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Tensor densities

The section "Tensor densities" seems to be about tensor fields, not specific tensors. I think it should move to the article on tensor fields. In general, anything that talks about a Jacobian belongs there, I think - there doesn't seem to be much reason to talk about Jacobians when, as in this article, our initial object is just a single vector space. Overall, the entire sections seems to be about tensor fields on manifolds. — Carl (CBM · talk) 00:01, 10 August 2017 (UTC)

Well, I for one do not think the restriction of this article to the linear algebra notion of "tensor" is helpful. That actually is (or should be) the subject of a separate article, tensor algebra. This article is intended to cover the more inclusive notion, including tensors in curvilinear coordinates. This having been said, a tensor density does make sense on an arbitrary vector space. A tensor density transforms by a usual equivariant transformation law, with factors of the determinant of the transformation. Sławomir Biały (talk) 01:26, 10 August 2017 (UTC)

I have always thought the purpose of this article was to talk about a tensor on a single vector space - which, from the viewpoint of manifolds, might mean a tensor at a single point of a manifold. At least that is what the hatnote says. When we start talking about assigning tensors to more than one point of a manifold (e.g. by talking about coordinate systems or Jacobians), we are talking about a tensor field, rather than a single tensor. (Of course tensor fields are sometimes called just "tensors" but for the purposes of this article we should be careful to distinguish tensors on a vector space from tensor fields.) Tensor algebra is a higher-level article, while this one is a more introductory article about single tensors.

I do agree that a tensor density makes sense as a concept on a single vector space, but that is not the way that the section currently in the article is written. Of course it can be rewritten, but rather than just removing the useful content there, I think it would be better to move it to the article on tensor fields. — Carl (CBM · talk) 12:10, 10 August 2017 (UTC)

I'm just not sure that the focus on the "tensor on a vector space" concept really captures quite what "tensor" means in applications. For example, consider the tensor $\mathbf {i} \otimes \mathbf {j}$, located at the point of the Cartesian plane $(1,2)$. Then, in polar coordinates, the vectors $\mathbf {i} \otimes \mathbf {j}$ changes to

${\begin{aligned}(\cos \theta \mathbf {e} _{r}-r^{-1}\sin \theta \mathbf {e} _{\theta })\otimes (\sin \theta \mathbf {e} _{r}+r^{-1}\cos \theta \mathbf {e} _{\theta })&=\left({\frac {\mathbf {e} _{r}}{\sqrt {5}}}-{\frac {2}{5}}\mathbf {e} _{\theta }\right)\otimes \left({\frac {2\mathbf {e} _{r}}{\sqrt {5}}}+{\frac {1}{5}}\mathbf {e} _{\theta }\right)\\&={\frac {2}{5}}\mathbf {e} _{r}^{2}-{\frac {2}{5{\sqrt {5}}}}\mathbf {e} _{\theta }\otimes e_{r}+{\frac {1}{5{\sqrt {5}}}}\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }-{\frac {2}{25}}e_{\theta }^{2}\end{aligned}}$

It's true that one can boil this all down to a computation in a vector space. But that's not usually how one would carry out the computation in practice, and it certainly isn't how a scientist would approach the problem. You need to use differential calculus, even if all you care about is the value of a tensor at a particular point of space. I oppose efforts to further minimize this important aspect of the tensor concept in the article. Tensor densities are of fundamental importance because of the behavior of the cross product and Levi-Civita symbol under coordinate transformations. Sławomir Biały (talk) 13:28, 10 August 2017 (UTC)

Tensors are often thought of as the value of a tensor field at a point of a manifold, so presentation of tensors in the context of a field does make sense. But just as this article stays away from the differential structure of a manifold, any mention of tensor densities should avoid detail that needs explanation of the motivation and utility to make sense of it, and is best left to the main article. Mentioning that tensor densities are a generalization that transform differently and that find some value as a calculation tool would suffice; even mentioning the metric tensor here seems excessive. —Quondum 22:45, 11 August 2017 (UTC)

Agreed, the discussion of tensor densities should be edited down. The Levi-Civita symbol and cross product should be mentioned, since these do not require differentiable structure to interpret and make use of. Sławomir Biały (talk) 11:06, 12 August 2017 (UTC)

The concept of a density (in the context of a field on a differentiable manifold) makes sense even without a metric tensor, but with only the lesser structure of a scalar volume element (or is an oriented volume form all that is needed?). This suggests that a scalar density should be used as the basis of mention/description of the general transformation by weights, rather than the determinant of the metric. —Quondum 14:59, 12 August 2017 (UTC)

Scalar densities seem to go against the trend in this discussion, that we should abandon differentiable structure. The main motivation for being interested in scalar densities is to describe things that can be integrated (thus transforming under the Jacobian of the coordinate transformation). Since we're apparently not allowed to say "Jacobian" in a conversation under the new regime, it may actually be more natural to consider things like current densities as pointwise tensor densities, than it does to consider scalar densities as pointwise objects. But I'm of the opinion that the article should include more discussion of the Jacobian, not less. Sławomir Biały (talk) 18:26, 12 August 2017 (UTC)

The determinant of a basis transformation is defined even without differential structure, though abandoning differential structure appears to make the word Jacobian too specific in the general statement. Even in the context of tensor fields, the Jacobian only applies with a holonomic basis. Here, "basis transformation" might be more helpful than "coordinate transformation"; in the article at large it is not clear whether "coordinate" refers to a vector coordinate or a manifold coordinate. The first paragraph of the section could reflect this with the change of only a few terms. The second paragraph relates to examples, where the utility within the differential structure of a manifold can be made clear, equating the Jacobian to the determinant of a coordinate(=holonomic) basis transformation. As such, it would amount to a reorganization of the two paragraphs with the first providing the algebraic definition, without removing mention of anything. —Quondum 19:53, 12 August 2017 (UTC)

I think the question of holonomic versus nonholonomic bases is not helpful here. A scalar density is only a useful concept because it gives something that can be integrated somewhere. That's what "density" means. The fact that it is required to transform by the Jacobian ensures compatibility with the change of variables formula. Sławomir Biały (talk) 21:49, 12 August 2017 (UTC)

Okay, yes; thinking through it, the basis does not seem to come into it. Further, the section does start by restricting the context to a fields on a manifold: "The concept of a tensor field can be generalized by ...". So I guess my reaction is to the immediate mention of the Jacobian. As I see it, the entire motivation of "weighted densities" is driven by the use of a volume element defined as dx¹∧...∧dxⁿ in expressions to be integrated (the intuitively correct "volume element" would be the n-form that integrates to the volume of a region), so this would be the appropriate way to further restrict the context. For example, on a 3-dimensional Riemannian manifold, a "number density" (e.g. 10²⁴ particles per cubic metre) is an invariant scalar, which can be integrated upon multiplication by the "correct" n-form). Or have I got it wrong? —Quondum 23:59, 12 August 2017 (UTC)

Yes, the treatment can be streamlined without referring to the metric at all, along the lines you suggest. Forms are integrated over oriented chains, while densities are integrable over arbitrary measurable subsets. Any Borel measure that is absolutely continuous with respect to the Lebesgue measure (or volume measure) has a density. Sławomir Biały (talk) 11:57, 13 August 2017 (UTC)

I've tried to capture this. Since it is quite a change, I might have put it in a way in which you feel it loses something that is needed here. —Quondum 04:06, 15 August 2017 (UTC)

Unless someone comes up with a source that actually discusses tensor densities in the context of tensors on a vector space, I see no reason why we should mention them at all in this article. There is no reason to discuss generalizations of generalizations. Instead of forcefully trying to cram things like this in this article, it is much more useful to expand the tensor field article with stuff like this.TR 07:05, 15 August 2017 (UTC)

This is the main tensor article. It is not entitled "tensor on a vector space". It is supposed to be an inclusive top level article. Generalizations like these certainly belong here. Sławomir Biały (talk) 10:18, 15 August 2017 (UTC)

This indeed is the main 'tensor' article. Hence stuff about 'tensor fields' does not belong here. Tensor fields are not tensors, never have been. This article should not be a collect all for anything involving the word tensor. A tensor density is not a generalization of a tensor. It is a generalization of a tensor field, given its close relation to integration over (parts of) a manifold, is uninteresting unless you are talking about tensor fields. There are very good reasons that we treat tensors and tensor fields separately. In particular, the later require much more conceptual baggage to understand. (Physics) literature often not distinguishing between the two things is a enormous source of confusion among students first learning about tensors.TR 11:12, 15 August 2017 (UTC)

Looking at some of you comments above, it seems you are one of the victims of the confusion in the literature as it seems that you have a hard time distinguishing a tensor from a tensor field (a tensor valued function if you wish). This confusion is witnessed by the use of phrases such as "consider the tensor ... located at ..." (tensors do not have a location and therefore cannot be "located at"), "tensors in curvilinear coordinates" (which makes sense when talking about a tensor field, but not when talking about a tensor).TR 11:19, 15 August 2017 (UTC)

