R} ^{N}\to \mathbb {R} ^{N}$ and considering the Jacobian of that mapping. This is a numerical matrix, not an endomorphism of the tangent space.

To think of it another way, suppose you insisted on R being an endomorphism of V. The transformation law

$v^{i}[R\mathbf {f} ]=(R^{-1})_{j}^{i}v^{j}[\mathbf {f} ]$

becomes ambiguous. Are the numbers $(R^{-1})_{j}^{i}$ represented in the old coordinates or the new coordinates? One can certainly resolve issues like this, but I hope it's clear that things will become less satisfactory if we try to do so. Sławomir Biały (talk) 14:02, 27 May 2011 (UTC)

I fully agree with you up to here. Mgnbar (talk) 17:57, 27 May 2011 (UTC)

Ok, so here is a full definition of a tensor in the differential geometry setting. I'm not going to use tensor products, because these are a somewhat different approach (that would be discussed in the "definition by tensor products" section of the article). Let PM be the bundle of linear frames of M. The fiber of PM at a point p in M consists of all invertible linear mappings $u:\mathbb {R} ^{N}\to T_{p}M$ (that is, it consists of all frames of the tangent space). There is a natural right action of GL(N) on each fiber, via $R_{g}u=u\circ g.$ Now, PM is the disjoint union of these fibers. A smooth local trivialization of the tangent bundle gives PM the structure of a smooth fiber bundle. Let $(W,\rho :GL(\mathbb {R} ^{N})\to GL(W)$ be a tensor representation of GL(N). That is W is a tensor product of copies of $\mathbb {R} ^{N}$ with its dual space, supporting the usual representation of the general linear group. By definition, a tensor of type ρ on M is a function $\tau :PM\to W$ that is equivariant with respect to the right g action, which is to say that if $g:M\to GL(N)$ is a smooth function, then

$\tau \circ R_{g}=\rho (g^{-1})\circ \tau .$

This definition is standard in differential geometry texts (especially where tensor bundles other than the usual ones obtained by a simple tensor product are important).

For example, when $W=\mathbb {R} ^{N}$ and $\rho =I$ is the identity, one obtains the tangent bundle. When $W=\mathbb {R} ^{N}$ and $\rho (x)=(x^{-1})^{T}$, one obtains the cotangent bundle. Sławomir Biały (talk) 14:26, 27 May 2011 (UTC)

Here's yet another way of thinking about it. What is a section of TM? It's a family of sections in a maximal atlas of a local trivialization U×R^N that are related by a GL(N) cocycle. The important thing here is that the cocycle takes values in GL(N) (numerical matrices). (This doesn't shed much light on the left-vs-right issue though.) Sławomir Biały (talk) 15:13, 27 May 2011 (UTC)

This stuff I need to process, on paper, because it's different from (and beyond) what little I know of this kind of geometry. Thanks for explaining. Mgnbar (talk) 17:57, 27 May 2011 (UTC)

New illustrations

Do any of the new illustrations have any value at all? The one on dyadic tensors seems a little bit offtopic for understanding the article, and it might be more suitable in the dyadic tensor article (if anywhere). I don't think it adds anything helpful or relevant to the Definitions section, so I have removed it.

The other recently-added graphics are more mystifying. The issues are numerous. First, the index conventions are wrong (what is $A_{\rho }\mathbf {e} _{\rho }$ supposed to mean, for instance?) Second, the vector itself should not be changed by a coordinate transformation: in fact, that's the whole point of having a covariant/contravariant transformation law, to ensure that the same vector is described with respect to the two different coordinate systems. Thirdly, there appears to be some confusion here between the basis vectors and the $x^{i}$ coordinates: one of these should be, roughly, "dual" to the other, so it's not correct to have the basis vectors pointing along the "coordinate axes" in this way as far as I can tell. So, given these various issues, it's unclear what meaningful information is intended. Sławomir Biały (talk) 01:35, 27 May 2011 (UTC)

Bad Grammar

I've tried to correct basic English in this article to no good effect, as another editor keeps re-introducing the same grammatical errors. The section "As multidimensional arrays" contains the following sentence:

Just as a scalar is described by a single number and a vector can be described by a list of numbers with respect to a given basis, any tensor can be considered as a multidimensional array of numbers with respect to a basis, which are known as the "scalar components" of the tensor or simply its "components".

Any tensor (singular) can be considered as a multidimensional array of numbers (singular) with respect to a basis, which are known (plural!)... As written, the sentence ties the plural components back to the singular array instead of the plural list of numbers. To tie the scalar components back to the numbers and not the array, introduce a new sentence "These numbers are known as the scalar components..." or else re-write the existing sentence to remove the misprision. Ross Fraser (talk) 02:48, 8 June 2011 (UTC)

Well, there was no grammatical error, since "which" referred to numbers not "array of numbers". The correction you made, resulted in a nonsensical statement, on top of which it introduced a grammatical error in numbers by connecting array and components. None the less your confusion shows that the sentence was somewhat ambiguous. As such, I've corrected it by splitting the sentence in two.TR 07:49, 8 June 2011 (UTC)

Künneth theorem

User:TimothyRias requests citation about how tensors enter into the Künneth theorem. But the Künneth theorem is all about how Cartesian products of topological spaces map to tensor products of graded modules under homology. Right? It couldn't be more about tensors. Am I missing something here? Mgnbar (talk) 12:37, 6 September 2011 (UTC)

1) No matter how clear you think the statement is, it still requires a citation.

2) Tensor product != Tensor. The Kunneth theorem involves tensor products of modules rather than vector spaces, i.e. does not involve tensors in the conventional sense. If you claim that an element of a tensor product of modules is referred to as a "tensor" then that certainly needs a citation.TR 12:48, 6 September 2011 (UTC)

Regarding citation: Would a statement that the Künneth theorem involves tensor products require citation in this article? Or is that obvious to anyone who follows the link to Künneth theorem? In other words, is your request for citation entirely separate from your other issue, that tensor product != tensor?

Regarding the other point: To me, a "vector" is an element of a vector space, and a "tensor" is an element of a tensor space, and there is no reason to restrict to tensor spaces over fields. But I will try to find some time to see whether this terminology is used by people other than me.

I should add that this small disagreement ties into my general dissatisfaction, which I have voiced on other occasions here, about how this article seems to favor the physics point of view over the math point of view --- even going so far as to shunt one of the primary aspects of tensors into a separate article. Mgnbar (talk) 15:02, 6 September 2011 (UTC)

To your first question, yes. (Although the statement that really needs a citation is that tensors play a basic role in algebraic geometry).TR 15:35, 6 September 2011 (UTC)

Okay, now I understand your point better. That tensor products play a fundamental role in algebra topology (not geometry, mind) would be very easy to support with citation. Any introductory algebraic topology book, such as Bredon or Massey, is stuffed with tensor products. Additionally, now that I read the paragraph in dispute more completely, I can see how whoever wrote it (it wasn't me, by the way) attempted to stave off complaints such as yours. It even discusses tensors over rings that aren't fields. So it seems to me that a little citation and clarification is all that's needed. Mgnbar (talk) 16:36, 6 September 2011 (UTC)

Understanding check on tensor type

I have some assertions based on my present understanding of tensors, and if they're wrong, could someone explain why?

