Talk:Tensor

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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)

Discussing a new lead

The above thread is getting too long to my taste

I think that the suggestion for a new lead packs too many diverging intentions together. Chatul rightfully addresses locality and mentions the cross product as leading to another vector space, which it is only in another ballpark. The same wrt ballparks I think about the concepts of manifolds, tensor fields, orientation, volume element and Hodge dual: I prefer not to see these mentioned in the lead of this article, and neither any symmetry groups and their representations. This is not to say that I would not wish to know all about these, but I think that this first contact article should not venture (in the lead!) into such spheres, unavoidably necessary in physics apps. I also see no advantage in mentioning the details of the tensor algebra, nor the finicky properties of the tensor product, not even of vector spaces and their duals, and also not the categorical aspects of universal properties as are fundamental for a mathy view on tensors.

I think that all the above mentioned concepts are necessary to talk about the full blown tensor experience in physics or math, but I plead for keeping the content of this lead restricted to the introductory level that is sketched in the hat-note, and that I tried to feel bound to. I see no urgent encyclopedic need to discriminate arbitrary number collections from tensors, nor is it to me necessary to have ordered bases for stress tensors, nor to talk more about the non-tensoriality of the cross product than I did. The Levi-Civita is just a near example for an order three tensor. BTW, for obvious reasons, we have no metric tensor and no index juggling either, no mentioning of this being free in Euclidean space, ... There is so much more, sensibly left out of this article.

I take the freedom to explain my reservations about details along this example snippet:

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 product, as shown in the figure (right).

Tensors have not been introduced as maps, now they map. There is no reasoning about the given order of the tensor. There is no "imaginary property" of the plane to which the result refers to. The stress -especially with tensors- is not across a plane, but in a point (the caption is still sloppy, but compare the content, please), just the orthogonal decomposition is wrt to the normal/the plane. A tensor product product of tensors more precisely 09:50, 27 September 2018 (UTC) is distinctly different to matrix multiplication, certainly not simply an alternative ("or"). I'd prefer to rather explain this example in detail than to give that lot of additional ones (WP:no textbook), and of non-tensors too.

As regards examples, I agree to JSpriggs that scalars (of any type) make mostly only poor illustrations, but I am incapable of presenting a good example that involves the curvature tensor beyond the fact that it is constant along all of a sphere, but that would be a non-local claim. Certainly, this would be an excellent example, touching areas also outside of physics. As an aside, I perceive terms like "hypercubic shape" and "tensor exponential" as paradigmata of bad expert lingo (vulgo jargon).

This ended up in a TL;DR, pardon me, please! Just one last suggestion, and then Good Bye:

I suggest to leave the scope of the lead roughly restricted to the current content and suggest to improve the text at places that do not meet an agreed upon level. Purgy (talk) 14:05, 26 September 2018 (UTC)

I've been too busy in real life to contribute to this discussion until now, but let me raise two points:

1. I don't see any of the proposals being better than the current lede.
2. Tensors are one of those classic math topics that can be approached differently by people with differing backgrounds (novice math students, experienced mathematicians, physicists, etc.). For such topics, there tends to be a lot of disagreement over "the right way" to do it. The current lede is already the result of compromises over many years.

So I agree with Purgy's last statement, that the current lede should not be rewritten massively, even as it is of course revised in places. Mgnbar (talk) 21:50, 26 September 2018 (UTC)

Purgy, thanks again for your thoughts. I have written a shorter proposal below in a new section, partly to meet your requests. I agree that there is some sort of balance to be reached between 1) too much info and context and 2) not enough info and context. It's a kind of art to provide information that gives a good amount of context and indication of relatedness to various topics, but not too much of it. I'm trying to strike the right balance, as I edit my proposal further. (And as I adjust, perhaps I'll be better prepared if, instead, I end up just tweaking the current lead.)

Regarding the snippet you quoted, I did introduce tensors (in the first sentence) as objects that can map. And I did define the order of a tensor. There *is* an imaginary property of the plane to which the result refers to: it's not a physical surface, it's a mathematical, hypothetical one. The stress *is* a stress over or across a surface: the surface provided by the directional surface-normal vector. (No disrespect, but you may want to review this topic to clear up this confusion. Or maybe I'll need some clarification if I am in fact mistaken.) I had taken the reference to the tensor product directly from what you had written in the caption, but I agree, you were mistaken to write "tensor product" there (and I was mistaken to copy what you wrote). I've changed it to "tensor contraction" now, but kept "matrix multiplication" too, since that's much more familiar to people.

I think scalars are interesting because they take on a new character in the context of tensors, but I have taken out an example for brevity. I'll try to consider how I could meet my need for clarity and unambiguousness in the current article by editing it, but if you have further thoughts on what I've written, I'd appreciate hearing them.

@Mgnbar: Thanks for your thoughts, too! I'll try to take to heart what you've said. It would be nice, though, if there was some kind of summary of conclusions that people made about where to draw lines in style and content. (I guess the style guide is the general best attempt at that.)

Zeroparallax (talk) 00:09, 27 September 2018 (UTC)

@Zeroparallax: as I tried to hint to already in my previous comment, I intend not to comment on this topic any further, at least for the time being. This includes valuations of your suggestions, as well as any efforts as to explicate who is mistaken where. As a final remark: In your RfCs you seem to largely underestimate the task of sensibly commenting on your suggestions, in face of already given arguments, and you barraging tangential hints. Sorry, Purgy (talk) 09:50, 27 September 2018 (UTC)

Purgy, thank you. (I think I'll withdraw even my shorter proposal now.) I appreciate the time and effort you've put in to this article and into my somewhat naive attempt at a proposed re-write. I may have more proposals in the future, but I'll try not to sap everyone's energy. Take care.

Zeroparallax (talk) 18:59, 27 September 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)

Thoughts re (short) new lead proposal

To keep things clear, maybe we could put thoughts regarding the (short) new lead proposal here.

Zeroparallax (talk) 23:07, 26 September 2018 (UTC)

So here's an example. Your proposal suggests that a tensor can be defined as an element of a tensor product. But other editors disagree. (Search for the phrase Not every element of a tensor product would be called a "tensor" by most mathematicians and physicists here.)

So you need to find reliable sources supporting this assertion and your other assertions. Of course you know that. What I'm trying to convey is that it's not just a little detail, that can be ironed out later. You have to do it now, preparing for each of your assertions to be challenged. The page is contentious.

In a section above, you asked for a succinct summary of the consensus on how this topic should be treated. The answer is: the current lede of this article. If that answer does not satisfy you, then the next answer is: the talk page of this article, including all of its archives.

I'm not trying to discourage you from improving the article. I'm just trying to convey that it's not going to be a quick task. Regards. Mgnbar (talk) 12:27, 27 September 2018 (UTC)

Here's another example. Your proposal contains a description of what a "geometric" vector is, which I don't understand, despite having a Ph.D. in geometry. (I mention this not to argue from authority or intimidation --- you can't verify that I have a Ph.D., and even with one I could be wrong --- but just to convey that the average reader may not understand either.) I interpret this passage to be something about how physicists apply tensors only in certain situations. However, it's worth noting that the current lede also suffers from a flaw like this. Mgnbar (talk) 13:07, 27 September 2018 (UTC)