One most certainly can have a tensor at a point, and does in practice. There is a vector space of all bound vectors at the point, and one then forms the tensor algebra as usual. The tensors in this vector space can then be represented in a Cartesian coordinate basis. Changing curvilinear coordinate systems, the basis vectors change by the Jacobian matrix of the coordinate transition functions. At a point, this Jacobian is just a matrix of constants. However, you and others like Carl seem to be arguing that, despite agreeing with the overwhelming weight of literature on the subject, this is not a case of the general "tensor" concept, but rather of the "tensor field" concept. At best, this is a false dichotomy, and at worst it is simply wrong. Marginalizing all discussion of curvilinear coordinates as not relevant for the article is not consistent with the neutral point of view policy, and it is also not helpful to likely readers of the article. Finally, if you wish to have a real discussion about the scope of the article, then this should be based on actual communication rather than false and ad hominem mischaracterizations of my own competence in this area. Sławomir Biały (talk) 11:49, 15 August 2017 (UTC)

The value of a tensor field at a point certainly is a tensor. That (in some sense) is the definition of a tensor field (and a tensor bundle). However, this is all additional structure that has nothing to do with tensors per se. The relationship between 'tensors' and 'tensor fields' is very one way though. You cannot talk about 'tensor fields' without talking about tensors. However, one can certainly talking about tensors without talking about tensor fields. And for the same reasons that Euclidean vector does not talk (much) about vector fields, and complex number is not about complex functions, this article should avoid tensor fields as much as possible.

Obviously, the most common and well known applications of tensors are in tensor fields and their applications in field theory (in particular continuum mechanics and general relativity). (However, not all tensors are the point value of some tensor field (multipole moments, and various conserved tensor quantities are obvious examples)). We cannot completely avoid mentioning tensor fields, certainly in the applications section. However, the discussion (of rather technical) generalizations of tensor fields such as tensor densities, certainly does not belong here as a general reader will be missing the appropriate context. Tensor densities are much better discussed in the tensor field article which provides the appropriate context.TR 12:15, 15 August 2017 (UTC)

One introduces the notion of vector at a point long before the idea of a vector field is introduced. I assume that you've drawn free body diagrams for freshman physics students, for example. I am puzzled why you think that tensors are different than this. Sławomir Biały (talk) 12:25, 15 August 2017 (UTC)

I have a similar viewpoint to Timothy Rias (of course I started this section on the talk page). This article should focus (as the hatnote says) on tensors on single vector spaces. When we start talking in detail about tensors "at a point" or about coordinate changes or Jacobians, we move into the scope of a different article, the one on tensor fields. This is not an issue of due weight, it is merely an issue of terminology: the word "tensor" has multiple meanings, and there are separate articles for the separate meanings of tensor on Wikipedia. The article tensor field is very minimal and, and really could use significant expansion, such as a concrete example before the general discussion of vector bundles. This does not mean that the content that should be there belongs here instead. — Carl (CBM · talk) 12:29, 15 August 2017 (UTC)

It really is about weight. This article should be about the primary topic "tensor". Do we disagree about this? If not, then we really should ask what that primary topic is, instead of assuming that it is "tensor on a vector space". There are entire textbooks on tensors (for example, Schouten's definitive treatise) that do not even mention vector spaces. Aris uses the word "vector space" a single time in his textbook. Another book on my shelf is Borisenko and Tarapov, which do not discuss vector spaces. Schaum's outline of vector analysis also introduces tensors without vector spaces. So, clearly there is a mismatch between the primary topic for this article, and editors' expectations of what it ought to be. I could see restricting the scope to tensors at a single point, or "tensors on vector spaces" in a much more permissive, big-tent sense, than is currently being entertained. But to insist in a strict sense that the concept of tensor in applied mathematics, physics, and engineering, is a fundamentally different one than the subject of this article, seems inappropriate. Sławomir Biały (talk) 12:52, 15 August 2017 (UTC)

It is certainly clear that many sources will use the word "tensor" as a shorthand for a tensor field. In fact, in field theory is similarly common use the shorthands "scalar" and "vector" for scalar fields and tensor fields, often without explicitly saying so. This is natural, but does not make "tensor field" the primary topic. (Similarly sources discussing "Java" the programming language vastly outnumber those that discuss the Indonesian island. Yet Java is about the island.) I would also hold that the vast majority of the authors of the works you name would agree that they are actually talking about tensor fields. "Tensor fields" are called "tensor fields" because at each point their value is a tensor (on the tangent space at that point). This article is about the type of object (tensor) meant when we say that a "tensor field" as a tensor valued function. All the stuff you want to include about curvilinear coordinates and Jacobians, is in fact not about tensors per se, but relates to the description of tangent bundles. Textbooks often treat this simultaneously with vectors (because the applications they have in mind are tensor fields). This does not mean that the two are not logically distinct things. And since this is not a textbook we should treat them as logically distinct.

An important reason (for me) to keep the two concepts in distinct articles is that explaining what tensors are requires significantly less conceptual baggage, than explaining what "tensor fields" are. Concequently, restricting this article to just tensors allows it to be a lot more accessible.TR 13:22, 15 August 2017 (UTC)

Well, I agree that in some areas "tensor" and "tensor field" are used interchangeably, but that is not particularly the case with the sources that I have cited. Aris, and Borisenko and Tarapov, both refer to tensor fields as tensor fields, yet neither discusses vector spaces. Schouten, in an extremely mathematically careful axiomatic treatment of the subject, clearly distinguishes between fields and tensors (and introduces densities in the chapter on tensors, not tensor fields), and also does not discuss vector spaces. In any case, our treatment of the fundamental concept of "tensor" should be based on the primary topic in sources, not on an artificially imposed puristic concept "tensor on a vector space" that only exists in a very marginal way in the literature (e.g., millions of scholar hits for "tensor" versus 128000 for tensor+"vector space".) Sławomir Biały (talk) 13:36, 15 August 2017 (UTC)

I had a look at what Schouten does exactly. The objects that he introduces in Chapter 2 are something slightly more general than the tensors discussed here as he considers "affinors" on an affine space allow general affine coordinates. This is somewhat unusual, but if you forget about the affine translations (by picking a point a the origin) the affine space becomes a vector space and the affine coordinates bases for this vector space, reducing exactly the notion we here describe as a tensor on a vector space. In particular, Schouten's notion of an affinor is a global concept on the affine space (i.e. there is no such thing as the affinor at some point in the affine space). There is also no notion of curvilinear coordinates (in Ch 2 anyway). So, as I see it, Schouten's chapter 2 can be seen as support (module exchanging affine structure for linear structure) for the approach taken here of considering tensors/affinors on vector spaces/affine spaces separately from tensor/affinor fields. (I'll grant that he indeed introduce the concept of affinor densities).TR 15:30, 15 August 2017 (UTC)

This begins to wander into true Scottsman territory. We're talking about tensor densities, and these make sense even under linear transformations. The affine translation is not especially relevant here. What is relevant is that a tensor is not a point of the vector space V on which the linear/affine transformations take place. So there is an important subtle feature of tensors that is lost under the assumption that these are just tensor product of some fixed vector space. (Compare the difference between a scalar and a number.) Sławomir Biały (talk) 16:41, 15 August 2017 (UTC)

The point was that Schouten's treatment is (modulo some irrelevant stuff about affine translations) is pretty much identical to the treatment of multidimensional arrays at the start of this article. In particular, no special reference to curvilinear coordinates, tangent spaces etc, which you seem to be claiming are integral to the notion of a tensor (partially basing yourself on Schouten's treatment). Are you trying to claim that the three definitions of tensor given in this article are not in fact equivalent?TR 10:35, 16 August 2017 (UTC)