• My understanding is that a tensor with all the indices raised is contravariant and with all the indices lowered is covariant, and that the two tensors are dual $T^{ijk}\in V\otimes V\otimes V=(T_{ijk}\in V^{*}\otimes V^{*}\otimes V^{*})^{*}.$ Because a space of one-forms is a vector space, the dual to the all-covariant tensor should be the same as looking at the all-covariant tensor as an all-contravariant tensor in the dual space - each covariant index is contravariant relative to the dual space. So, if you represented each contravariant index by a vector, the covariant version would be that same set of vectors in the dual space. And vice versa - the level set representation of each covariant component is the same after taking the dual tensor and observing it in the dual space.

Going on this I imagine raising and lowering indices as looking at pieces of an same object being pushed through a door between the collective vector and dual spaces, and when all the pieces are to one side or the other it looks the same from that side of the door (the two products of all non-dual spaces). ᛭ LokiClock (talk) 22:50, 19 December 2011 (UTC)

• Since the object itself isn't altered by a change of basis, only its matrix representation, lowering a component $T^{ij}\rightarrow T_{j}^{i}$ and then applying a rotation will show the contravariant component moving against the rotation, and the covariant component moving in advance of the rotation (or is it equal?), but when you raise the index again it will be the same as if the rotation was applied with both indices contravariant $R_{j}^{i}T^{ij}=g^{jj}R_{j}^{i}T_{j}^{i}.$ ᛭ LokiClock (talk) 22:50, 19 December 2011 (UTC)

I've moved this discussion to User_talk:LokiClock#Understanding_check_on_tensor_type. — Quondum^t_c 05:54, 20 December 2011 (UTC)

first sentence

To me, the first sentence of this article is nonsensical: "Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors". So tensors describe relations between tensors? To me that sounds like circular logic and that is not a definition. Wouldn't it be better to say that tensors are objects that transform in certain ways (when going from one coordinate system to another)? I won't correct this because I'm no expert on the subject, but as a reader I find the sentence confusing. — Preceding unsigned comment added by 128.178.54.250 (talk) 14:50, 5 April 2012 (UTC)

I'm not an expert, either, but I believe the statement follows from the duality of vector spaces. The dual space of a vector space, which represents all scalar-valued linear transformations from the vector space, is itself a vector space. Similarly, rank-2 tensors (matrices) represent vector-valued transformations. In other words, tensors represent linear transformations between other tensors. Schomerus (talk) 16:13, 14 June 2012 (UTC)

Actually, the statement is not circular, rather it is inductive: Higher rank tensors describe linear relations between lower rank tensors. Also note that this is not a definition, but a description of what tensors are. (Also, the trouble with saying that "tensors are objects that transform in a certain way" is that is not as much a statement about what tensors are, but about how they are represented.)TR 16:25, 14 June 2012 (UTC)

"The old version, is better suited to the context (multi-dimensional arrays)"

With this edit is being suggested that tensor fields, in the context of an article, have more relevance to as multidimensional arrays than as fields on a manifold? I beg to differ, and suggest that the heading should be raised one level, as the "fieldness" has only incidental relationship with arrays inasmuch as the transform is a tensor as well, which can be expressed, like a tensor, as an array when broken into components. Opinions? — Quondum^☏ 14:11, 13 June 2012 (UTC)

Unfortunatley I can't really comment. Maybe TR wanted the next section tensor fields to follow directly from the previous section on the multidimensional arrays? I would think discussing manifolds would be better suited to the tensor fields section as written by Quondum, but again its better if I stay out of it, it'll end up wrong... F = q(E+v×B) ⇄ ∑_ic_i 14:31, 13 June 2012 (UTC)

Actually, my point was the more that the section on tensor fields currently is a subsection of the section on the definition of tensors in terms multidimensional arrays. As such, it makes most sense to give the definition of tensor fields in that same context (since no other definition of tensors has been discussed up to that point in the article).

If you want to introduce tensor fields in the context of any of the other definitions of tensors, the section on tensor fields needs to moved further down the article. However, I feel that this possible source of confusion (due to the word tensor also being used for tensor fields) needs to be addressed as soon as possible.

As to the actual text of the change you introduced, actually falls right in the uncanny valley of accessibility. It uses enough jargon to be rather inaccessible to a lay reader (engineering undergrad maybe), on the other hand it missed the mathematical rigour to satisfy our more mathematically inclined readers.TR 22:07, 13 June 2012 (UTC)

As I've already said/implied, I think the introduction of the concept of tensor fields does not belong in a subsection of multidimensional arrays. On the matter of accessibility you have a good point. I do not think the mathematically inclined reader needs to be addressed per se: the only point that really needs to be made is that the concept of a tensor field is different from that of a tensor, in the same way that a function is different from a value. And of course a link to tensor field is needed. — Quondum^☏ 07:00, 14 June 2012 (UTC)

To me the main use of having the section is saying: "Hey, there is this related concept called a tensor field, which is sometimes also called a tensor, and may have been what you were looking for. Here is a link to that article." As such, I think it should come as early as possible. This basically means, that it should come as soon as we have explained what a tensor is. The way the article is currently written this means right after of the explanation of tensors in terms of multidimensional arrays satisfying a transformation rule. It therefore makes sense to use the same language to introduce tensor fields (as multidimensional arrays of functions with a modified transformation rule).

This is in fact how many engineering books treat tensor fields (and not unimportantly how Ricci introduced tensor fields!). In that sense, it is also the form that may be most familiar to our readers with the least mathematical sophistication. Conceptually, however, I think you have a good point, that we need to be more explicit that in this case a "function-valued tensor" can also be viewed as a "tensor-valued function".TR 07:55, 14 June 2012 (UTC)

The use you describe is what we use hatnotes for; IMO to have to wade through the bit on arrays and a semi-unintelligible math expression to get there subverts this function. I'm not familiar with the textbook approach, but I'd suggest a textbook can take liberties with someone's patience that an excylcpedia should not. The historical approach is inciidental to this purpose, and should be elaborated only in the main article. I would have been happy to put it later in a section on generalizations or related concepts, but a hatnote works better in some ways. — Quondum^☏ 13:27, 14 June 2012 (UTC)

There already is a hatnote at the top of the article, so that's not really a problem. The current arrangement works for me. Typically when one uses the multidimensional arrays + transformation laws approach to tensors, one is in fact working with tensor fields where the transformation law comes from the coordinate transitions. This fact has to be pointed out, but it isn't our task in this article to give a general definition of tensor fields. Sławomir Biały (talk) 14:12, 14 June 2012 (UTC)

