Talk:Cartesian tensor

Meaning of the concept

Maschen, do not conserve words please on explanation of its core meaning. Are these (any) tensors over vector spaces with an Euclidean structure, and the article explains those rules of transformation which are specific to orthogonal coordinates? Are these tensors with some special properties? Are these a special calculus which explicates an Euclidean structure in some unusual way? And I am very unhappy with this title, because it is not WP:PRECISE. Incnis Mrsi (talk) 10:10, 9 June 2013 (UTC)

The sources in the ref section state Cartesian tensors transform according to orthogonal transformations, and the article says this over and over.

It starts from the absolute basics of vector algebra in the most elementary possible treatment using Cartesian coordinates, in 3d Euclidean space, which most people reading the article will be familiar with, then introduces the permutation and identity symbols in this familiar context which apply in vector calculus, then after discussing the position vector in any finite number of dimensions, the idea of the invariance of the position vector under orthogonal transformations is developed, then the idea extends to tensors.

As to the stuff about special properties and associated calculi, that can be explained in another section. One such section is already Cartesian tensor#Second order reducible tensors in 3d if this is what you mean.

So what's your angle? Why is the meaning unclear? And would the title include affine tensors (transforming under affine transformations) also? M∧Ŝc²ħεИ_τlk 10:42, 9 June 2013 (UTC)

Your referenced two books with this title, of G.F.J. Temple and Sir H. Jeffreys. Did you actually read at least several introductory paragraphs from there? You explanations still do not satisfy me. If you want tensors (not something more complicated like geometric algebra) in an Euclidean space, then there is only two things to learn:

1. Tensor calculus (unspecific to Euclidean structures);
2. Raising and lowering indices.

Of course, you can restrict yourself to orthonormal bases only, but it is not a new theory: it is just a restriction of the tensor calculus to bases only of a special kind. There is too little sense in it to write a dedicated article. Incnis Mrsi (talk) 11:10, 9 June 2013 (UTC)

I think there is possibly scope for a separate article. For most applications of tensors to engineering (e.g., rigid body dynamics and stress analysis), it is not necessary to accommodate the general coordinate systems of tensor calculus. Usually there is some fixed background frame and at worst one needs to know how things transform when that frame is rotated. Then raising and lowering of indices is also not required, and all of the tensoriality is essentially quite trivial (so suitable for engineers). It seems to me that this is the intended scope of the article, but it might be helpful if it stated this plainly somewhere, although it is possible that I might have misunderstood what is meant by "rectangular coordinate system".

But assuming that the tensoriality of things is supposed to be trivial here, then the article should not dwell too much on this aspect of things (as it presently does). The likely audience of the article will only get confused by this, and those looking for a more comprehensive treatment of tensors have other articles to select from. I think some time should be devoted to discussing examples. A good one is the inertia tensor and Sylvester's law, in my opinion, because it's very easy to visualize (for instance, in terms of a spinning plate or something). Sławomir Biały (talk) 12:23, 9 June 2013 (UTC)

Incnis Mrsi: Yes I did read the refs, though the leading ones I wrote from were the Schuam books (the ones I actually have). Also the topic by this title is notable and there is some substance to the subject, so it should have an article.

Sławomir Biały: Thanks, I know some sections have too much inessential detail and they'll be reduced/removed, and I've been meaning to add examples and applications. However I'd like to keep at least most of Cartesian tensor#transformations of Cartesian vectors (any number of dimensions) since it collects all the different ways of writing the transformation matrix elements and illustrates why co/contra-variance is equivalent. M∧Ŝc²ħεИ_τlk 13:18, 9 June 2013 (UTC)

Clarifications accepted, except one shortcoming. Only in an orthonormal basis are raising and lowering trivial. In an orthogonal basis (a.k.a. rectangular coordinates) they are not. Please, watch for the difference carefully to avoid mistakes. May I try to reformulate the lead to emphasize that it is actually a simplified tensor calculus? Incnis Mrsi (talk) 13:37, 9 June 2013 (UTC)

Edit by all means, I'll try more trimming and examples later to prevent interfering with others. M∧Ŝc²ħεИ_τlk 13:52, 9 June 2013 (UTC)

