Talk:Tensor field

Twisting by a line bundle and half-density

The section on "twisting by a line bundle" and "half-density" is quite opaque as it currently is written. linas 00:21, 5 May 2005 (UTC)

I was redirected to this page from half-form (by clicking the link in the article on geometric quantization) but on this page there is no explanation to what a half-form is, if it is the same as a tensor field or something else. I have not found information on this anyewhere else. Erik J 10:41, 5 October 2005 (UTC)

I'm planning on fixing this, but at the rate I'm going, it'll take another 6 months. Mean time line bundle and Hopf bundle may help. linas 14:02, 5 October 2005 (UTC)

OK, a half-form would be something like a square root of an n-form on an n-dimensional manifold. The question is, what is the geometric meaning of such a thing? It is not obvious that geometrically taking square roots is harmless (it isn't for complex numbers). But you are right that it is time to support this all by references (Guillemin-Sternberg, Geometric Asymptotics, p. 251). Charles Matthews 21:28, 5 October 2005 (UTC)

Sounds like a mangled volume form, a jacobian-like thingy. Anyway, the difficulty of understanding geometric quantization will be far beyond resolving this issue. linas 23:17, 5 October 2005 (UTC)

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 14:12, 12 Jun 2005 (UTC)

WikiProject class rating

This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class. BetacommandBot 10:04, 10 November 2007 (UTC)

Poor lead

The lead doesn't actually say what it is. It talks around it, but without any attempt at definition. Leads are supposed to define.- (User) Wolfkeeper (Talk) 23:39, 27 April 2009 (UTC)

Illustrations

I made an image demonstrating the concept if anyone wants to use it. LokiClock (talk) 23:04, 4 September 2009 (UTC)

Tensor field graphic

By analogy with the tensor field arrows, shouldn't the vector field illustration have 2 arrows for each cell. One arrow pointing out of the cell and one in the cell and their product is the resulting arrow? I think that is the sense of the final illustration. Each pair of arrows defines a product. YouRang? (talk) 02:23, 23 February 2010 (UTC)

My understanding (and this is more from intuition) is that in a plot of tensors any number of arrows greater than one represent a 2nd-order tensor of that dimension. Note that there are three arrows with three degrees of freedom; the tensors in question can be represented by 3x3 matrices, with each vector representing a separate three indices. To represent a 3rd-order tensor requires groups of such arrows (multiple 2nd-order tensors). To differentiate you could use color coding or make them the vertices of a polytope (with multiple polytopes being assigned to each point). Further methods are available, but every visualization method must make compromises, and in general higher-order tensors are notoriously difficult to visualize. I can understand why this would be the case much easier if the geometric complexity of the units of representation grows as the order grows. But if a 3rd-order tensor can be parametrized by three vectors, what then is the point of using the higher-order form? ᛭ LokiClock (talk) 23:46, 14 September 2010 (UTC)

I agree with User:YouRang?, and as you can see I have removed the graphic. For one thing, 3rd-order tensors are not parametrized by three vectors (you are confusing the tensor product with the direct sum). Anyway, the graphic was supposed to illustrate a 0th order, 1st order, and 2nd order tensor in two dimensions. But 2nd order tensors in two dimensions are represented in a basis by 2×2 matrices, not 3×3 matrices, and so it would be more natural to display a pair of vectors at each point. In any event, if there is to be such a graphic, then it would be better to also explain the meaning of the representation of a 2nd order tensor as a pair of vectors, lest readers leave with the false impression that an nth order tensor field is nothing more than an association of n vectors to each point of space. Sławomir Biały (talk) 13:04, 21 September 2010 (UTC)

Amazing Introduction

I am trying to teach myself the basics of fluid mechanics in order to implement wind flow in a video game, and I have been intimidated all night by the constant references to tensors. The sentence in the introduction about how a tensor is a generalization of a scalar field and a vector field was exactly what I needed. I'm not scared anymore. That is a much simpler and more concise explanation than is found in the actual article on tensors. —Preceding unsigned comment added by 65.50.39.118 (talk) 05:31, 7 September 2010 (UTC)

The C∞(M) module explanation a bit conservative?

Is there any particular reason that it only assumes that ${\mathcal {T}}(M)$ is a module over a ring, instead of a vector space over a field? It seems to me that C^∞(M) would definitely qualify as a field (just using point-wise addition and multiplication), and that ${\mathcal {T}}(M)$ could just as easily be a vector space. It could be that I'm overlooking some subtleties, but if so, I'd love to hear what they are. (Perhaps it would be good in general to add one or two references.) --Jaspervdg (talk) 10:29, 28 October 2011 (UTC)

The function f(x)=x is a C^∞ function on the real numbers. Its inverse, if it had one, would be g(x)=1/x. But this is not even continuous on the real numbers, i.e. C^∞(R) is not a field. RobHar (talk) 14:13, 28 October 2011 (UTC)

Tangent space?

A tensor field is almost universally (or at least in physics) assumed to be a function from the manifold to the tensor algebra of the tangent space. It is possible to have a tensor bundle on a manifold that is not related to the tangent space. The description, as given in this article, makes little reference to the tangent space, and is thus more general. Is the more general definition intended? —Quondum 04:26, 21 July 2017 (UTC)

Terrible writing

"For example, a vector space of one dimension depending on an angle could look like a Möbius strip as well as a cylinder."

I finally figured out what this means, but for anyone who is unfamiliar with fibre bundles, this will make no sense at all. Avoid vague sentence like that. Almost no readers will have any idea what "a vector space ... depending on an angle" means. And nobody will understand how it might be possible that "a vector space ... could look like a Möbius strip."

Such "helpful" statements that are so very poorly written do more harm than good.2600:1700:E1C0:F340:ED8B:9E01:6222:F4CC (talk) 19:56, 20 January 2019 (UTC)

Circular reference

In the section „Tensor bundle“ there is a link to tensor bundle which redirects here again, and there is no actual definition of a tensor bundle given in the section. --2001:67C:10EC:5749:8000:0:0:EAB (talk) 16:07, 5 February 2021 (UTC)

A tensor field is a tensor

See this math stack exchange:

https://math.stackexchange.com/questions/270297/difference-between-tensor-and-tensor-field

The difference is all in your head. Literally. The difference in calling the same object A a "tensor over X(M)" as opposed to "a tensor field over M" is that the former emphasizes the fact that we have an algebraic object: a tensor over some module, while the latter emphasizes the fact that underlying the module there is some manifold and geometry is going on there. Calling something a tensor field instead of a tensor forces you to remember that is not just some arbitrary module, but that its elements can be identified with smooth sections of the tangent bundle of some manifold. These additional structures are occasionally useful.

The opening to this article is therefore incorrect. The Riemann curvature tensor is a tensor. It is also a tensor field as it does occur over a manifold. 71.33.153.31 (talk) 06:41, 8 December 2022 (UTC)