I'll defer to your opinion on this, though I'll make a final suggestion: In the limited context described, should the focus then not be on the components of the transform itself, thus the simple formula

$R_{i}^{j}={{\partial x^{j}} \over {\partial {\hat {x}}^{i}}}$? — Quondum^☏ 15:17, 14 June 2012 (UTC)

(ec) Yes, this is also the function of hatnotes. However, there is the problem that a user confused about tensors (and deciding to look it up on wikipedia) might not know what he is looking for. In particular, he may not understand the difference between tensors and tensor fields. For such a user, a single line hatnote telling him that the term tensor may also refer to a tensor field, is useless, because it won't tell him what the difference is. For this you need to first explain what a tensor is, and then how the concept of tensor field relates to this.TR 15:23, 14 June 2012 (UTC)

Regarding an old reply

I posted a comment three years ago about the apparent lack of mentioning of the multilinear map formulation of tensors. (I'm not sure if it actually was there back then and I simply didn't notice, but in any case there is a subsection on it now so whatever.) I got the reply, "There is no royal road to tensors." I mean, I guess I can understand such a comment from a general point of view, and I am much more familiar with the subject now than I was back then, but it still puzzles me as to how this was meant to be relevant to my past grievance. The multilinear map formulation is a common and useful one, and I merely asked about its inclusion, not about any sense of superiority over other formulations. I know this is kind of off left field, but after checking now whether I did get said reply, it just bothered me and I didn't think this comment would make sense anywhere but here. I've had Wikipedia comments misunderstood before and I just wanted to clear this up, even if it is three years after the fact. Capefeather (talk) 17:04, 1 March 2012 (UTC)

From what I can tell, multilinear maps are objects in the direct product of vector spaces, which form a quotient space of the tensor product. See Tensor product#More than two vector spaces, and mentions of quotients in Tensor product. ᛭ LokiClock (talk) 22:01, 1 March 2012 (UTC)

No. One definition of tensors is in terms of sums of formal products and one is in terms of multilinear maps; they are distinct entities, although there is a canonical isomorphism. Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:40, 11 September 2012 (UTC)

People who are very comfortable with the idea may see no distinction, but I agree that the distinction is important to people just learning the material. Anyway, this discussion is substantially obsolete. The article now has a presentation using multilinear maps. Mgnbar (talk) 13:52, 11 September 2012 (UTC)

dyadic tensor in the table?

Added a couple more links to the table.

Also for completeness, I think we should add dyadic tensor, yet confused where. This is a covariant tensor of the form A_ij so its type is (0,2), but this clashes with bilinear forms and inner products which would cause confusion since dyadics are formed from the dyadic product of two vectors (not collapsed to scalars) - not the same as an inner product.

Even worse is the immense overlap of terminology between "outer product" and "dyadic product" and "tensor product" of vectors when a reader goes on to read about these things, in addition the matrix product comes into the matrix representations of them...

For now I'll not persue this, its fine without. Thanks in advance for any clarification.

F = q(E+v×B)⇄ ∑_ic_i 09:40, 2 July 2012 (UTC)

Not quite at home in the terminology of dyadics, but isn't a Dyadic a tensor a contravariant tensor of rank 2?TR 14:53, 2 July 2012 (UTC)

According to dyadic tensor, its covariant. I'm not too familiar with the terminology either, Dyadics just seem to be matrix representations of rank-2 tensors... =( F = q(E+v×B)⇄ ∑_ic_i 15:33, 2 July 2012 (UTC)

If I understand correctly, a dyadic tensor is a tensor product of two vectors, i.e. (1, 0)-tensors. So it is a (2, 0)-tensor, with two upper indices and zero lower indices. I would look for an error in the usage of the jargon "covariant" and "contravariant", before deciding that the tensor product of two (1, 0)-tensors is somehow a (0, 2)-tensor. But maybe I'm missing something. Mgnbar (talk) 15:41, 2 July 2012 (UTC)

Regardless, I don't think dyadic are in use any more. I've only ever seen them in very old-fashioned books. Sławomir Biały (talk) 15:46, 2 July 2012 (UTC)

Agreed. Let's just leave it out. =) F = q(E+v×B)⇄ ∑_ic_i 15:57, 2 July 2012 (UTC)

Just in case people consider this in the future, I added a hidden note to prevent people adding dyadics to the table. (Dyadics in tensor terminology have confused me and presumably others, which is why I don't want it to continue within this fundamental article). F = q(E+v×B)⇄ ∑_ic_i 16:14, 2 July 2012 (UTC)

Talking about Dyadics. The history section may need a mention of them (and Gibbs). But I still haven't found a source I can access about this.TR 16:29, 2 July 2012 (UTC)

Good point, though I don't have sources which go through dyadics in detail and history either (they only mention in passing). The Dyadics article does claim that Gibbs introduced them, though without reference... F = q(E+v×B)⇄ ∑_ic_i 17:13, 2 July 2012 (UTC)

Determinant example

An edit today removed the determinant as an example of tensors, because it is not an example. But the determinant on R^n is an alternating multilinear n-form. That is, the determinant is a particular antisymmetric (0, n)-tensor. Its array of coefficients is the Levi-Civita symbol. So I don't understand why this example was removed. Mgnbar (talk) 13:55, 16 August 2012 (UTC)

A tensor is, by definition, invariant (often referred to as covariant) under a change of basis. Any object that does not conform is not a tensor, and the Levi-Civita symbol is not invariant. The Levi-Civita tensor, on the other hand, differs from it by a (basis-dependent) multiplying factor, and would be a candidate example. Strictly speaking, the determinant of a type (1,1) tensor may be found using the type (n,n) generalized Kronecker delta. The type (k,k) generalized Kronecker delta is invariant for any k and n, and may be regarded as a tensor. — Quondum^☏ 14:22, 16 August 2012 (UTC)

Please help me find the defect in the following argument. A determinant on $R^{n}$ is a function $R^{n}\times \cdots \times R^{n}\to R$ that is linear in each argument. (It also obeys an alternating property, and it is often normalized with respect to some inner product, but that's not essential to my argument.) Therefore a determinant is a (0, n)-tensor, according to the subsection "As multilinear maps". Alternatively, it is a map $R^{n}\otimes \cdots \otimes R^{n}\to R$, and hence a (0, n)-tensor, according to the subsection "Using tensor products". Mgnbar (talk) 15:07, 16 August 2012 (UTC)

More commonly, the determinant is a map: $V\otimes V*\to C$. Since that map not linear, it is not a tensor. That alone is reason enough that include determinant as an example is more confusing than enlighting.TR 15:32, 16 August 2012 (UTC)