I made my edit. Sorry if its value is less than you expected, but see #Cross product. Incnis Mrsi (talk) 15:43, 9 June 2013 (UTC)

Point noted. I was unsure whether rectangular system here was being used in the more restrictive sense of an orthonormal basis. (Otherwise, it's an error to say that the associated transformation is an orthogonal transformation.) Either way, it's an issue that needs clearing up. Sławomir Biały (talk) 14:35, 9 June 2013 (UTC)

If the intent is to cover coordinate systems that are orthogonal but not orthonormal, then it is indeed difficult to see the need for a separate article. (Indeed, since we're apparently only concerned here with linear coordinate systems, it's very difficult to fathom where such a thing might be usefull. It would presumably entail different coordinate axes using different units of neasurement—a rather bizarre notion from the point of view of applications.) Sławomir Biały (talk) 15:13, 9 June 2013 (UTC)

Yes - orthonormal bases and not orthogonal are the subject of this article. M∧Ŝc²ħεИ_τlk 16:25, 9 June 2013 (UTC)

The core concept of this article appears to be clear now: essentially, manipulation of tensor components that are restricted to orthonormal bases. As it is an extremely common to restrict bases in this way, and component manipulation is often the approach taken to tensors (rather than a more abstract approach), I agree fully that the topic is notable and valuable. I do have a quibble though: whether the title is appropriate, in that it should convey the correct meaning. My first reaction was: what proper subclass of tensors can be labelled "Cartesian" (example: "symmetric tensor")? Upon looking more closely, it is only when you regard the collection of components as the tensor that the phrase makes sense, but I consider this to be a misnomer, one which I think perpetuates some confusion and impedes the step in learning that involves grokking a tensor for what it is: not something defined in terms of its components (and the transformation laws thereon), but an entity with certain abstract properties independent of any concept of basis, although it can be expressed in terms of components should one identify a basis. One consequence of this starting point is that the transformation laws of the components of a tensor are inherently satisfied, not imposed. Okay, long story by way of setting the scene. If the name is notable, I withdraw my case. But a Google books search gives me the impression that the term Cartesian tensor only sees reasonable use in fields such as studies of elasticity and fluid mechanics (in short, certain fields of engineering). The lead should restrict the fields appropriately, alternatively the article should be renamed to something that would be understood across a broader swathe of disciplines, including linear algebra. Such a title would be clumsier, but right now (unless my impression is wrong), it creates the IMO incorrect impression that this term is standard across all the disciplines in which it is a notable concept (and my guess is that linear algebra is one where the concept is notable but the name is probably not). Perhaps Cartesian tensor components? — Quondum 18:21, 9 June 2013 (UTC)

It's apparently the case that there is something called a "Cartesian tensor" and that (apparently, without having been able to look into it myself) means what you have described. Of course, it's naturally a bit of a misnomer, since a "Cartesian tensor" isn't a special kind of tensor, but rather a way of thinking about tensors in Euclidean spaces that does not account for arbitrary coordinate transformations. Insofar as the terminology already seems established by tradition, I don't think it's consistent with our role to change it. But I do think that it could be made clearer in the first few sentences what is meant by the term. Sławomir Biały (talk) 19:00, 9 June 2013 (UTC)

I was not suggesting that there is no such term (it seems to be notable), nor that it does not have this meaning (it seems to be used this way), only that its use may not be general across the fields that this article attributes it to. I will take your reply to mean that its use is standard within linear algebra, in which case I withdraw the suggestion. — Quondum 19:24, 9 June 2013 (UTC)

Thanks for your edits Quondum. The article is clearer in places now. M∧Ŝc²ħεИ_τlk 14:57, 10 June 2013 (UTC)

Cross product

Although the article says well-known things about determinants, the orientation dependence of the cross product is never mentioned. Incnis Mrsi (talk) 15:43, 9 June 2013 (UTC)

If you mean antisymmetry then that's already illustrated. M∧Ŝc²ħεИ_τlk 16:45, 9 June 2013 (UTC)