My point is not that the definitions are not "equivalent" in some abstract sense, but that waving the magic wand of tensor products and multilinear maps does little to clarify the concept of a tensor, and that the idea of tensor-on-a-vector-space, abstracted from the geometrical context, is not consistent with the primary topic here. Take the example of an electric quadrupole moment $Q_{ij}$, which is a tensor with units of $Cm^{2}$. It is certainly the case that there is a vector space V, such that $Q\in V\otimes V$. The "vectors" in V have units of $C^{1/2}m$. So in this way, what we mean by "The quadrupole moment is a tensor" is indeed "equivalent" to the statement $Q\in V\otimes V$ for some vector space V. But there is no connection of the vector space V with anything geometrical: no one should be forced to interpret a quantity with units of $C^{1/2}m$. The geometrical approach is that one has a fixed physical space, and a class of coordinate systems on that space (Veblen and Whitehead axiomatize such coordinate systems, which may be linear coordinates, affine coordinates, holomorphic coordinates, smooth coordinates, etc). The tensors are then derived objects that transform in a certain way under a change in the coordinate systems. In this view, they are not tensor products of some vector space, because there is no vector space (!). It is perhaps possible to identify some natural geometrical vector space, and then identify tensors with tensor products of that (maybe with a one-dimensional vector space to account for the units). That natural geometrical vector space is often the "tangent space", thus my reference to tangent spaces below. That may involve calculus, and thus curvilinear coordinates are very natural objects of consideration, or it may be the translation group associated with the affine structure of Schouten, or to a Euclidean configuration space of classical mechanics. I also note that this actually is consistent with the sources we cite for our discussion of multilinear maps and tensor products: Lee discusses tensors in the context of smooth manifolds, in which all tensors are based on the tangent space to a smooth manifold (yet we have apparently abstracted his definition away from this important context in the name of enforcing the article's tensor-on-a-vector-space idea). Likewise, the Hazewinkel source refers to the work of Dubrovin, Novikov, and Fomenko, where again it is the transformation point of view (and smooth (!) theory) that is emphasized. The closest that comes to the perspective of our article is Dodson and Poston (also referenced in Hazewinkel), which develop the theory of tensors via tensor products, but with the following important caveat: "That all of these wrap up in the same algebraic parcel is a great convenience, but it does mean that geometrical interpretations must attach to particular types of tensor, not to the tensor concept." Sławomir Biały (talk) 11:33, 16 August 2017 (UTC)

The unit issue you quote arises from the fact that physical quantities typically have some additional physical/mathematical structure, for example, the electric potential is actually the "time" component of a u(1) valued one-form. The dimension if "C" is inherited from the "charge" of the u(1) representation used. So yes, "It is perhaps possible to identify some natural geometrical vector space, and then identify tensors with tensor products of that (maybe with a one-dimensional vector space to account for the units)." not only possible but this is how it works. This is not unlike the fact that we commonly claim that velocities "live" in the tangent bundle, they do not have the correct units to do so. This is just a manifestation of the fact that (naïve) physics is more than just geometry.TR 12:23, 16 August 2017 (UTC)

Ok, I will try again to clarify my point, since you still seem to be missing it. The vector space V is not the same thing as physical space, but tensors are characterized by their relation to the coordinate transformations of physical space, not of V. Sławomir Biały (talk) 13:02, 16 August 2017 (UTC)

The role played here by the vector space V is completely analogous to the role played by affine space in the book by Schouten you quote. The only difference is that in the case of a vector space, there exists a natural identification of the space of rank 1 tensors on V and V itself. So in some sense yes V is (identified with) the "physical" space. (PS. it is not my fault that your own sources contradict your arguments.)TR 13:34, 16 August 2017 (UTC)

Barring your own novel interpretation of the source, I see no contradiction. I remain underwhelmed by the sources that support the supposedly primary topic of this article. Sławomir Biały (talk) 13:46, 16 August 2017 (UTC)

I think I can clarify the issue. The concept of a tensor, as it is used in mathematics, physics, and engineering, is based on the tangent space, not simply an abstract vector space. Thus, ignoring questions over "fields", a physicist will call one thing a tensor and another a spinor. These are both elements of an abstract vector space, so both are tensors-on-a-vector-space, but only one is really deserving of the designation tensor. It is still necessary to say in what sense the "vectors" are themselves tensors, an issue which is apparently swept under the rug with the abstract perspective. It is true that you can find treatments in the differential geometry literature that discuss tensors on a general vector space, but these are promptly then confined to tangent spaces at a point (usually followed by the tangent bundle, sometimes by its sheaf of sections). A differential geometer would also not call a "spinor" a "tensor", even though the definition of spinor that he gives in Chapter 9 may technically be a special case of tensor that he gave in Chapter 2 (in the context of abstract vector spaces). Sławomir Biały (talk) 14:23, 15 August 2017 (UTC)

Finally, I should think that a physicist should support the idea of including densities (or at least some discussion of the dimensions of tensors vis-a-vis the dimensions of the coordinates). Under the definitions currently in the article, neither the force vector, nor the stress tensor, nor any of the multipole moments you just added, are tensors with respect to the physical space, because they have dimensions other than length (so they are not actually vectors in the physical Euclidean space). Sławomir Biały (talk) 14:37, 15 August 2017 (UTC)

Why would we need tensors to be vectors in the physical Euclidean space? That seems a rather pointless requirement?TR 14:46, 15 August 2017 (UTC)

You're looking at tensors with respect to coordinate transformations of a set of reference axes. Multipole moments, forces, and stress, are all tensors with respect to the same set of coordinate transformations, despite having very different dimensions. (You're invited to compare with the concept of a scalar.) Sławomir Biały (talk) 14:48, 15 August 2017 (UTC)

Compare also the treatment of Veblen and Whitehead's The foundations of differential geometry. There they define tensors via their transformation law under point transformations of a fixed linear manifold. Sławomir Biały (talk) 18:46, 15 August 2017 (UTC)

I know this is a completely different point (I'm not that worried about whether this belongs in the article), but the section as it has now been reworded tries to motivate tensor densities, but motivation itself IMO definitely does not belong here. As a side issue, it is apparent to me that weighted densities add no power to weight-0 densities whatsoever: every valid construction that is formulated in terms of weighted tensor densities can immediately be restated in terms of only weight-0 densities, and thus weighting is merely an algebraic tool: a different representation of the same thing. In a coordinate-free formulation, there is not even a way to express the volume element you have in your example. I don't have the credentials to be challenging you, but I'm surprised that others have not picked up on this. —Quondum 02:42, 16 August 2017 (UTC)

The new content even more certainly belongs at tensor field. In this article, we are looking at a tensor on a single vector space, so there is no notion of integrating around. In order to talk about integrating, we need to have a tensor at each point of the domain of integration - we need to have a tensor field. — Carl (CBM · talk) 11:33, 16 August 2017 (UTC)

Tensor-on-a-vector-space is not the primary topic for the article. That a single (constant) scalar value is not an example of a tensor in the sense that you think this article is about should give you immediate cause for reassessing this viewpoint. Sławomir Biały (talk) 11:47, 16 August 2017 (UTC)

Concensus clearly is that "Tensor-on-a-vector-space" IS the primary topic of this article. Also a single scalar value clearly is a rank zero tensor on a vector space.TR 11:50, 16 August 2017 (UTC)

I strongly disagree that there is any such consensus. The cited sources in this discussion very strongly indicate otherwise. If you wish to claim consensus, please present supporting reasons. Also, you have just told me that a scalar is the same thing as a number. Is this something that you would like students to believe? Thanks, Sławomir Biały (talk) 12:04, 16 August 2017 (UTC)

I don't think that sources that use "tensor" as a synonym for "tensor field" are in conflict with this article, which explicitly points out that some sources use 'tensor' to mean "tensor field". However, there is no word I know other than "tensor" to refer to a tensor on a single vector space, while the term "tensor field" is well established to refer to tensor fields. So I think that the naming used by this article and the article on tensor fields makes sense. As a stated below (feel free to reply there) I think the division between tensors-on-a-vector-space and tensor fields makes sense for the Wikipedia presentation of tensors. This does mean that much of the material on tensor fields will go into that article, but I don't see that as a negative in any way. The only thing we can do with the conflicting terminology is to point it out and try to be precise ourselves. — Carl (CBM · talk) 12:32, 16 August 2017 (UTC)

The sourcing for the current usage of "tensor" to refer to "tensor on a vector space" is quite weak. We cite Lee, who immediately specializes to the tangent space, thus reproducing the geometric notion of a tensor, and Hazewinkel for the tensor product, a tertiary source at best. I've added the secondary source Dodson and Poston (cited by Hazewinkel), who actually do define the concept of "tensor on a vector space". The conflict, as I see it, is not between tensors and tensor fields, but between tensors (the geometrical notion and the one from physics) and tensors on a vector space. The issue of fields versus non-fields is a red herring. But what I see here is editors pushing the idea that tensor-on-a-vector-space is the same thing as the primary topic of "tensor", but it really isn't quite. There should be a separate article for tensor on a vector space, summarized here appropriately. Sławomir Biały (talk) 12:53, 16 August 2017 (UTC)

So that I don't misunderstand you, could you explain the difference you have in mind between the geometrical notion of a tensor, as different from both the notion of a tensor field on a manifold and the notion of a tensor on a vector space? — Carl (CBM · talk) 12:56, 16 August 2017 (UTC)

A geometrical object is a function of a coordinate system belonging to a certain class of coordinates, that transforms in a certain way when the coordinate system is changed to another one belonging to the same class. A tensor, in the usual sense, is a function of the coordinate system that takes values in a rational representation of GL(n), and transforms by the Jacobian of the coordinate transformation. Sławomir Biały (talk) 13:08, 16 August 2017 (UTC)

A scalar is certainly (identifiable with) a constant multilinear map from $V^{0}\times (V^{*})^{0}\to \mathbb {R}$, so it is a tensor in the sense of this article. However, if we associate a scalar with each point of a manifold (even the same scalar), so that we can integrate, then we have a tensor field (and a scalar field) rather than just a scalar. — Carl (CBM · talk) 11:53, 16 August 2017 (UTC)