A volume form is a tensor by definition. It is a section of the bundle of n-forms. That is, a section of a tensor bundle. That is, a tensor. Sławomir Biały (talk) 15:14, 16 August 2012 (UTC)

Yup, you beat me to reverting my own edit on this one. I was misled by the use of dx∧dy∧dz as defining a volume form, which does not transform as the same expression in a new coordinate system (it requires an additional Jacobian determinant). While this leads to confusion, a volume form correctly defined is of course a tensor. — Quondum^☏ 15:42, 16 August 2012 (UTC)

Also, I'd not say that the determinant is a tensor. This is, at least, a vague statement. The determinant either is a tensor or isn't one, depending on the point of view you adopt. Sławomir Biały (talk) 15:24, 16 August 2012 (UTC)

One thing is pretty certain: the determinant in any interpretation as a tensor does not belong in the (0,M) entry in the table. As I suggested, it could fit into the the (M,M) entry, but then I would not call it a determinant, as this usage would be non-notable. — Quondum^☏ 15:50, 16 August 2012 (UTC)

I don't see how the determinant could possibly be in the (M, M) entry; wouldn't there then be too many coefficients? Is the issue here that people regard the determinant as a nonlinear operator on linear transformations (endomorphisms), rather than a linear operator on M-tuples of vectors/covectors? Although of course I can be outvoted here, let me just point out that the latter viewpoint is not my invention. For example, the classic Hoffman and Kunze linear algebra text defines determinants in essentially this way. Mgnbar (talk) 16:30, 16 August 2012 (UTC)

It depends on what you mean by determinant. The usual definition of the determinant of an endomorphism is the induced endomorphism of the highest exterior power of as vector space. This identifies it naturally as an (n,n) tensor. The definition you are working with is the multilinear map that generalizes the scalar triple product to higher dimensions (I.e., a multilinear function of the columns of the matrix). This is a tensor of type (0,n). (Except there's then more than one "determinant", one for each reduction of the structure group to SL(n).) Sławomir Biały (talk) 17:30, 16 August 2012 (UTC)

If I understand correctly (it's been a long time since my coursework), the determinant, according to your definition, is a function from the $M^{2}$-dimensional vector space $V\otimes V^{*}$ to the one-dimensional vector space $V^{\wedge M}\otimes (V^{\wedge M})^{*}$? It's nonlinear, and hence certainly not a tensor of any kind, right?

Long ago, I was the editor who added this Examples section to the Tensor article. My hope was to connect tensors (which are somewhat abstract and confusing to learn) to more pedestrian linear algebra (which anyone learning about tensors has already studied). The matrix determinant, as discussed by the Determinant article, can naturally be viewed as a nondegenerate, alternating (0, n)-tensor on the vector space R^n. In fact, except for a normalization factor, this can be taken as the definition of the matrix determinant. So I thought that it was a good example. And it's not original research, because textbooks do this. But if the consensus is that this example is misleading, then let's keep it out. Mgnbar (talk) 18:28, 16 August 2012 (UTC)

I wouldn't necessarily say it is misleading. It certainly is confusing though, because many readers will think of the determinant as a nonlinear map.TR 18:37, 16 August 2012 (UTC)

I would add to this. The Levi-Civita symbol must be applied twice, once with lower and once with upper indices, to get the determinant. In this form, as the product with n lower and n upper indices, it is equivalent to the (M,M) entry. Why would you want to identify half the operation with the name determinant? Oh, I see – it is in combination with a selection of one index value per copy of the matrix, in direct association with the index of the Levi-Civita symbol. This is IMO highly unnatural. Anyhow, these are all very much non-tensorial operations, and the Levi-Civita symbol does not qualify as a tensor in isolation. — Quondum^☏ 01:05, 17 August 2012 (UTC)

I confess that I don't know much about the Levi-Civita symbol. I employ the "tensors as arrays of numbers" viewpoint only reluctantly. Perhaps I have applied it incorrectly. Anyway, I have already explained repeatedly, with citation, how the matrix determinant can be viewed as a tensor, and you have never refuted this argument. But, because consensus is against me, I've stopped arguing about the content of the article. Cheers, all. Mgnbar (talk) 01:53, 17 August 2012 (UTC)

The intention is not to beat you into submission. Going back to your earlier description, the determinant is not as you described it V ⊗ ... ⊗ V → F (V being the vector space over the scalars F, e.g. V=Rⁿ and F=R in your example). It is a map (V⊗V^∗) ⊗ ... ⊗ (V⊗V^∗) → F. I too prefer the abstract view to components. You mention a citation (which I presume is the section in Determinant), but I fail to see that supporting the abstract perspective as you present it. I apologize if this is frustrating you; you needn't pursue it if you don't wish. — Quondum^☏ 02:57, 17 August 2012 (UTC)

No, the citation I meant was Hoffman and Kunze, a classic, highly respected linear algebra text that defines matrix determinants in exactly the manner that I have described. Mgnbar (talk) 12:17, 17 August 2012 (UTC)

The determinant is a multilinear form of the columns of a matrix. In this sense, it is a (0,n) tensor. Sławomir Biały (talk) 12:32, 17 August 2012 (UTC)

Exactly. Slawomir (throughout this thread) understands and acknowledges my point, even while explaining how the other viewpoint is better. Cheers. Mgnbar (talk) 12:55, 17 August 2012 (UTC)

For my understanding in more abstract terms, the ith "column" of a (1,1) tensor T is the vector obtained when the tensor transforms the ith basis vector: col_i : V⊗V^∗→V : T↦T(e_i). In this way we can decompose a tensor into n vectors corresponding to the matrix columns, which we can then act on with the (0,n) fully antisymmetric tensor to produce the determinant of the original T. This tensor would, of course, depend (for its scaling factor) on the choice of decomposing basis to arrive at an invariant determinant. — Quondum^☏ 18:10, 17 August 2012 (UTC)

As I have repeatedly said, I'm talking about the determinant of matrices, not of (1, 1)-tensors. Equivalently, I'm talking about a particular function defined on $\left(\mathbb {R} ^{n}\right)^{n}$, using the fact that $\mathbb {R} ^{n}$ has a standard basis. The function is normalized so that its value on the standard basis is 1. I did not expect to have to explain this concept so many times, over so many days. It's not supposed to be difficult. Mgnbar (talk) 11:54, 18 August 2012 (UTC)

Specific to finite dimensional vector spaces

The equivalence between the definition in terms of sums of formal products and the definition in terms of multilinear maps holds only in finite dimensions. In addition to the issues discussed in Topological tensor product, there is the problem that the dual is normally restricted to continuous (bounded) functionals and that V and V** no longer have a canonical isomorphism. Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:55, 11 September 2012 (UTC)

Currently, the article implicitly only treats the case of tensors on finite dimensional vector spaces. Since, this is by far the most common case in practical applications, I think it is wise to keep it that way and not make the article any harder by including the necessary "ifs and buts" needed for the infinite dimensional vector spaces. An explicit note somewhere, that this is the case seems warranted though.TR 15:23, 11 September 2012 (UTC)

What hatnote would be appropriate for pointing to articles discussing the infinite-dimensional case, e.g., Dual space#Infinite-dimensional case, Dual space#Continuous dual space, Topological tensor product?