Do you not understand what means “orientation dependence”? Cartesian tensor #Cross and product and the Levi-Civita symbol states:

“ For the cross product × of two vectors, the results are (almost) the other way round, cyclic permutations in perpendicular directions yield the next vector in the cyclic collection of vectors: e_x × e_y = e_z … ”

Is it an invariant definition of cross product? You did not restricted your bases only to those with correct handedness, where e_x × e_y = +e_z (or, the same, your transforms only to proper rotations) ⇒ contradiction. Is it a definition of cross product with respect to given basis, which would be invariant up to sign? Then, you should explicate it. BTW, Levi-Civita symbol is a pseudotensor of order n, for exactly the same reason: it is invariant up to sign. Drop these obscure “tensor densities” which rely on more complicated concepts of differential geometry, please. A pseudotensor it is, due to pure finite-dimensional linear algebra over an ordered field. Incnis Mrsi (talk) 17:11, 9 June 2013 (UTC)

I was conflating the left/right-handed orientations of the basis vectors with plain exchanges of basis vectors, so explicitly stated the system was a right-handed Cartesian coordinate system linked to Orientation (vector space). Also changed tensor density to pseudotensor. M∧Ŝc²ħεИ_τlk 14:57, 10 June 2013 (UTC)

Another possible reason for keeping this page:

There is no article on spherical tensors or spherical bases, which was pointed to me by Teply that it's red-linked in numerous places on WP. Spherical tensors are apparently a special case of Cartesian tensors (see for example B. Baragiola, unless the pdf is wrong). Perhaps an article on Cartesian tensors including reducibility (like the section in this article, taken from Baragiola) may help these red articles ? (In addition the original intentions stated above).

But, I'm not insisting on anything. M∧Ŝc²ħεИ_τlk 15:55, 9 June 2013 (UTC)

Seems reasonable to me. I think I recall seeing special coordinate systems (orthogonal coordinate systems like spherical, cylindrical, etc) dealt with in one of our tensor articles. Try to survey what's already there to avoid needless duplication (and extra work). One thing that such a hypotherical article would benefit from is pictures (that you're so good at). Sławomir Biały (talk) 00:30, 11 June 2013 (UTC)

The particular reference given here (Baragiola) seems to be particularly poor. I would suggest that this article should illustrate examples of Cartesian bases including a spherical basis. Exactly the conformal mappings of the Cartesian coordinates will yield a Cartesian basis when normalized. I suspect that the (rectilinear) Cartesian coordinates are unique in producing a coordinate basis, though. The meat of this example may be found at Spherical coordinate system#Integration and differentiation in spherical coordinates. — Quondum 03:20, 11 June 2013 (UTC)

OK. Yes, Baragiola apparently gets the definition of the spherical basis wrong and its just a pdf he wrote and not an actual paper, still looking for better refs for this article.

BTW Quondum, you mentioned in an edit summary if the section should be in this article, where else could it be? M∧Ŝc²ħεИ_τlk 05:33, 11 June 2013 (UTC)

Here is one better pdf - M.S. Anwar (2004) Spherical Tensor Operators in NMR. M∧Ŝc²ħεИ_τlk 12:32, 11 June 2013 (UTC)

I merely glanced through this paper, so can't comment in detail. If you can see that the reduction is of particular value when dealing with spherical tensors, then it makes sense to make mention of this here (I have no idea whether this is the case here). In the more general case, the reduction seems interesting, and may belong somewhere such as under Tensor#Operations. A more general treatment might want to deal with reduction of tensors of higher order too (which, of course, is more complicated and might be treated better via symmetries). A connection that I find intriguing is that when mapping the full tensor algebra to the geometric algebra (as in the universal construction), for second-order tensors the T⁽¹⁾ part becomes the grade-0 part, the T⁽²⁾ part becomes the grade-2 part, and T⁽³⁾ simply vanishes. — Quondum 12:52, 11 June 2013 (UTC)

