Talk:Tensor
|
|
This page needs serious trimming. Taral 17:09, 19 Jun 2004 (UTC)
| Contents |
Disambig needed
The article says:
The word tensor was introduced by William Rowan Hamilton in 1846, but he used the word for what is now called modulus.
which is fine except that modulus is a disambiguation page. So which of the 6 possible meanings did he use? I can discount 3 of them straight off, but that leaves another 3 still. --Phil | Talk 15:03, Sep 21, 2004 (UTC)
Generalized Hooke's law
Hi,
I have a question here about the tensor notation, if anyone can help me... Thanx
Cdang 15:38, 26 Nov 2004 (UTC)
Abstract index notation?
As a physicist with a mathematical bent, I have for some time had a strong preference for the "abstract index notation" for tensors introduced by Roger Penrose. (This is the notation used by Robert Wald in his textbook General Relativity, for instance.) It applies to the modern component-free approach to tensors, but it looks like a component formalism. For instance, raised and lowered indices represent whether each "slot" of a tensor acts on elements of the vector space or its dual. And contraction between vector and dual vector slots is represented by a repeated index (which looks like the Einstein summation convention for components, even though no component sum is implied).
I feel like using that notation could help to simplify the article: it's a fully modern mathematical approach which should satisfy mathematicians, but it is in general very easy to convert it to a component formalism (just substitute component labels for the abstract indices) which makes it directly useful to physicists and engineers who haven't studied the mathematics in depth. (In fact, those who don't care about the mathematical details could probably get through the less modern parts of the article without even realizing that they weren't looking at component expressions.)
Do other people out there have experience with this notation? Are there pitfalls in using it that I haven't considered?--Steuard 22:30, Apr 18, 2005 (UTC)
- Well, we are duty-bound to include all major points of view in this, the central tensor article; favouring just one approach goes against our charter. This does cause problems, which are particularly obvious here, in this case. But we can't address them by assuming that the right way to teach tensors is ...; there just are a number of aspects. Charles Matthews 07:04, 19 Apr 2005 (UTC)
Most physicists who have done 2 years maths methods can (and need to) handle the expansion of (curl (curl u)) (and the equivalent expansions for eg u. del (u) needed in fluid mechanics, eg for 'Crocco's relation') - can someone write down/derive these expansions for me in modern notation? After a few weeks (admittedly casual) acquaintance with Penrose, I still wouldn't wish to, (and even less, would I expect to be able to teach it to a fellow physicist). Linuxlad 09:15, 19 Apr 2005 (UTC)
- Curl is supposed to be done with Hodge duals. Curl of a curl would be like *d*d with the exterior derivative, so related to a Laplacian like *d*d + d*d*. It is perfectly true that some mutual incomprehension results from divergent (sorry) ideas about how to get to the needed bits of vector calculus. A mere encyclopedia article is unlikely to sort out schisms, such as existed between J. W. S. Cassels and George Batchelor in Cambridge. Being a mathematician, I am always going to stick up for a sensible answer to 'what a tensor is', coming near the beginning. One thing I feel is needed is to get tensor density off this page, and treated properly on its own. There is a good reason for that, namely that it 'breaks' the Bourbaki approach to tensors (comes back when one moves onto tensor fields). Charles Matthews 09:36, 19 Apr 2005 (UTC)
It would be fun to have your take on the Cassels/Batchelor interaction. (I remember going to a joint CEGB/DAMTP 'Problems Drive' where GKB was paricularly heavy on our lead mathematician, who spent too long going through his new FE fluids code - GKB wanted only to hear the essential science of it...)
discussion at Wikipedia talk:WikiProject Mathematics/related articles
This article is part of a series of closely related articles for which I would like to clarify the interrelations. Please contribute your ideas at Wikipedia talk:WikiProject Mathematics/related articles. --MarSch 14:10, 12 Jun 2005 (UTC)
Article is not approchable
This article attempts to provide a non-technical introduction to the idea of tensors, but fails, because it seems rather caught up on the philisophical/pedagogical nature of tensors, rather than a concrete description of what they are. While it may be true that While tensors can be represented by multi-dimensional arrays of components, the point of having a tensor theory is to explain further implications of saying that a quantity is a tensor, beyond that specifying it requires a number of indexed components, an article that doesn't actually list what the properties of tensors are, does neither.
I'm sorry to sound a little negative, but although I have a passable understanding of scalars and vectors, I can't even begin to understand tensors from this article - to me it appears you have to know what a tensor is before you can learn what a tensor is. I'd try to fix it myself, but I don't know what tensors are. (Hence my issue.)
Although tensors can be treated as an abstract quantity, it might be beneftial to talk about the concrete aspect of tensors first, so that a rudimentary understanding can be gained by non-experts. Then you can move on into the abstract generalization. E.g. with vectors: in a vector space a vector is much more general than the conventional "list of numbers" view, but that view is usually presented first, and once that is understood, generalization proceeds from there.
At the vary least, the first paragraph or two should precisely define what a tensor is, instead of weasling around it. (Consider: "In mathematics, a rectangle is a certain kind of geometrical entity. The rectangle concept includes the idea of a square. Rectangles may be written down in terms of coordinate systems, or as a set of points, but are defined so as to be independent of any chosen representation. Rectangles are of importance in physics and engineering. In the field of surveying, for instance, .... While rectangles can be represented by coordinates, the point of having a rectangle theory is to explain further implications of saying that a quantity is a rectangle, beyond that specifying it requires a number of points. In particular, rectangles behave in specific ways under geometric operations. The abstract theory of rectangles is a branch of Euclidlean geometry." All true, but if you didn't know it before, there is no way from that that you'd discern that a rectangle is a planar figure with four sides and four right angles - you certainly wouldn't understand any difference between a rectangle and a rhombus.) Specifically, I might reccommend moving the approaches in detail toward the top of the article, certainly before the examples, and probably before the history (Which, by the way, is woefully lacking in discussion of *why* tensors were developed.
- Great comment. Did you also look at the other tensor articles: Classical treatment of tensors, Tensor (intrinsic definition), Intermediate treatment of tensors and hopefully that's all. I really hate that we have such a mess of articles and I would try to clean it up, if the consensus wasn't against me. Maybe you can help change that, see also the above section. --MarSch 10:25, 22 Jun 2005 (UTC)