A scalar is an invariant of the affine group, per the cited source. Sławomir Biały (talk) 12:03, 16 August 2017 (UTC)

Yes, and an element of $V^{0}\times (V^{*})^{0}\to \mathbb {R}$ is invariant under the natural action of GL(V).TR 12:10, 16 August 2017 (UTC)

Yes, all numbers are scalars. But you apparently believe that the opposite is true, that every scalar is a number. If a student said "A scalar is just a number", what would your response to that statement be? Sławomir Biały (talk) 12:13, 16 August 2017 (UTC)

I would ask them what definition of scalar they are using... which is likely the source of discussion here. If the student says that "scalar" means a tensor of type (0,0) on a vector space, then every such tensor on a vector space can be immediately identified with a single element of the underlying field of the vector space. I am sure that we all know this. The real question is how to present tensors on Wikipedia. I think that the current division is the most reasonable: we have one article for a tensor on a particular vector space, and another article for a tensor field, so that the latter article can point to this one. We certainly want to a void a situation where we have to say "a tensor is a map that assigns a tensor to each point of a manifold". This article can talk about how the array for a tensor changes under a change of basis; the other one can move to the situation where the change of basis is induced by the Jacobian of a coordinate change. — Carl (CBM · talk) 12:27, 16 August 2017 (UTC)

I don't really object to discussing tensors on vector spaces. But I think there are important differences between a tensor-on-a-vector-space and what most people would call a tensor. For example, a spinor is a tensor-on-a-vector-space, but we would not call it a tensor because it doesn't transform like one on coordinate transformations of the physical space. I think the emphasis on fields versus pointwise values misses this main conceptual problem. Sławomir Biały (talk) 12:56, 16 August 2017 (UTC)

To respond to your example. A spinor on V is not a tensor on V, it maybe a tensor on some vector space, but not on the vector space V. I starting to become convinced that the conceptual problem you see, only exists inside your head.TR 13:40, 16 August 2017 (UTC)

I'd admonish you a second time in this discussion to knock it off with the ad hominem personal attacks. Yes, a tensor-on-a-vector-space is a thing. But that does not mean that tensor-on-a-vector-space is the primary topic of this article. The primary topic is geometrical, not algebraic. Thus a spinor is not a tensor in the sense that anyone would actually use that word, even though it is a tensor-on-a-vector-space in the sense that you believe is the primary subject. This should give you pause: could it perhaps be that you're trying to fit a square peg into a round hole by identifying the primary topic to be tensor-on-a-vector-space? Furthermore, you here seem to be asserting that the vector space V is the physical space. Is that really consistent with what physicists do? Does one take the tensor product of physical space with itself? I'm unable to find any evidence that physicists actually do this. Sławomir Biały (talk) 14:10, 16 August 2017 (UTC)

I am asserting that V is both the physical space (i'd rather say geometric space as we are not necessarily talking physics) and the space of rank-1 tensors. The spaces of higher rank tensors are obtained by taking tensor products of the later (and its dual).TR 14:34, 16 August 2017 (UTC)

Yes, ok. It is the general linear group that acts, rather than the affine group. Sławomir Biały (talk) 15:20, 16 August 2017 (UTC)

Several points.

* There would be no need for this article if it were about tensor fields, since there is already a tensor fields article.

* The natural way to discuss densities here is with antisymmetric tensors of maximal rank and their products with other tensors.

* The cross product requires a metric on a three dimensional vector space. The natural way to define it is as the dual of the exterior product. Shmuel (Seymour J.) Metz Username:Chatul (talk) 16:03, 16 August 2017 (UTC)

Missing information

A lot of the language used in the article is not explained or linked (e.g. linear map, basis, etc ...) It reads like one math professor lecturing another (who is the actual audience? what foundation is needed to understand the subject matter?) Jwo7777777 (talk) 13:50, 28 August 2017 (UTC)

The items you mention (linear map, basis) are linked on first occurrence (lead). YohanN7 (talk) 14:00, 28 August 2017 (UTC)

Since tensors are some sort of generalization of vectors, some (basic) familiarity with vectors (and therefore bases) is assumed.TR 14:36, 28 August 2017 (UTC)

Requested move 25 October 2017

The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review. No further edits should be made to this section.

The result of the move request was: not moved. No prospect of consensus to move. But lots of good discussion on other ways forward. Andrewa (talk) 20:35, 1 November 2017 (UTC)

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Tensor → Tensor on a vector space – A review of the literature shows that the artificially imposed restriction on the topic of the article, to be only about tensor on a vector space, is not the primary topic of this article. For example, of the two and a half million scholar hits for "tensor", only a little over a hundred thousand also use the phrase "vector space". I note that this is a comparable number to the number of sources that use tensor along with the exact phrase "at a point" [1], a phrase which earlier discussion shows is reviled by those in the camp that believes that "tensor on a vector space" should be the primary topic. The presumptive topic of this article, which we have been largely told by a single editor, is "tensor on a vector space", yet that concept itself has a very small footprint in the available literature (see, for example, this search.) I note that even the most basic engineering calculation, that of a stress tensor computed at a point in spherical coordinates, is not permitted under the regime here. (See the reactions in an earlier thread where I mentioned the Jacobian.) A more specific literature review shows that such calculations are indeed fundamental to the primary topic of a tensor. Arfken and Weber, for example, consider tensors in curvilinear coordinates. So do Borisenko and Taparov, who introduce tensors first in a rectangular coordinate system, and then in curvilinear coordinates. Spiegel introduces tensors using the Jacobian of the coordinate transition in the second chapter of Schaum's outlines. I therefore propose that "tensor on a vector space" is indeed not the primary topic. It is reasonable to have an article "tensor on a vector space", ideally sourced to the literature where exactly that phrase is used, but it is unreasonable (and against the neutral point of view policy) to insist that the primary topic of tensor is that of tensor on a vector space (whilst citing sources for which this is plainly not the case). Sławomir Biały (talk) 19:10, 25 October 2017 (UTC)

• Oppose I make no comment on the issue of the scope of this article.

But for today, we need an article which answers the question "Tensor". This article is as close as we have as yet. It ought to stay at Tensor.

If you want to advocate expanding this article, then go for it. If the expanding article would then be worth splitting, you could advocate that in turn. But none of these support renaming the main article away from Tensor, which would anyway be a redirect back to wherever this article was at the time. We need an article at Tensor, and today we only have this article. Andy Dingley (talk) 19:37, 25 October 2017 (UTC)

Unfortunately, we have a regime here that feels (contrary to all cited sources) that the topic of this article is the artificially constructed "tensor on a vector space". Can I take this as a support for generalizing the subject of the article? Sławomir Biały (talk) 20:55, 25 October 2017 (UTC)

You mean the scope that you renamed it as? And now you're against that? Andy Dingley (talk) 10:40, 26 October 2017 (UTC)

No, that's already the scope of the article. I was renaming the article in accord with what TR feels is the subject of the article, as opposed to the primary tensor concept. (See the hatnote at the top of the article, although it does feel a little Orwellian being told the subject of an article in a hatnote.) Sławomir Biały (talk) 11:04, 26 October 2017 (UTC)

• Oppose The google statistics mean very little, as also when using "tensor" exactly in the sense as currently described in this article the term "vector space" is unlikely to be used. Moreover, the first search will also hit all uses of "tensor product", "tensor algebra", "tensor bundle", or any of the dozens of other scientific terms involving tensor.