Should the article be renamed to, e.g., Tensor (finite dimensional case)? Shmuel (Seymour J.) Metz Username:Chatul (talk) 12:49, 14 September 2012 (UTC)

No, because the common usage also restricts to the finite dimensional case. In fact, I don't think I have ever encountered a tensor on an infinite dimensional vector space (that was actually referred to as a "tensor". This probably related to the fact that the tensor product of two separable Hilbert spaces is a separable Hilbert space and hence isomorphic. Consequently, there is little reason to move beyond linear functionals and linear operators.

It think the best way to deal with this, is to simply add a remark at the of the tensor product section that this definition has the advantage of extending to the case of infinite dimensional vector spaces. Of course, provided a suitable reference is found.TR 13:59, 14 September 2012 (UTC)

Do you have a reference for the general LCTVS case? In any case, that probably deserves its own article (with a mention from here, of course) with a good dose of motivation and details. For Hilbert spaces, I suppose the extension is straightforward. But tensor products of even just Banach spaces, seems to me, would take you into a delicate situation where even the appropriate norm has to be picked carefully, and different norms gives you different duals. If someone resorts to this, there has to be a pretty good reason. Mct mht (talk) 03:53, 15 September 2012 (UTC)

Maybe the hatnote shouldn't be flatly ruled out. One can talk about vector fields on a Hilbert manifold. I don't know this but pretty likely that people have constructed tensors over, say, Banach manifolds, to some extent. Question is whether the community of folks who do this stuff have agreed to a standard way and the technology is sufficiently mature to warrant WP coverage. Mct mht (talk) 06:46, 20 September 2012 (UTC)

1)Does would be tensor fields

2)The fact that you could in principle construct does fields does not mean that

a)people use them.

b)people call them tensors.

In practice (for example in quantum mechanics) people frequently use elements of tensor products of Hilbert spaces. However, if never seen anybody call these elements tensor.

The page that the hatnote was linking to was about tensor products of infinite dimensional vector spaces. (Note that the issues with those involve the construction of the topology, not the construction of the elements.) Again, nobody seems to call the elements of these tensor product spaces "tensors". Consequently, it seems an extreme stretch to presume that a reader looking for information about the tensor product of infinite dimensional vector spaces, would look for "tensor". (Instead of "tensor product")TR 08:38, 20 September 2012 (UTC)

Wolfram Mathworld Tensor Space and TENSOR ALGEBRAS OVER HILBERT SPACES. I should be enough to show that people do speak of infinite dimensional tensor spaces. Shmuel (Seymour J.) Metz Username:Chatul (talk) 16:35, 24 September 2012 (UTC)

That wasn't the point of Tim's argument. Indeed, it does make sense to form tensor products of infinite dimensional spaces. But neither I nor Tim have ever heard such things referred to as "tensors" (i.e., the geometrical notion, which is the subject of this article). Mentioning the infinite dimensional case would be appropriate in the tensor product article. Sławomir Biały (talk) 16:50, 24 September 2012 (UTC)

You didn't read the references that I gave. Both of them use the term tensors for elements of the tensor space. If you would prefer a reference to a text book, I can provide that as well: Abraham, R.; Marsden, J. E.; Ratiu, T. (1988) [First Edition 1983]. "Chapter 5 Tensors". Manifolds, Tensor Analysis and Applications. Applied Mathematical Sciences. Vol. 75 (Second ed.). Springer-Verlag. pp. 338–339. ISBN 0-387-96790-7. Elements of T^r_s are called tensors on E, {{cite book}}: Cite has empty unknown parameters: |laydate=, |separator=, |trans_title=, |laysummary=, |trans_chapter=, |chapterurl=, and |lastauthoramp= (help); Unknown parameter |month= ignored (help) Shmuel (Seymour J.) Metz Username:Chatul (talk) 17:18, 30 September 2012 (UTC)

Indeed, I didn't read the sources, since you seemed to be making a different point than the one you now are. However, I think that a separate subsection is preferrable to adding hatnotes about the infinite dimensional case. Sławomir Biały (talk) 22:44, 15 October 2012 (UTC)

Perhaps infinite-dimensional tensor algebra can be treated as a generalization - thus it would be up to an infinite-dimensional article to show how much of the theory generalizes. Then whether the people using that theory call them tensors doesn't affect the discussion of tensors proper. ᛭ LokiClock (talk) 19:56, 30 September 2012 (UTC)

As you may or may not have actually noted, I did add a sentence or so to the end of the "tensor product definition" section reflecting the fact that this definition can be used for infinite dimensional vector spaces as well if a tensor product is defined. This seems a much more elegant solution, then the very awkward hatnote you proposed. The sentence itself could use some more work though.TR 09:27, 1 October 2012 (UTC)

The problem is that the title and leadin suggest that the article is generic while it really is about the finite-dimensional case, although as you note there are some caveats later on. That's why I consider a hatnote appropriate. Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:58, 5 October 2012 (UTC)

There's a lot you can't take for granted about infinite-dimensional linear algebra. Vector spaces and their double duals are only naturally isomorphic in finite dimensions, the axiom of choice is required to say all vector spaces have a basis and all bases give it the same dimension, Hilbert spaces only generalize inner product spaces, &c. It makes more sense to talk about how finite-dimensional tensor algebra works and for those concerned with Hilbert spaces to talk about how much of the theory generalizes. ᛭ LokiClock (talk) 18:47, 10 October 2012 (UTC)

I don't have a problem with an article limited to the easier, finite dimensional case. I do have a problem with an article of limited scope that has a generic title and doesn't have an appropriate hatnote mentioning that it covers only a special case. Shmuel (Seymour J.) Metz Username:Chatul (talk) 21:26, 15 October 2012 (UTC)

The construction from the Marsden reference is very different in nature from topological tensor product. They skirt around the issue of topological completion completely and the construction is pretty much identical to the finite dimensional construction. I am assuming something by Marsden is considered standard. Doesn't take much to integrate that point of view and is probably a good idea for the article. Mct mht (talk) 11:37, 1 October 2012 (UTC)

To clarify, Marsden et al deal with the case where there is a weak pairing of V with V* and V can be identified with V**. Their nomenclature defines V* as a subspace of the full algebraic dual in order to make that possible. This is similar to the dual used for Hilbert spaces in Functional Analysis. If there is no online copy and someone sees a need, I will key in their definition from my dead tree copy. Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:58, 5 October 2012 (UTC)

Proposal for new Transformation article

I've proposed at Wikipedia_talk:WikiProject_Physics#General_article_on_Transformation_may_be_needed that we should perhaps have a new article which summarizes transformations and the transformation rules for vectors, tensors and spinors. Count Truthstein (talk) 16:45, 16 March 2013 (UTC)

Misleading phrasing at the end of section "2.1. As multidimensional arrays"

Though not an expert of tensors, the paragraph just before the boxed definition of a tensor is misleading, specifically the part "If an index transforms like a vector with the inverse of the basis transformation, it is called contravariant and is traditionally denoted with an upper index, while an index that transforms with the basis transformation itself is called covariant and is denoted with a lower index." This reads as if the indices of a tensor change under a coordinate transformation. I propose to rewrite this part and to speak of /components/ of a tensor that get changed by a coordinate transformation instead. The article makes this distinction previously. Unless there is a different notion of "index" that I am not aware of, the phrasing "an index transforms like" seems at least confusing to me.