Or perhaps just spherical basis when created. In this draft, I'm sort of getting to grips with the idea of spherical tensors but it's a tricky subject with plenty of room for confusion, and some authors use really confusing notations/conventions like for the infinitesimal rotation operator like in that previous pdf. Here's another pdf [cc.oulu.fi/~tf/tiedostot/pub/kvmIII/english/2004/09_tensop.pdf Tensor operators]. I'll look for proper refs. It's taking time since my internet connection is diabolical. M∧Ŝc²ħεИ_τlk 13:40, 11 June 2013 (UTC)

I can see from your draft (and memory of one reference that I glanced at), that my assumption of what a "spherical basis" is is an unrelated concept (your draft makes this pretty clear). So disregard my comments in this regard for the time being, until I actually figure out why this name was chosen and what the utility of this basis is. — Quondum 15:04, 11 June 2013 (UTC)

My draft could be filled with errors and should not yet be taken as correct, but there are some good references eventually found and inserted, accessible through google books (have a look in the "notes" section if you want to). I'm going to try and tie up all the loose ends and then just launch into mainspace, it's been in my userspace for too long. The diagrams for it can come later once others have had a look in the mainspace version.

As for this article (Cartesian tensor), the particular section on reducing second order tensors can be deleted anytime since it's going to reappear in spherical basis. M∧Ŝc²ħεИ_τlk 16:53, 11 June 2013 (UTC)

Yes, put it in mainspace. I'm not sure it qualifies as a Cartesian basis (what I was thinking of before was, but this basis uses complexified vectors and might not qualify), so the last sentence in the lead might have to go. — Quondum 18:44, 11 June 2013 (UTC)

section: Difference from the standard tensor calculus

This section appears to restate what is already written earlier in the article. Considering that Incnis Mrsi said he would edit the lead, some of this section should be moved up to the lead. Also, it's blatant the section needs expansion, so the "expansion" tag was removed. M∧Ŝc²ħεИ_τlk 08:33, 13 June 2013 (UTC)

I have no objections to its removal or merging elsewhere – I agree that it is pretty redundant. — Quondum 22:22, 13 June 2013 (UTC)

Up to the lead, really? Please, recall provisions of WP:MOSINTRO. Incnis Mrsi (talk) 08:39, 15 June 2013 (UTC)

Hope this edit is not too offensive. Nothing was moved to the lead. M∧Ŝc²ħεИ_τlk 09:23, 15 June 2013 (UTC)

Notation for inverse of a set of transformation components

Given the linear transformation $\mathbf {L}$, represented by the set of components $L_{ij}$, the notation for the set of components representing the inverse linear transformation $\mathbf {L} ^{-1}$ is a bit tricky. In particular, the notation $(L_{ij})^{-1}$ is ill-defined. It actually suggests taking the reciprocal of each component value, which is not the intention. Even interpreted as "the components of the inverse of the linear transformation specified by the full set of components $L_{ij}$" (which is the intention), the indices are not in principle directly accessible. A far better notation (and one that I've seen) is $[\mathbf {L} ^{-1}]_{ij}$, and I suggest that it should be adopted in this article in preference to the existing notation (unless an even clearer exposition is given). Here the ordering of the indices is still not really defined, but that is the only problem with this notation. At least it is calling a spade a spade. — Quondum 17:01, 15 June 2013 (UTC)

Perhaps, I've seen both notations used, and don't find either more confusing than the other. The notation $(L_{ij})^{-1}=L_{ji}$ can't be ill-defined, it seems to just be writing preferences. I see what you mean: $(L_{ij})^{-1}=1/(L_{ij})$. Feel free to change it, or I'll do it in time.

On a related note, should we use something else from boldface for the L matrix? Plain italics is not ideal since it looks like a scalar without indices, and bold makes it look like a tensor. How about \mathsf: ${\mathsf {L}}$ ? This seems to be at least slightly standard for tensors and matrices (including row and column vectors). M∧Ŝc²ħεИ_τlk 17:14, 15 June 2013 (UTC)

P.S. Thanks for changing all the Cartesian subscripts to upright, just me being lazy (on top of my inane number of already clumsy edits). However, more often than not, I've seen coordinate labels as italic, especially for the usual Cartesian, spherical polar, and cylindrical polar - partially the reason for leaving it that way. M∧Ŝc²ħεИ_τlk 18:27, 15 June 2013 (UTC)