Further the proposed target "tensor on a vector space" clearly does not pass WP:COMMONNAME which is policy, with GScholar providing just 70 hits for that phrase. (Because everybody describes the thing discussed in the current article "tensor".TR 19:42, 25 October 2017 (UTC)

Here are the scholar statistics for "tensor algebra", and here are the scholar statistics for "tensor bundle". Both are less than 20000 hits. Furthermore, tensor algebra, tensor product, and tensor bundle are subjects of other articles, not this one. Furthermore, against this evidence, for some reason it is "tensor field" that is excluded from your preferred revision of this article, while "tensor bundle" and "tensor algebra", despite having separate articles, are fair game for the subject of this article. (Tensor products are not even the same thing as this article, by the way.) Sławomir Biały (talk) 21:19, 25 October 2017 (UTC)

• I think the mathematical issue is well understood: the word "tensor" is used to mean two things. The first is a particular kind of multilinear map. Let's invent a new term and say that, given a vector space V over F and natural numbers p and q a tensor-1 of type (p,q) is just a multilinear map from $(V^{*})^{p}\times V^{q}\to F$. The second is a field that assigns a tensor-1 to (the tangent space of) each point of a manifold. These fields are called either "tensor fields' or just "tensors" (as our article says, "In some areas, tensor fields are so ubiquitous that they are simply called 'tensors'.") Because the only standard term for a tensor-1 is "tensor", while the field kind can be called either a "tensor field" or a "tensor", it makes more sense to me, keeping with WP:RS, to name our article about a tensor-1 with the name "tensor", and to name our article about the field with the name "tensor field". — Carl (CBM · talk) 19:58, 25 October 2017 (UTC)

I agree completely. Also to be clear (to other readers), this is the current status quo. We have two articles tensor (discussing "tensor-1") and tensor field (discussing tensor fields).TR 20:04, 25 October 2017 (UTC)

There is a third too, which is prevalent in sources. A tensor at a point transforms under the Jacobian of the coordinate transition at the point. A tensor on the tangent space is of this kind. Yet we've been repeatedly told by a single editor that "tensor at a point" is meaningless, and Jacobians don't come into it. That's contrary to sources and the neutral point of view policy. We should summarize sources. If sources don't enforce the artificial "tensor on a vector space" ideal, then neither should the main article on the subject. But, just so we're clear: If the sources cited in the article do not concern the "tensor on a vector space" concept, is it appropriate for the article to specialize to that case? If so, how is presenting these sources out of context consistent with the neutral point of view. If not, what is the way forward? Sławomir Biały (talk) 20:55, 25 October 2017 (UTC)

"A tensor at a point" clearly is a tensor on a vector space (the tangent space in this case). This wider context however is only sensible in the sense that a "tensor at a point" is the value at a point of some tensor field, and is more naturally discussed in that article. TR 21:47, 25 October 2017 (UTC)

This view is inconsistent with most sources I am familiar with in this subject, as well as many of the sources in the current article. If you wish to specialize the subject of this article, could you please compile a list of approved sources that you believe support your vision of the primary topic here? Sławomir Biały (talk) 22:00, 25 October 2017 (UTC)

Also, I don't think invoking the tangent space at a point makes the Jacobian go away. One still needs to say in what sense tangent vectors at a point are tensors (with respect to coordinate transformations of the manifold). Sławomir Biały (talk) 02:45, 26 October 2017 (UTC)

It is only your view that the current presentation of the article is not supported by the sources cited. You have yet to convince anybody of that fact.

As two your second point, if you uses a coordinate basis for the tangent space, then clearly the basis transformation induced by a change of coordinates is the Jacobian, and yes if you are using that coordinates basis to represent a tensor on the tangent space, then the tensor will transform with the Jacobian. However, the tensor could have also been represented in a non-coordinate basis and then there is no link with the Jacobian. My point here is that the notion of a Jacobian isn't intrinsically linked to that of tensor, but rather with coordinate bases for the tangent space. My objection to discussing Jacobians here, is that they require a lot of additional context (tangent spaces, coordinate bases) that are only relevant to (a fairly common) subset of tensors (namely those defined on a tangent space), while being completely irrelevant to tensors on other vector spaces such as commonly used in machine learning or quantum information theory. Diverting in to such a context would be (IMHO) but distracting and confusing for general readers. More so, when we have several other related articles where this context is much more natural such as tensor field and tensor bundle.TR 07:29, 26 October 2017 (UTC)

We don't currently cite any sources that discuss tensors in machine learning or quantum information theory. In quantum information theory, it seems that it is the tensor product of Hilbert spaces (or possibly that tensor product) that is relevant, not the primary topic of tensor. Not every element of a tensor product would be called a "tensor" by most mathematicians and physicists, niche uses of the word notwithstanding. But surely it is less confusing to the reader to point out that this niche use is different from the majority of the literature, rather than pretend that there is uniformity. But if, on the other hand, you feel that the article should be specialized to "tensor on a vector space", then portions of the article that use sources which are about the traditional notion of tensor (what I would call the primary topic) and sourced to (what I would call) standard references, should be moved to a different article because they are about a different topic than this article. But I am a little uncertain what would remain of the article in that case. So I think it would be very helpful if you could list some sources you feel directly support your vision of the primary topic of this article. Most of the citations that I added to the article do not. Sławomir Biały (talk) 10:25, 26 October 2017 (UTC)

There already is a different article, Tensor field. Shmuel (Seymour J.) Metz Username:Chatul (talk) 21:57, 26 October 2017 (UTC)

• Clarification/Support: Am I correct to understand that this is a proposal to split the article into two: one about a tensor on a vector space and the other on the notion of tensor in general? Certianly tensor makes sense in more general setup (e.g., tensor product of modules) so the proposal makes sense to me. Alternatively, it seems, we can also broaden the scope of the article, though I don't if that's workable. -- Taku (talk) 21:41, 25 October 2017 (UTC)

Partly correct. At issue here is what the primary topic is. When the literature says "Let T be a tensor," what is usually meant by that? One wouldn't ordinarily call something in a random tensor product of modules a "tensor". But TR feels, against all textual evidence, that this sentence usually means that T is a tensor on a vector space. In particular, that it is meaningless to involve Jacobians. I think there are really two concepts: one is the primary topic "tensor", and the other is "tensor on a vector space". It's a mistake with no support in reliable sources to identify these.

Tensor product of tangent sheaves (and cotangent sheaves) might approximate the modern view. "Natural sheaves" might plausibly be included in the tensor panoply, as much as tensor products would be. (See, for instance, this paper by Eck.) I don't know what the current state of the art is. Sławomir Biały (talk) 02:04, 26 October 2017 (UTC)

It seems to me that the issue with Jacobians only gives certain kind of tensor field (much as we could also look as continuous or smooth tensor fields), and doesn't really give a different kind of object. In any case, it seems to me that the key issue is first whether there should be an article on the tensor-1 concept, and another one on the tensor field concept, and if so what they should be titled. I think you are saying you prefer this article to be about tensor fields. But then what sourceable title would we use for the tensor-1 concept? On the other hand, I don't personally see it as a large issue to call the article on tensor fields by the name "tensor field", even is many sources just say "tensor" when they mean a tensor field or some particular kind of tensor field. When the same name is used for two different things, at least one of the two articles will need to have a different title. — Carl (CBM · talk) 00:08, 26 October 2017 (UTC)

I hope this edit clarifies the sense in which the distinction between "tensor" and "tensor field" is artificial and unhelpful. Sławomir Biały (talk) 01:34, 26 October 2017 (UTC)

@Sławomir Biały: Am I correct to summarize your position as saying that the notion of "tensor on a tangent space" is/should be the primary topic of this article? If not, I still don't properly understand you position is, which clearly hampers our ability to have a constructive discussion. (From your various comments above it is also clear that you misunderstand what my position is, which we also need to clarify.)TR 07:48, 27 October 2017 (UTC)

My position is that the primary topic of the article should include the idea that tensors are geometric objects, not just algebraic objects. So, a tensor is a function of the coordinates that transforms in a certain way when the coordinates are subjected to a transformation belonging to a certain class. I believe this is consistent with most treatments in elementary sources, like Arfken and Weber, Borisenko and Taparov, or Spiegel, but also in many more advanced sources, like Schouten, McConnell, Sharpe, or Kobayashi and Nomizu. Furthermore, the idea of tensor on a vector space is not terribly consistent with the elementary idea of a vector one encounters in elementary physics, because there one is concerned with the Galilean transformations of an affine space rather than a vector space. But also, in the Galilean picture of physics, vectors can moonlight as (constant) vector fields, so that we may (for example) compute the relative velocity of two spatially separated particles.

Regarding a tensor as a geometrical object in this way also blurs the distinction between tensor-on-a-vector-space and tensor field. Some sources do scrupulously enforce this distinction, but most do not. This is not mere laziness on the part of some physicists (that were until recently disparaged by the article). The difference between one and the other is the class of allowed transformations: on the one hand, only affine (or linear) transformations of the coordinate system are permitted, and on the other hand more general smooth/differentiable/analytic/algebraic transformations are permitted. But it's still the same essential geometrical tensor idea, and at least part of what this article should be about that, rather than trying to enforce the hatnote artificially and disparaging all sources that do not use one of the phrases "tensor product" or "multilinear function" because these do not match our preferred definitions of things.

On an affine space, the tensor transformation law is the general rational functorial one (densities exist too, but these are either already tensorial or non-rational, depending on the degree). So in that case, one does not need to specify what sort of transformation law counts as a "tensor". But conventionally one does, and that requires the Jacobian: a tensor transforms under the linear part of the affine transformation. This is useful in practice because it allows us to write down formulas, but also it's precisely the same thing needed to make sense of the tensor transformation law in curvilinear coordinates. In the Galilean picture, a fixed, constant tensor can be expressed in terms of a general coordinate system, such as spherical coordinates. So, in both cases, the Jacobian is an essential ingredient: without the Jacobian, there is no practical way to write down the transformation law for a tensor, and also no way to distinguish between a tensor and more general kind of geometrical object like a spinor or jet.