Assisted by an expert on the topic, I am sure this article can be improved easily. This new section on the talk pages intends clarify the phrasing without editing the article prematurely. --194.95.26.162 (talk) 16:04, 9 April 2013 (UTC)

I think it's clear from the context what "An index transforms..." is supposed to convey. Ideally, it should be made more precise, but I don't see an easy way to make it more precise without introducing new awkwardness. Sławomir Biały (talk) 12:40, 19 May 2013 (UTC)

Excuse me, for the question, I am not sure right or wrong: Does the statement covectors w transform with R matrix is always valid, or it applies only when we transform from one ONB to another ONB? Twowheelsbg (talk) 12:33, 19 May 2013 (UTC)

It's correct as stated. Covectors transform with the same matrix that transforms the basis to the new basis. One source of confusion is that this matrix is the inverse of how the coordinates themselves transform, since the coordinates are themselves contravariant with respect to changes of basis. Sławomir Biały (talk) 12:38, 19 May 2013 (UTC)

Thank you, Slawomir, for the answer. I agree and my question changes: Then vectors should transform with the inverse transponse of R matrix, not just the inverse as written here? I explained myself the correct equation of vector components transformation this way. Twowheelsbg (talk) 23:48, 31 May 2013 (UTC)

Yes that's right. Sławomir Biały (talk) 00:40, 1 June 2013 (UTC)

This depends on how you regard the vectors and covectors. When translating into matrix notation, normally a vector would be represented as a column, and a covector as a row (as stated in the article). Then only the inverse is used, not the inverse transpose, but of course post-multiplication is needed with the covector (the row vector):

${\hat {\mathbf {v} }}=R^{-1}\mathbf {v}$

${\hat {\mathbf {w} }}=\mathbf {w} R$

— Quondum 01:44, 1 June 2013 (UTC)

Yes, but the action of premultiplication is the transpose of the action of postmultiplication and vice-versa. The notion of "transpose" is not dependent on a particular realization in terms of matrices. Sławomir Biały (talk) 01:57, 1 June 2013 (UTC)

When swapping pre- and postmultiplication, you need to transpose everything, including the vectors/covectors. So I'd not say that so readily. And I'm not too sure what you mean by "transpose" in the context of tensors – its is a concept that makes sense to me for type (2,0) and (0,2) tensors, but not for type (1,1) tensors. — Quondum 02:46, 1 June 2013 (UTC)

I've found Dual_space#Transpose_of_a_linear_map, which gives a general definition of transpose that should apply to any order-2 tensor, even if not a type (0,2) or (2,0) tensor or in the presence of a metric tensor (which would be natural to use for one definition of this transpose). I need to see how this applies in the context of abstract tensors before commenting. — Quondum 11:28, 1 June 2013 (UTC)

I concede: in the abstract realm (where left- and right-multiplication both become merely a linear transform action), the inverse transpose is necessary, not as obvious in terms of tensor components. I should know better than to take issue with something said by you, Sławomir. — Quondum 13:26, 1 June 2013 (UTC)

Introduction is inadequate.

I was very surprised NOT to see anything about Cartesian Tensors in the Introduction. The word 'tensor' is commonly used to signify either the general or the Cartesian concepts, rarely both. Many engineering students are ONLY exposed to Cartesian tensors. Much of that literature - in my experience - uses the term without qualification, making many statements and conclusions that are NOT true in the general case. It would be worthwhile, IMHO, to add a paragraph detailing the differences between a general tensor and a Cartesian tensor. The subjects are approached very differently in the applied literature vs the mathematical (and theoretical Physics) literature. I also think a section dealing exclusively with the Cartesian tensor would be useful. Unfortunately, I lack the competence to do this myself.216.96.76.236 (talk) 16:13, 22 June 2013 (UTC)

It's true that the first exposure to tensors in applied maths, physics, engineering are usually Cartesian tensors, and that results are stated in this simplified formalism which are not general. But since this article is the general scope, a lead paragraph or even a section on Cartesian tensors here would detract from it. That's why we have another article for such explanation. I added a link at the top and it was moved to the see also section, which is fine - the important thing is this article (Tensor) links to Cartesian tensor. M∧Ŝc²ħεИ_τlk 16:34, 22 June 2013 (UTC)

A second talk thread in the same section. In an attempt to make the page more accessible to laypeople, I have done my best to untangle the explanations, moving each to its own paragraph and also adding simplifying interpretations where I felt understanding of the article depended way too much on the reader understanding other elements of mathematics - elements that are not, in themselves, crucial to understanding the article. I have, next to the previous text, added references to things a reader can be expected to know (most obvious is the addition of basis ≈ coordinate system analogy).

However, I have not been able to figure out how to break down the introductory text without removing information from it. The problem with the current text is that it assumes the reader knows a whole bunch of stuff the average person may very well have never heard of. Somehow, the rigorous mathematical terms should be replaced with their "common tongue" equivalents. Triklod (talk) 01:09, 13 August 2013 (UTC)

The problem is in the opening sentence, tensor are not necessarily geometric objects, and as a working mathematician the best way to grasp the concept is understanding them as a multilinear maps, and then make exemplification with determinants, metrics, change of basis in vector spaces, change of coordinates on manifolds, and so kmath (talk) 02:08, 13 August 2013 (UTC)

This point has been disputed for several years now. The article emphasizes the engineer's (?) perspective on tensors, and relegates the mathematician's perspective to other articles. As a mathematician, I'm okay with this. The engineer's perspective requires less background, and hence is probably a better introduction to the topic for an encyclopedia. I do agree with you that the "geometric" part is rather unnecessary and probably biased, although it may reflect the historical motivation for the concept.