Fixed the brackets. M∧Ŝc²ħεИ_τlk 20:23, 15 June 2013 (UTC)

Tried in this edit sans serif for L also. \mathsf can be removed by a global find/replace on notepad/wordpad or whatever. M∧Ŝc²ħεИ_τlk 20:33, 15 June 2013 (UTC)

It is looking much improved. One could try for exact semantics conveyed by typeface (not an uncommon approach in textbooks). I am inclined to stick with a non-bold italic root when dealing with explicit (indexed) components, but a different typeface when referring to a complete tensor, e.g. $L_{ij}\equiv [{\mathsf {L}}]_{ij}$, ${\mathsf {L}}=L_{ij}\mathbf {e} _{i}\otimes \mathbf {e} _{j}$. In this, the subscripted square brackets denote the operation of finding the components with respect to a basis, not the same as simply subscripting a tensor (which would denote an array of complete tensors, as is the case with $\mathbf {e} _{i}$. Thus one could legitimately say $[\mathbf {e} _{i}]_{j}=\delta _{ij}$. Anyhow, this is a bit of intellectualizing, and should not be taken too seriously yet.

The non-italic x/y/z labels could be regarded as another of my crusades: an attempt to establish a semantically consistent style on WP regardless of the prevalence of less consistent/"pleasing" styles elsewhere. WP guidelines are semi-self-contradictory here: prefer a given style, but also prefer well-established styles. So please do not feel that you need to take much note of my changes: consider whether you think there is any sense in them, and ignore, comment, adopt or revert as you see fit. I've had my non-italic x/y/z labels reverted before... — Quondum 22:30, 15 June 2013 (UTC)

It's a shame when editors like yourself make mass-formatting changes throughout articles only to be reverted.

The notation ${\mathsf {T}}=T_{ij}\mathbf {e} _{i}\otimes \mathbf {e} _{j}$ is nice for tensors, bold sans serif looks rounded in the standard default LaTeX font ${\boldsymbol {\mathsf {T}}}=T_{ij}\mathbf {e} _{i}\otimes \mathbf {e} _{j}$ which is less nice, and hypocritically I have tried using this in places such as Navier–Stokes equations to see what it's like within text. For this article, this would conflict with the transformation matrix L. As you say, the current formatting is probably fine as is. M∧Ŝc²ħεИ_τlk 12:38, 16 June 2013 (UTC)

The (my) purposes of mass-formatting are sometimes still achieved, even if reverted (besides, doing it feels a bit zen). To me, bold sans serif comes across as double bold and not easily distinguished from non-bold, and so seems a little clunky (it is really only the fonts that are not great, and I try to ignore this type of thing). I normally would argue that to distinguish the order of a tensor through font is not helpful (e.g., I disagree with the tradition in GA of distinguishing scalars and vectors from anything else in this way), and thus feel that only one font variation should be used (e.g. either ${\mathsf {T}}=T_{ij}{\mathsf {e}}_{i}\otimes {\mathsf {e}}_{j}$ or $\mathbf {T} =T_{ij}\mathbf {e} _{i}\otimes \mathbf {e} _{j}$). My general preference is that if the semantics associated with a notation can be made exact (in the sense that a compiler can assign exact semantics to parsed source code) combined with orthogonality (don't worry about what I mean by this), then it is to be encouraged since it will facilitate clarity of thought and communication.

I don't think there is the conflict that you mention. I have had a heated debate in the past about the interpretation of a passive transformation. I contend that such a transformation is a tensor, and thus that to denote it as a tensor using the same notation is appropriate. It is easy to show this is valid if you allow distinct basis vectors for each of the vector spaces in the tensor product underlying the tensor algebra, but some people seem not to understand that allowing such mix-and-match bases is necessary in a general tensor algebra. If you like, I can make the L notation/font consistent by my interpretation here; my main concern is that I'm distracting you from your work on the content of the article. — Quondum 14:19, 16 June 2013 (UTC)