Many things can be forced into the created mould of tensor on a vector space, but I think that obscures an essential geometrical aspect of the tensor concept. In the Galilean picture of physics, there is a vector space of equivalence classes of affine vectors, modulo parallel transport. In differential geometry, one has the option of constructing the tangent space intrinsically first, and then regarding tensors at that point through the tensor algebra. In algebraic geometry, one often constructs the cotangent space first, using maximal ideals. My own point of view though is that such constructions provide models of the geometrical concept of "tensor", but tensor-on-a-vector-space, tensor-on-an-affine-space, tensor-on-a-manifold, tensor-on-an-algebraic-variety, regardless of the model, are merely special cases of this general geometrical tensor concept.

There is a danger of falling down the slippery slope into regarding every nominal phrase in which "tensor" appears as a compound modifier is then equally well a tensor: "tensor product", "tensor algebra", "tensor bundle". For example, the tensor product of Hilbert spaces used in quantum information theory may or may not be regarded as a "tensor". Perhaps an even more extreme example is the buzzword "tensor" as it is applied in machine learning, to refer simply to anything with indices. While my own preference would be to exclude such niche uses of the term as not really a part of the primary topic, as they bear a very limited relation to geometry, I am fine with a somewhat inclusive view on such matters. However, I do feel that the primary topic of this article should focus mostly on what appears in high quality reliable sources as "A tensor is..." (or "Such and such generalizes tensors...") without judging its suitability for this article based on a hatnote, particularly not when things of far more questionable provenance apparently get a pass. Sławomir Biały (talk) 12:11, 27 October 2017 (UTC)

"So, a tensor is a function of the coordinates that transforms in a certain way when the coordinates are subjected to a transformation belonging to a certain class." - I would view that as an algebraic viewpoint, in a different sense, because it focuses on coordinates instead of geometry. For me, the geometric viewpoint would be that a tensor-1 is just a kind of multilinear map (no need to mention coordinates) and that a tensor field is a section of some appropriate tensor-1 bundle (again, no reason to mention coordinates). Of course, many texts do present tensors in terms of coordinates, which are needed to perform calculations, and so we should discuss coordinates here are well. But definitions that directly refer to coordinates are not very geometrical in my opinion. This is one reason why I think this article should keep to the tensor-1 concept: because it is more straightforward to talk about the geometrical nature of the map when we are not simultaneously trying to talk about coordinate transforms. Perhaps this is the difference in viewpoint that we have. — Carl (CBM · talk) 15:58, 27 October 2017 (UTC)

I think that view is a bit naive, and not really supported by the best sources. "Geometric object" is something that has a precise mathematical definition, and was formalized in the 20th century, though mostly associated with Albert Nijenhuis; for a modern account, see [2]. What makes a tensor geometrical is functoriality (in the coordinates, or under linear transformations of the coordinates, diffeomorphisms, etc), not the fact that there is a "tensor product" appearing in the definition somewhere. This is also true of the standard definition of tensors in differential geometry, as equivariant functions on a principal bundle. Even in the most rudimentary applications, tensors are usually regarded as living in an affine space (or even on a manifold), where there isn't a linear structure on which to base "multilinear map". (One still needs to identify a basic space of tensors: in one case, it's the translation group; in the other it's the tangent space, or infinitesimal translation group, which itself is characterized by its invariance properties under a pseudogroup of transformations.) So, you simply have it backwards (per reliable sources) that "multilinear map" is geometrical, while transformation law under the coordinates is algebraic. Geometry isn't about the vector space to start with; there's a vector space of tensors because of the geometry. Sławomir Biały (talk) 16:35, 27 October 2017 (UTC)

I was not referring to any kind of functorial definition of "geometrical", but to the usual informal terminology in which (for example) a linear map on a vector space is a geometrical object, unlike the matrix that represents that map relative to a basis. Similarly the vector space itself is a geometrical object, even before any basis is chosen for it. I'm finding this discussion difficult because I feel that your responses don't get quite to the point of mine, and perhaps you feel the same way. In any case, I don't have a dog in the fight here, and I am just going to walk away from this RFC. However, I think that the current emphasis in this article on a coordinate-based viewpoint does a disservice to the geometrical picture of tensors, and the proposal would move it farther from my vision of how this article might be arranged; for that reason I can't bring myself to support the page renames. — Carl (CBM · talk) 17:24, 27 October 2017 (UTC)

Yes, and my point is that even though one can formulate the idea of a tensor-on-a-vector-space, the vector space may or may not be a geometrical thing. There is a difference between a geometrical vector and the idea of a vector space. Yet the philosophy of the regime at this article seems to be to regard vector spaces as the same thing as geometrical vectors, which is wrong in my opinion. These have separate entries, for example, in Ito's Encyclopedic dictionary of mathematics, as well as the Springer Encyclopedia of mathematics (in addition to Wikipedia). Sławomir Biały (talk) 17:45, 27 October 2017 (UTC)

I want to chip in a bit since I think I understand (little bit) what by "geometric" Sławomir Biały wants to mean. I would personally prefer the adjective "natural" to what he (or she?) means by "geometric"; that is, a tensor is a sort or example of a natural object, "natural" in mathematicians' jargon. The notion of "tensor field" for instance can be derived from the naturality requirement (i.e., being a section); in concrete terms, that's about Jacobian, I think. For me, geometry must involve a space or a variant; otherwise, everything natural must be geometric, which is an arrogant supposition :( (Algebraic approaches can also be capable of expressing required naturalness mentioned above.) Anyway, how about creating an article tensor (geometric sense)? That article can cover what you referred to as "general geometrical tensor concept". If I understand correctly, a tensor field is just one example of such a notion. (Again, I personally think "geometric" here can be subsumed into "natural" but that's another matter.) For the readers, what truely matters is content not title. We can then debate which article is a primary one to acquire the coveted title "tensor". -- Taku (talk) 23:11, 27 October 2017 (UTC)

I think forking the article would do a disservice to our readers, as despite our disagreements about various aspects such as defining features and scope, the main objects that we wish to describe are still the same. I reader mostly unfamiliar with tensors looking it up on Wikipedia should not be presented with the choice of two article about almost the same topic which he/she should read both before knowing which was the one he/she was looking for.TR 08:03, 1 November 2017 (UTC)

@Sławomir Biały: I think I better understand your position. My position is that the current "tensor on a vector space" scope of the article encompasses the notion of a "tensor on a tangent space" (a tangent space is a vector space after all). I appreciate your view the "tensor on a vector space" concept does not include some the naturalness features of "tensors on a tangent space" that you view as central to the definition of a tensor. (However, it should be noted that those features are in fact naturally recovered when applying the "tensor on a vector space" concept to a tangent space.) I also appreciate that there many sources sharing your view (to varying degree). There are however also many notable sources that first define "tensors of vector spaces" before specializing to the case of "tensors on a tangent space" (some examples currently note cited in the article include Wald's General Relativity and Nakamura's Geometry, Topology, and Physics both considered standard texts in modern theoretical/mathematical physics.)

My main objection to many of your previous additions was that they would specialize to the narrower scope of "tensors on a tangent space" without warning to the reader or providing the appropriate context. My proposal would be to remedy this by adding section on "tensors on tangent spaces" (we may need a better name) that both provides this context and can explain a bit about naturalness aspects of these objects to you (and some sources) place at the center of the tensor concept. This would also make it easier to allude to this notion of tensor later in the article. Would this solution work for you as well?

Finally, I think you are being a little to dismissive about the use of tensors in quantum information and machine learning. Although not always apparent these uses do tend to build on the "tensor on a vector space" concept. See for example [3] for a review on the use of tensor networks in quantum information. Very similar tensor network construction appear in machine learning. We probably should make some mention of these in the Applications section, although I don't feel at home enough in either field to appropriately summarize their use of tensors.TR 09:06, 1 November 2017 (UTC)

It is in some sense already the status quo, so I don't think any special rewrites are needed. Sławomir Biały (talk) 18:21, 1 November 2017 (UTC)

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The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.

