Talk:Tensor (intrinsic definition)

I am now trying to make this page more easily maintainable, by replacing Wiki markup + Unicode with TeX markup for the math stuff. This makes this page visible in many more browsers, and should make the page more editable for experts; of which I am not one. Could any mathematicians here proof-read this article, please?

Can I protest (a) about calling things in mathematics 'formalisms' (which is too much like Serge Lang for me) - one might as well say 'unmotivated stuff'; and (b) calling things 'modern' - unlike calling things classical, which is fair enough?

Charles Matthews 09:11, 12 Nov 2003 (UTC)

Charles, what name would be suitable?

• Tensor (differential geometry treatment) ?
• Tensor (component-free treatment) ?
• something else?

-- The Anome 22:44, 12 Nov 2003 (UTC)

Tensor (abstract algebra) is good. The old argument (six decades ago now) was: don't say modern algebra (perishable), say abstract algebra. In fact there is an argument now to go further to tensor (category theory) (as well, not instead of) for monoidal categories.

Charles Matthews 17:55, 13 Nov 2003 (UTC)

Tensor (differential geometry) is better, except that even the component-treatment is differential geometry... I don't quite think abstract algebra covers tensor bundles, differential manifolds, connections, sections, etc... Phys 17:25, 14 Nov 2003 (UTC)

Connections? Keep on topic, please. This is the old argument (for tensors here) that you have to introduce tensor fields at the same time as tensors. Why? There is a page for tensor fields.

Charles Matthews 18:50, 14 Nov 2003 (UTC)

I've looked around a bit - not hard to find ten 'tensor' pages. I think this tells me something a little more positive, namely that this isn't really a name-space problem any more. It's more a question of licking the hypertext into shape. I've not moved any pages up to now, and don't intend to start.

Charles Matthews 20:17, 14 Nov 2003 (UTC)

Wow, this page has mobilized! What is intrinsic definition supposed to mean? aren't tensors in general intrinsic definitions of a space? isn't that what riemannian geometry (classical tensors) is known for: intrinsic definition of curvature? how else would one use the phrase "intrinsic definition" other than refering to the topology of the space or curve? does one mean by this "component-free"? if so, why not just say "component-free" instead of the ambiguous, mysterious, and confusing (because of its allusion to the intrinisic definition of curvature in riemannian geometry (classical tensors)) "intrinsic definition".

regarding "abstract algebra": (and someone mentioned differential geometry) there was discusion earlier in the Talk:Tensor page regarding the topical arrangement of the mathematics page, and where in that tensors should go (why they are on top), and it was concluded that they belong in both the abstract algebra section (for the modern treatment) and the differential geometry section (for the classical treatment), which makes a lot of sense. maybe "abstract algebra approach" would be more fitting, informative, and helpfull, seeing that the approach is based on abstract algebra, comes from the perspective of abstract algebra, and one needs to know abstract algebra in order to learn it.

--Kevin Baas Still suggesting:

• Tensor (absract algebra treatment)

-- Kevin Baas 14:41, 24 Feb 2004 (UTC)

This article needs re-hauling. The title is inadequate and suggests confusion with the notion of tensor fields in differential geometry. The header, as others have above suggested, ought to be: "Tensors (algebraic treatment)". While listing the correct properties of a tensor space, the standard explicit construction (in terms of a quotient of some free module) is not provided. Tensor products from the viewpoint of category theory (as covariant functors) should at least be briefly hinted at.

--Anon 14 July 2006.

(First talk post for me; experienced posters please correct me re: posting conventions.) Suggestion: Display a commutative diagram of VxW, Vx_FW, Q, Q', and X in the definition of the tensor product. I find commutative diagrams to be helpful, for both algebra-minded and non-algebra-minded readers. 24.155.243.76 21:30, 20 December 2006 (UTC), Nooj.

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 14:11, 12 Jun 2005 (UTC)

This article seems to completely miss the point of the algebraic approach to tensors. A tensor is simply a multi-linear map. I think this article presents this idea in a manner that is more complicated than it needs to be.

-- Anon 2 Jan 2007 —Preceding unsigned comment added by 68.197.9.185 (talk) 03:20, 3 January 2008 (UTC)

Alternate notation

In the section "Alternate notation", what do the wavy approximate equal signs mean? Shouldn't they just be equal signs? —Preceding unsigned comment added by 128.103.54.116 (talk) 17:18, 20 May 2008 (UTC)

They aren't approximation signs but mean "is naturally isomorphic to", as the article states, hence the link to natural isomorphism. Dependent Variable (talk) 20:19, 24 September 2009 (UTC)

Tensor products

Is this correct, and if so, what does it mean?

"An element of this tensor product is referred to as a tensor (but this is not the notion of tensor discussed in this article)"?

The link to the "Tensor product of vector spaces" section of Tensor product" leads me to think that perhaps whoever made the link misread it as "This tensor product (i.e. the tensor product of vector spaces) is referred to as a tensor" or that, at the time when the link was made, the sentence actually said something equivalent to "This tensor product is referred to as a tensor", as opposed to "An element of this tensor product ..." As it stands - as far as I can see - the sentence refers to exactly the notion of tensor that the rest of the article goes on to define: "A tensor on the vector space V is then defined to be an element of (i.e., a vector in) a vector space of the form ..." Dependent Variable (talk) 20:43, 17 September 2009 (UTC)

Rank: Disambiguation

I suggest that Tensor Rank direct to a disambiguation page, and not directly to the subsection, so that inline disambiguation is not necessary and conflicting terminology can be accounted for by an explicitly defining link. Either that, or the definition of the term used on Wikipedia should be standardized. LokiClock (talk) 13:15, 23 January 2010 (UTC)

Contravariance and covariance

Since this is the second time I've needed to revert the wording of "contravariant" and "covariant" in the definition, I thought I'd include an explanation of my edit here. I think the confusion comes from the fact that one way of defining a type (m,n) tensor T is as the multilinear map

$\begin{matrix} T: & \underbrace{ V^* \times\dots\times V^*} & \times & \underbrace{ V \times\dots\times V} &\rightarrow \mathbf{R}. \\ & \text{m copies}& &\text{n copies} & & \end{matrix}$

However, the definition used in this article is, instead, that a type (m,n) tensor is an element of the tensor product of vector spaces

$\begin{matrix} T\in & \underbrace{V \otimes\dots\otimes V} & \otimes & \underbrace{V^* \otimes\dots\otimes V^*}. \\ & \text{m copies}& &\text{n copies} \end{matrix}$

In either case, though, a type (m,n) tensor is said to be contravariant of order m (denoted with upper indices) and covariant of order n (denoted with lower indices). I hope that clarifies the naming convention.

- Rundquist (talk) —Preceding undated comment added 03:53, 14 July 2011 (UTC).