Feel free to make any changes you like, I'm busy for now and will not be editing as frequently. M∧Ŝc²ħεИ_τlk 14:31, 16 June 2013 (UTC)

Okay, so now we have the following notational convention:

${\boldsymbol {\mathsf {L}}}$ means the entire tensor (or entire transformation matrix, if you prefer)

$({\boldsymbol {\mathsf {L}}})_{ij}$ means extracting the i,j scalar component with respect to a chosen basis

${\mathsf {L}}_{ij}$ means the i,j scalar component of a matrix

The difference between the last two (the latter not being bold) reflects different semantics: extraction of a scalar component from a tensor (or matrix), and simply giving a scalar component. We are now left with a notation that is slightly clumsy, and is repeated often:

$({\boldsymbol {\mathsf {L}}}^{-1})_{ij}$

being the extraction of a scalar component of the inverse of a matrix. I'm going to leave that as is for the nonce, because I want to fix something else first: indicating onto which basis an index sums. (I have vague memory that the need for explicitly indicating the inverse goes away when one does that first.) The notation I'm thinking of using is given in Ricci calculus#Reference to coordinate systems. — Quondum 22:58, 16 June 2013 (UTC)

Looks OK, thanks. Surely the indices are as correct as possible but I could easily be wrong. M∧Ŝc²ħεИ_τlk 06:33, 17 June 2013 (UTC)

In this instance, tagging the basis can be very helpful. Thus, we would have to distinguish between sets of coefficients transforming between different bases e.g. ${\bar {\mathbf {e} }}_{j}={\mathsf {L}}_{ji}\mathbf {e} _{i}$ and $\mathbf {e} '_{k}={\mathsf {M}}_{kj}{\bar {\mathbf {e} }}_{j}$. It is natural to write this as something like $\mathbf {e} _{\bar {j}}={\mathsf {L}}_{{\bar {j}}i}\mathbf {e} _{i}$ and $\mathbf {e} _{k'}={\mathsf {L}}_{k'{\bar {j}}}\mathbf {e} _{\bar {j}}$. In this notation, the need for the clumsiness $({\boldsymbol {\mathsf {L}}}^{-1})_{ij}$ disappears, because it becomes clear that ${\mathsf {L}}_{ij'}$ is a different set of coefficients from and indeed the matrix inverse of the set ${\mathsf {L}}_{j'i}$. I know that you knew this, but I'm saying that it will get rid of the "slight clumsiness", and is exactly the way it was handled in my own course way back. — Quondum 10:45, 17 June 2013 (UTC)

Although both conventions of using different letters, or bars/primes, are used in the literature. I think it's just typographically easier and visually clearer to use different letters, and not overbars or primes on indices, since it clutters already fine-detailed expressions. HTML also becomes a nightmare because of the extra wikicode for templates for the overline, and primes are also a nuisance using an extra apostrophe or other coding every time. Why is $({\boldsymbol {\mathsf {L}}}^{-1})_{ij}$ clumsy, anyway? Everything you said in your previous post seemed coherent and is standard enough to be used. M∧Ŝc²ħεИ_τlk 13:19, 17 June 2013 (UTC)

Using different letters also works, and you have a point about the cluttering. Using different letters builds less "foreknowledge" into the notation (which is probably good idea here, maybe making differnt letters preferable over primes etc.). The clumsiness largely relates to the fact that these expressions do not use pure index expressions; the inverse (when L is thought of as a matrix) implicitly simultaneously uses all the components in a complex way, and would negate the benefit of index-based tensor notation that it gets rid of this kind of expression. It would be normal to unabiguously define $M_{ij}$ as the components that satisfy $M_{ij}L_{jk}=\delta _{ik}$, and then use M throughout. — Quondum 15:01, 17 June 2013 (UTC)

It's a nice idea to use a different letter for the inverse, which would be typographically cleaner, but less easy to keep track of (i.e. the reader may miss the statement "where M is the inverse of L" and get confused elsewhere). The inverse notation L⁻¹ has an immediately clear and unambiguous meaning. I'll leave it to others if they prefer a different letter for the inverse. M∧Ŝc²ħεИ_τlk 16:25, 17 June 2013 (UTC)