A different question

Should this article follow the sources, or is it appropriate to restrict the scope based on our editorial opinion of the primary subject? Sławomir Biały (talk) 21:36, 25 October 2017 (UTC)

At least we can hopefully all agree that integrating over a vector space or affine space is meaningful, so that the definition of a scalar density makes sense for the subject of this article (whatever that may be). Sławomir Biały (talk) 12:08, 30 October 2017 (UTC)

No, we can't; a scalar density is a field on a manifold. See Tensor field#Tensor densities and Tensor density. Shmuel (Seymour J.) Metz Username:Chatul (talk) 20:59, 31 October 2017 (UTC)

So is a tensor on a vector space, though. A vector space is a manifold too. Sławomir Biały (talk) 11:32, 1 November 2017 (UTC)

A vector space is not a manifold, although a manifold structure can be imposed upon it. For finite dimensional real or complex vector spaces there is a canonical manifold structure, but that is not true in the infinite dimensional case. Shmuel (Seymour J.) Metz Username:Chatul (talk) 22:41, 2 November 2017 (UTC)

This article is very explicitly about finite dimensions, so this is just a red herring. Also, the failure of scalar densities to make sense in infinite dimensions has nothing to do with manifolds, but rather the failure of the determinant to make sense in infinite dimensions. In any case, excluding scalar densities from this article because they do not generalize in a standard way to infinite dimensions is really begging the question. Neither does much else in tensor analysis. Sławomir Biały (talk) 14:46, 3 December 2017 (UTC)

Units

It seems to me that we should have a paragraph on the units associated with tensors. This was removed as unclear rather than improved. I think it should be improved instead. Here is the text that I added, improvements are of course welcome:

In classical mechanics, tensors have units the spatial (length) dimensions associated with a tensor of order (p,q) are length^q−p. For example, force is a tensor of order (0,1), and has units of kg m/s², whose length dimensions are length¹. Likewise, the stress tensor takes a vector (units of length) as input, and produces a pressure (units of force per area) as output. So the spatial dimensions of stress are length⁻².

--Sławomir Biały (talk) 14:36, 3 December 2017 (UTC)

It pains me to disagree with one of the regular and very productive members of the math and physics projects. However, your section is simplistic and wrong.

For example, consider velocity and acceleration. They are contravariant vectors in 3D (freezing out the time dimension), i.e. they are (1,0) tensors. But their units include Length⁺¹. This contradicts your formula. On the other hand, force (regarded as the negative of the gradient of a potential energy) is a covariant vector, i.e. a (0,1) tensor as you said. (Potential) energy itself has units including Length⁺² and yet it is a scalar, i.e. a (0,0) tensor.

There is no royal road here. You must transform all your physical equations to a curvilinear coordinate system and see what you get. JRSpriggs (talk) 05:06, 4 December 2017 (UTC)

Ok, fair enough. It seemed prudent to have some discussion of dimensions, but it probably does more harm than good. Sławomir Biały (talk) 16:59, 4 December 2017 (UTC)

Tensor

"Given a reference basis of vectors, a tensor can be represented as an organized multidimensional array of numerical values". The term "basis" does not seem correct. Other sources use the term "indices" to refer to the ... indices. However AFAIK an indices and a basis of a vector space are different things unless I just can't see how they are connected? -- 136.186.248.251 (talk · contribs) 06:56, 28 January 2018‎ (UTC)

Please look up the notion of a matrix of a linear transformation. In particular we don't mean just a basis, but an ordered basis: a basis paired with an indexing of its members.--Jasper Deng (talk) 07:42, 28 January 2018 (UTC)

What is "isospace"?

A lot of papers about tensors seem to talk about isospace. I would like to define it for Wiktionary. Any hints? Equinox ◑ 11:44, 25 April 2018 (UTC)

A shorter version of my proposal

[I'm withdrawing this shorter proposal as well.]

A (shorter) proposal for a new lead section. I'd like to continue this discussion with a shorter proposal, which (at the very least) will help me consider how I might be able to improve the current version of the lead section, if my proposal is not adopted. (It seems that the Wikipdedia style manual prefers "lead" to "lede". I'm also trying to follow the guidance of this style guide, by providing a clear, constructivist definition of "tensor" as the first sentence.) I'd appreciate any further comments or suggestions that people have.

(short version 1)

In mathematics and physics, a tensor is a geometric object, constructible as an element of tensor products of a geometric vector space and its dual, that can act as a (multi-)linear map from one or more kinds of tensors to other tensors. The term "geometric" here means that the object relates to a geometric or spatial space (rather than an abstract space with no geometric meaning) and that the properties of the object are independent of coordinate system or basis vectors. The number of spaces $n$, both original and dual, that are tensor-multiplied to describe a tensor is called the order (also called degree or rank) of the tensor.

The concept of a tensor is a generalization of geometric scalars and geometric vectors (not simply scalars and vectors as aspects of general vector spaces) as well as the duals of geometric vectors, which are linear mappings. Geometric scalars and geometric vectors and their duals are the simplest kinds of tensors. The purpose of the tensor concept is to encapsulate many related geometrical linear relationships and invariants into an efficient abstraction, data structure, and notation.

A tensor is expressible as an array of numbers or quantities, possibly a single-element array or a multi-element array, described in more detail further in this article. One point of confusion to clarify is that although a tensor can be represented as an array, an array is not necessarily a tensor; the array must be of or expressible in a particular shape and must have particular transformation properties under change-of-basis to qualify as a tensor. (See below for a more general mathematical term for an arbitrary array.)

Examples of tensors include:

• the dot product (also known as the scalar product) of two (geometric) vectors, which is a (geometric) scalar -- a tensor of order zero;
• the charge of an elementary particle, a scalar (which can be negative) that can act as a linear map from electrical field vectors to a space of electrical force vectors ($\mathbf {F} =q\mathbf {E}$);
• the velocity of an object, a first-order tensor and geometric vector;
• the Cauchy stress tensor T, a second-order tensor that can take a directional unit vector n as input and map it to the stress vector T⁽ⁿ⁾ across an imaginary surface orthogonal to n -- this relationship between vectors is expressible as a matrix multiplication or tensor contraction, as shown in the figure (right).
• measures of curvature of a smooth curve, surface, or manifold, which can be a scalar, vector, or higher-order tensor quantity.

Examples of objects that are not tensors include:

• the vector cross product of two (geometric) vectors, which seems to be a tensor (geometric vector) under basis rotations (since it doesn't doesn't change under rotations) but is called a pseudotensor (and pseudovector) because it is altered by a change of basis when altering the right-hand rule convention to a left-hand rule;
• the magnetic flux over a surface, a dot product between a vector (the surface normal) and pseudovector (the magnetic field), which is called a pseudoscalar because it also is not a true geometric scalar as it is altered by certain changes of bases;
• an array containing data pertaining to some non-geometric domain, such as the ages of people in a sequence of people;
• an arbitrary array, which does not necessarily have the same shape or transformational properties of a tensor.

Although tensor fields and tensor densities are sometimes simply called tensors, they are technically not tensors. A tensor field is a function that returns a tensor at every point in a geometric vector space or a manifold, and a tensor density is a generalization of the tensor field concept. (This is similar to the case of geometric scalars and geometric vectors ambiguously being called scalars and vectors.)

Tensors were conceived in 1900 by Tullio Levi-Civita and Gregorio Ricci-Curbastro, who continued the earlier work of Bernhard Riemann and Elwin Bruno Christoffel and others, as part of the absolute differential calculus. The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.

Zeroparallax (talk) 23:03, 26 September 2018 (UTC)

Proposal for new lead

[I'm withdrawing this long version of my proposal, and I'll post a shorter version in a new section.] I'd like some feedback, particularly from Purgy Purgatorio and JRSpriggs. Here's an idea for a rephrasing of the lead (but will require some verifying references):

(Edited: version 10, with more links, clarification, rearrangement, and incorporating more of original):

In mathematics and physics, a tensor is a geometric object, constructible as an element of tensor products of a geometric vector space and its dual, that can act as a (multi-)linear map from one or more kinds of tensors to other tensors. The term "geometric" here means that the object relates to a geometric or spatial space (rather than an abstract space with no geometric meaning) and that the properties of the object are independent of coordinate system or basis vectors. The number of spaces $n$, both original and dual, that are tensor-multiplied to describe a tensor is called the order (also called degree or rank) of the tensor.

The concept of a tensor is a generalization of geometric scalars and geometric vectors (not simply scalars and vectors as aspects of general vector spaces) as well as the duals of geometric vectors, which are linear mappings. Geometric scalars and geometric vectors and their duals are the simplest kinds of tensors. The purpose of the tensor concept is to encapsulate many related geometrical linear relationships and invariants into an efficient abstraction, data structure, and notation.

A tensor is expressible as an array of numbers or quantities, possibly a single-element array or a multi-element array, all having hypercubic (that is, $n$-cubic) shape with $n$ indices to label the element or elements, where $n$ is the tensor's order. Given that the tensor's associated vector space is of dimension $D$, the tensor's array will have $D^{n}$ elements. For $n>0$, a set of basis vectors must be chosen to determine the expression of the tensor as an array; for $n=0$, no choice is necessary. For a 0th-order tensor ($n=0$), the single-element tensor is called a (geometric) scalar, and the "tensor product" of the vector space is like a "tensor exponential" with exponent zero, yielding the vector space's field. (The scalar is an element of that field.) For a 1st-order tensor ($n=1$), the tensor is called a (geometric) vector. For $n\geq 2$, there is no special term other than "tensor". (Note that the term "matrix" does not refer to a geometric object; instead it refers to a particular kind of data structure: a rectangular array, usually arranged in rows and columns. So a 2nd-order tensor can be expressed as a matrix, but a matrix isn't necessarily a tensor.) Since the behavior of a tensor is independent of choice of basis vectors, that implies that the array representations of tensors must transform in particular ways under a change of basis. That means that although a tensor can be represented as an array, an array is not necessarily a tensor; the array must be of or expressible in a particular shape and must have particular transformation properties under change-of-basis to qualify as a tensor. (See below for a more general mathematical term for an arbitrary array.)