In Wikipedia articles with heavily disputed intros, it is common to propose rewrites on the talk page, that other editors can critique. So give us a rewrite? Mgnbar (talk) 02:55, 13 August 2013 (UTC)

I disagree that the article emphasizes any particular perspective. The purpose is to discuss all significant perspectives, per WP:NPOV. I also don't see what's wrong with the word "geometrical". Even in pure mathematics, tensors are defined as equivariant functions on a G-torsor with values in a representation of G; that's an intrinsically geometrical perspective, a la Klein. Finally, I fail to see how the first sentence is in any way inconsistent with the view of tensors as multilinear maps. Indeed, it seems to accommodate this view specifically (without going into details). Sławomir Biały (talk) 11:39, 13 August 2013 (UTC)

The modern mathematical viewpoint is relegated to Tensor (intrinsic definition), and only briefly mentioned here. The "As multidimensional array" viewpoint is given first and in greatest detail, suggesting to the reader that it is primary. Tensor fields are introduced before the multilinear and intrinsic definitions of tensor non-fields. All of this is what I meant about "engineer's perspective". But something has to go first, and the emphasis is not as strong as it once was (if I recall correctly), and anyway I'm not upset about it.

A tensor can be defined on a vector space in a purely algebraic way, without reference to geometry or topology (although historically those subjects were major motivators). That's what I meant by "geometric". Again, i do not consider this a major problem.

I agree that the first sentence is consistent with "multilinear maps". Mgnbar (talk) 12:17, 13 August 2013 (UTC)

Could it be that in this context, what You call "algebraic" and others call "geometric" is actually the same thing, namely the coordinate-free resp. index-free treatment of tensors?--LutzL (talk) 13:09, 13 August 2013 (UTC)

That could be, although coordinates are also a purely algebraic concept (scalars arising in linear combinations, which are algebraic). I mean, coordinates and coordinate-freeness do not require an inner product, a topology, or any of the other core concepts of "geometry" in its various manifestations. But we're arguing over a term ("geometry") that has no precise definition anyway. So it's probably not worth our effort. Mgnbar (talk) 14:15, 13 August 2013 (UTC)

Re: rewrite, if I had an idea for a rewrite (of the introduction), I would have proposed the text. But I don't. However, since my edits to the meat of the article were reverted for introducing errors, for me the introduction is a moot point now. And so is the continuance of me editing Wikipedia since, if I make errors on simple articles like tensors, what will happen if I move to more complicated articles? Whelp, nevermind. There are other things to do. Triklod (talk) 00:46, 14 August 2013 (UTC)

Sorry for no suggestions for improvements, but I think the entire article (not just the lead) reads extremely well and coherently, and the article could at least be rated to B class (if not A class). M∧Ŝc²ħεИ_τlk 12:52, 13 August 2013 (UTC)

Not All Vectors and Scalars are Tensors

I'm certainly no expert in tensors, but it seems to me that the article says that all vectors and scalars are tensors. I have been led to believe this is not true due to the tensor's rules concerning coordinate transformations, which, as far as I know, do not apply to the general vector and scalar.

No this is always true (for the appropriate notion of a tensor). A scalar is invariant under changes of the coordinate system (note that a scalar is not just a number, but a number times its unit of measurement: this product is invariant). Likewise, the components of a vector transform contravariantly under changes of coordinates in the vector space that it lives in. Sławomir Biały (talk) 19:58, 1 July 2013 (UTC)

Not position vectors. -- — Preceding unsigned comment added by 86.184.230.128 (talk) 16:58, 18 September 2013 (UTC)

Vectors are tensors of type (1, 0). Scalars are tensors of type (0, 0). The coordinate transformations operate exactly as described. Mgnbar (talk) 03:48, 2 July 2013 (UTC)

What might underlie the perception is that some sets of quantities are defined with a dependence on the choice of coordinate basis and that are incompatible with the coordinate transformation rules. In this context, however, the terms "scalar" and "vector" are reserved for concepts that in some sense are independent of the choice of basis and thus inherently fit the tensor transformation rules. There are therefore quantities that look like scalars and vectors, but the terms may not be used to refer to them in this context. As examples, you might want to look at pseudotensor, Levi-Civita symbol and Christoffel symbols. — Quondum 13:04, 2 July 2013 (UTC)

Thank you very much for the replies, guys. So does that mean texts such as this: https://www.grc.nasa.gov/WWW/k-12/Numbers/Math/documents/Tensors_TM2002211716.pdf on page 7 are wrong or am I simply reading it and interpreting it incorrectly? Perhaps it is more in line with what Quondum is saying and I need to get my contexts straight. --204.154.192.252 (talk) 14:04, 2 July 2013 (UTC)

The treatment there is very far from the mathematical idiom (and rigor), so I have trouble reading it. But the relevant example is actually on page 10: The frequency of a light source is a scalar, but two observers moving with nonzero relative velocity will observe different numerical values for this scalar, and hence this scalar does not obey the (0, 0)-tensor transformation law. I'll let someone with better physics explain this. Mgnbar (talk) 15:46, 2 July 2013 (UTC)

Frequency in Relativity is not a scalar; it is the time component of a 4-vector, and as such changes with coordinate changes. Shmuel (Seymour J.) Metz Username:Chatul (talk) 18:58, 2 July 2013 (UTC)

I guess you mean one divided by the time component, since time has the dimension T while frequency has the dimension T ^-1? —Kri (talk) 21:33, 10 November 2013 (UTC)

There are serveral definitions for tensors, e.g., Tensors#As multilinear maps and for vectors, and with the most common definition of tensors a vector is not a tensor. However, there is a natural map from vectors into tensors, and it is common to ignore the distinction in informal prose. Shmuel (Seymour J.) Metz Username:Chatul (talk) 18:58, 2 July 2013 (UTC)

The linked document appears to use the terms scalar, vector, dyad in the "wrong" sense in this context: to mean a collection of real numbers used to represent something. So, where they say "scalar", read "real number", where they say "vector", read "tuple of real numbers". There is nothing inherently wrong about different use of terminology (I should know, I've spent much energy on trying to decipher what the defining terms really mean in various WP articles), but it helps hugely to figure out the exact intended meaning of the author. This WP article takes the approach that each of the names (scalar, vector etc.) applies to a class of tensor, which is useful in a more abstract approach, whereas the linked text uses the terms in a more school-text approach. On a point of order: this is no longer discussing anything that might relate to editing the article, so if taken further, it should be at Wikipedia:Reference desk/Mathematics‎‎. — Quondum 02:45, 3 July 2013 (UTC)

Tensor#As multilinear maps defines tensors in terms of vectors, so it is not true that the article uses vector interchangeably with tensor of rank 1. Shmuel (Seymour J.) Metz Username:Chatul (talk) 19:50, 3 July 2013 (UTC)

I don't quite see what you are driving at. This section effectively defines two types of tensor of rank 1: as a vector, or as a covector. They are thus the same thing by definition, and are interchangeable except for the purpose of making this definition. Also, the concepts vector and covector being used here each refer to an element of an abstract vector space, independent of any concept of basis, so there is no room for confusing this with the "tuple" concept dealt with earlier. — Quondum 23:50, 3 July 2013 (UTC)

No, by the definition in Tensors#As multilinear maps, a tensor of type (1,0) is a map $T:V^{*}\rightarrow \mathbf {R}$, not an element of V. Shmuel (Seymour J.) Metz Username:Chatul (talk) 17:38, 5 July 2013 (UTC)

... which is naturally isomorphic to a vector, and in the context of tensors, the two are identified (by defining the action of a vector on a covector, thus making V^∗∗ = V). But, as I alluded, this aspect (i.e. a discussion about the abstract definitions) has no relevance in this thread. If this should be addressed, start a new thread so that it can be addressed as a topic on its own. — Quondum 19:01, 5 July 2013 (UTC)

Of course there is a natural map from V to V**; otherwise I wouldn't have written However, there is a natural map from vectors into tensors, and it is common to ignore the distinction in informal prose.

The subject of this thread is Not All Vectors and Scalars are Tensors ; the distinction between V and V** is most definitely relevant to that. Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:20, 8 July 2013 (UTC)

If V and V** are not identified, then arguably it is wrong to define tensors as multilinear maps. From the point of view of linear algebra, these are the same space, even if they are strictly not the same set. Sławomir Biały (talk) 16:30, 8 July 2013 (UTC)

I think that this is understood by all. As far as the article is concerned, notability is primary. Sławomir, in your experience, are there any notable sources that do not make this identification in their treatment, and if so, are they relevant to the article? — Quondum 16:39, 8 July 2013 (UTC)

Yes, right. In the rare occurrence that these spaces are not identified, we are in infinite dimensions and the relevant notion of tensor product becomes considerably more nontrivial. The relevant notions were developed in the 1950s by Alexander Grothendieck, fwiw. Sławomir Biały (talk) 23:11, 8 July 2013 (UTC)

I guess that the identification is sometimes/often made might be worth a mention. Another point that could be improved is the assumption of a holonomic basis in Tensor#Tensor fields. I'm not about to tackle this at this point, but might at some stage. — Quondum 02:18, 9 July 2013 (UTC)

Can a tensor have different dimensionality for different indices?

When reading this article, I can't' find whether a tensor has to have the same number of dimensions for every index. That is, can all indices go up to the same number or can some indices go up higher than others? For example, can a linear map between two-dimensional vectors and three-dimensional vectors be considered a (1,1)-tensor? If so, then the contravariant index of that tensor could go to three while the covariant index only could go to two. —Kri (talk) 21:22, 10 November 2013 (UTC)

In the abstract mathematical treatment of tensor products, it is certainly possible to take a tensor product of two vector spaces of differing dimension. The elements of this tensor product are "tensors", in which the indices do not all go up to the same number. However, there has been some argument on this talk page in the past, about whether the term "tensor" is really used in this generality.

This article tends to focus on tensors from the engineering/physics perspective. In that context (for example fluid dynamics textbooks), I have never seen the indices differ in dimension, because the tensors in question are constructed from a single vector space (and its dual). But someone more expert than me may comment differently. Cheers. Mgnbar (talk) 22:00, 10 November 2013 (UTC)

To make a more strict statement, of course you can have tensors of the type $(V\otimes V)\otimes (W\otimes W\otimes W)^{*}$. But firstly, in the index-notation, this would be counted as a 5-fold tensor product, preferably with different types of letters for V and W. Which actually happens in physics for space and spinor indices. And "But" secondly, the (r,s) classification is, as Mgnbar says, only used for tensor product of a single vector space and its dual. So ${\mathcal {T}}^{(2,0)}V\otimes {\mathcal {T}}^{(0,3)}W$ could be used, if necessary.--LutzL (talk) 01:48, 11 November 2013 (UTC)

If one wants to be really pedantic, braiding can also be taken as significant, so if being thoroughly pedantic, one might put for example the type of a tensor as $V^{*}\otimes V\otimes W\otimes V\otimes W$. Tensor (intrinsic definition) is more the place for this, though. — Quondum 02:29, 11 November 2013 (UTC)

Why can't the (r,s) classification be used for a tensor product of two or more vector spaces? What is wrong with the example I provided for example, where a linear map from two to three dimensions is written as a tensor — can't that be said to be a (1,1)-tensor? It takes one contravector and converts to another contravector, so there has to be equaly many covariant as contravariant indices in that tensor, doesn't it? By the way, do r and s in (r,s) classifications stand for anything, or are they just arbitrary variable names? In the article they use m and n. —Kri (talk) 21:21, 11 November 2013 (UTC)

The r and s are just variable names. They are positional placeholders in the "type-( , )" terminology. One reason why you cannot apply the "type-(r,s)" terminology for multiple independent vector spaces is that choosing which to call covariant and which to call contravariant is completely arbitrary, independently for each pairing of a vector space and its dual. Choosing V as "covariant" (and thus V* as "contravariant") doesn't stop you choosing W as "contravariant" (and thus and W* as "covariant"). You'd need a separate (r,s) pair for each new vector space in the product. — Quondum 23:11, 11 November 2013 (UTC)

Raising and lowering indices

There is a problem with the description of Raising and lowering indices in Tensor and related articles. In the traditional braided index notation, you get the original tensor after you raise and lower a particular index, but with the definition in the article you get a braided equivalent instead. Using a definition of the tensor algebra that uses mixed products, e.g., $V\otimes V^{*}\otimes V^{*}\otimes V$, with either the quotient vector space definition or the multilinear function definition of the tensor product, allows for clearer and unambiguous raising and lowering of indices of mixed tensors like $T_{\alpha \beta \gamma },\ T_{\alpha \beta }{}^{\gamma },\ T_{\alpha }{}^{\beta }{}_{\gamma },\ T_{\alpha }{}^{\beta \gamma },\ T^{\alpha }{}_{\beta \gamma },\ T^{\alpha }{}_{\beta }{}^{\gamma },\ T^{\alpha \beta }{}_{\gamma },\ T^{\alpha \beta \gamma },$. at the expense of more complicated notation for the type of a tensor and a more complicated definition of the tensor algebra. I believe that this isue should be discussed at least briefly. Shmuel (Seymour J.) Metz Username:Chatul (talk) 19:27, 13 December 2013 (UTC)

I don't really follow your specific objection in relation to this article, but I do feel that the notational conventions for braiding in this context get glossed over unduly. This is in part because braiding is so naturally handled by the index notation, but also in part because so many people think of the set of tensor components as the tensor. It would be nice if these subtleties are made clearer. For example, I've seen a suggestion that lexicographical order of the indices can be significant in some conventions with respect to braiding, but this is rarely mentioned. This should also be addressed in more explicit detail in Ricci calculus. —Quondum 03:43, 15 December 2013 (UTC)

My concern is that the definitions given do not allow for braiding of indices, which is still common in the literature. Shmuel (Seymour J.) Metz Username:Chatul (talk)