Examples of tensors include:

• the dot product (also known as the scalar product) of two (geometric) vectors, which is a (geometric) scalar -- a tensor of order zero;
• the mass of an object, a scalar that can act as a linear map from a space of gravitational (would-be) acceleration vectors to a space of gravitational force vectors ($\mathbf {F} =m\mathbf {g}$);
• the charge of an elementary particle, a scalar (which can be negative) that can act as a linear map from electrical field vectors to a space of electrical force vectors ($\mathbf {F} =q\mathbf {E}$);
• the velocity of an object, a first-order tensor and geometric vector;
• the dual of a geometric vector, a first-order tensor that acts as a map from vectors to scalars via the dot product;
• the inertia tensor of a rigid object, a second-order tensor relating angular acceleration to torque;
• the Cauchy stress tensor T, a second-order tensor that can take a directional unit vector n as input and map it to the stress vector T⁽ⁿ⁾ across an imaginary surface orthogonal to n -- this relationship between vectors is expressible as a matrix multiplication or tensor contraction, as shown in the figure (right).
• the (multi-linear regime) stiffness tensor, a fourth-order tensor that when negated and multiplied with the strain (deformation) tensor yields the stress (force per area) tensor: ${\boldsymbol {\sigma }}=-\mathbf {c} {\boldsymbol {\varepsilon }}$ -- a multi-dimensional, anisotropic analogue of the one-dimensional Hooke's "law" (linear regime for stiffness of an elastic medium: $\mathbf {F} =-k\Delta \mathbf {x}$).
• measures of curvature of a smooth curve, surface, or manifold, which can be a scalar, vector, or higher-order tensor quantity;

Examples of objects that are not tensors include:

• the vector cross product of two (geometric) vectors, which seems to be a tensor (geometric vector) under basis rotations (since it doesn't doesn't change under rotations) but is called a pseudotensor (and pseudovector) because it is altered by a change of basis when altering the right-hand rule convention to a left-hand rule -- further vocabulary distinguishes a geometric vector (a "true vector" or "polar vector") from a pseudovector (or "axial vector");
• the magnetic flux over a surface, a dot product between a vector (the surface normal) and pseudovector (the magnetic field), which is called a pseudoscalar because it also is not a true geometric scalar as it is altered by certain changes of bases;
• an array containing data pertaining to some non-geometric domain, such as the ages of people in a sequence of people;
• an arbitrary array, which does not necessarily have the same shape or transformational properties of a tensor.

Although tensor fields and tensor densities are sometimes simply called tensors, they are technically not tensors. A tensor field is a function that returns a tensor at every point in a geometric vector space or a manifold, and a tensor density is a generalization of the tensor field concept. (This is similar to the case of geometric scalars and geometric vectors ambiguously being called scalars and vectors.)

Tensors were conceived in 1900 by Tullio Levi-Civita and Gregorio Ricci-Curbastro, who continued the earlier work of Bernhard Riemann and Elwin Bruno Christoffel and others, as part of the absolute differential calculus. The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.

Zeroparallax (talk) 11:52, 25 September 2018 (UTC)

In my recent rewrite of the lead I tried to cling as much as I could both to the structure, as to the content of the previous version. I admit that I personally prefer a view on tensors as multilinear maps to the view as arrays of numbers, even when being fully aware that every concrete calculation requires some numbers. In my estimation I added just connections between these views (stress tensor), explicated the examples (order), and hinted to a generous interpretation of the cross product (Levi-Civita); and, yes, I expanded the links to physical apps. Asking me to criticize your draft ... is like "Hey, madam, throw away your baby, I make you a new one, much prettier ..." :p

Seriously, I perceive your suggestion

• as way too much biased in favor of arrays of numbers (too much literature on merates in holors? ;) ),
• containing too much physics (charge, stiff, strain, stress, Hook, acceleration, electrical, gravitational force, flux, EM-field, ...),
• introducing all those pseudo/axial/polar/anisotropic/... objects and leaving them unexplained,
• as only sloppily discerning the paradigmatic tensors from the objects they act upon in the given examples (these are all tensors in the end),
• as being sometimes off the appropriate math rigor and consistency in terminology (e.g.: "geometric scalar"; BTW, do you remain undecided about "geometric" at all: "simple/geometric scalars"?)

I am aware that my list is not really nice, but, please, rest assured, I'm just describing frankly my impression and do not intend any offense. I am also aware of weak points in my version. Purgy (talk) 15:27, 25 September 2018 (UTC)

Before using the term local you need to explain the difference between a tensor and a tensor field.

The cross product of two 3-d vectors is a vector, but it lives in a different vector space. The details are too complicated to belong in the lede.

Similarly, axial, polar, wedge product and such pertain to fields and IAC are too complicated to belong in the lede. I was tempted to refer to line bundles, but, again, that does not belong in the lede. Shmuel (Seymour J.) Metz Username:Chatul (talk) 19:16, 25 September 2018 (UTC)

Purgy, thanks for your feedback! I can tell that you would like to see a clear, accurate, and well-balanced article. I appreciate your efforts of re-writing the article. Contributing to a collective writing process requires that we all have some humility to see our writing rejected outright (perhaps my case) or changed (as those who came before you saw their writing change, and as we will see it inevitably change again later). The hope and the effort is that over time the writing will become better than we can now imagine, and even we will be later enlightened by the new phrasings. Maybe together we can write something better than either of us could alone.

I'll add some links now to help clarify some of my meanings. (For instance, there is a difference between (geometric) scalars as tensors and scalars defined simply/generally as elements of a field, used in defining a vector space. And there is a difference between (geometric) vectors as tensors and vectors defined simply/generally as elements of a vector space, whether that vector space refers to a geometric, spatial space or some abstract space.) I'm also adding some clarification regarding "geometric": that for a "geometric object", "geometric" implies that the object relates to a geometric or spatial space.

I'll respond to more of your points soon. (And Chatul, thanks! I'll respond to you as well.)

Zeroparallax (talk) 19:41, 25 September 2018 (UTC)

Purgy and Chatul, first I'll explain my overall perspective. It seems that the lead section should cover what a tensor is, what a tensor isn't, examples, and maybe some reference to related topics. The fundamental definition of a tensor is geometric and not related to arrays, so I think it's proper to start with that (unlike many authors). Regarding arrays, I think it's very important to discuss this aspect in the lead for a couple reasons: it makes the ideas more concrete, and most people who work with tensors will predominantly be seeing and using arrays of some kind. (I think showing the particular transformation rules is secondary to knowing *why* there are particular transformation rules in the first place.) Additionally, the term "tensor" is a fairly common mathematical term that seems to have more confusion surrounding it than any other common term. People loosely know the definition and then they contradict themselves when describing a tensor concretely. This is because they actually have two contradictory definitions in mind: one as a (multi-)linear mapping (with implicit or explicit independence of basis), another as an arbitrary array. So I find it to be very important to confront common confusion immediately in the lead, so that people don't just read the overview as usual and walk away with the same contradictory "understanding" they had when they arrived to read it.

Purgy: If you have some non-physics examples, feel free to suggest them. Most of the interesting non-physics examples I know of are actually tensor fields in the context of differential geometry. Regarding any of my sloppiness, could you point out a specific example and maybe suggest how it could be cleaned up?

Purgy and Chatul, I mention and very quickly explain pseudotensors because they are very common (like the cross product), and it is extremely easy to be under the false impression that they are tensors. I find it to be reasonable to make a quick, concise reference to these ideas and leave the details in the links to the full articles.

Chatul, thanks for your suggestions. I removed "(local)", because this article is about tensors, not tensor fields. I actually think a quick reference to line bundles or some other differential geometry terms would be entirely valid, if they are made concisely in a manner that gives a sense of the full context of tensors.

Zeroparallax (talk) 22:31, 25 September 2018 (UTC)

Scalars are tensors, but they are much better known to people as just ordinary real numbers. So they should not be used as examples (except maybe once) of tensors. Similarly for ordinary vectors. Transformation matrices are much better examples of tensors. And make sure to mention the creme-de-la-creme of tensors, the Riemann–Christoffel tensor. JRSpriggs (talk) 03:11, 26 September 2018 (UTC)

@JRSpriggs: Thanks for your thoughts! In my shorter version (now posted below), I'm following your advice by removing a scalar example and adding reference to curvature tensors (and maintaining the link to the Riemann curvature tensor / Riemann-Christoffel tensor). However, since the nature of a scalar (relationship) like a dot product and mass seem to be quite different, it seems useful to refer to both of them to show some of the richness of the tensor idea.

Zeroparallax (talk) 23:19, 26 September 2018 (UTC)