Talk:Tensor

Change Algebraic Object to Algebraic Structure in Tensor Definition

I propose changing the wording in the definition of tensor from using the term algebraic object to algebraic structure or to change the link destination to the entry for algebraic structure. As of now the 'algebraic object' links to the mathematical object entry, which is not relevant to the the definition of a tensor in a direct way. Linking to the algebraic structure entry and perhaps changing the text itself would have more utility and makes more sense. 69.162.230.42 (talk) 21:47, 14 December 2023 (UTC)

I would reserve "structure" for the system into which a tensor fits: $(V^{*})^{\otimes q}\otimes V^{\otimes p}$, for a (p, q)-tensor. So I disagree, but I don't feel strongly about it. Mgnbar (talk) 23:16, 14 December 2023 (UTC)

An algebraic structure is a set equipped with some operations. Not being a set, a tensor cannot be an algebraic structure. So the lead is correct by saying that a tensor is an algebraic object. Indeed, tensors are mathematical objects that belong to some algebraic structures (tensor algebra or geometric algebra). So, I strongly disagree with your suggestion that goes against the standard mathematical terminology. D.Lazard (talk) 09:51, 15 December 2023 (UTC)

Seeming definition unclarity

In the section "As multilinear maps", a (p,q) tensor is defined w/r/t the (direct) product of p copies of V* and q copies of V, but in the section "Using tensor products", a (p,q) tensor is defined as the tensor product of p copies of V and q copies of V*. Can an explanation about why p and q flip roles be added to one of these sections? Makhnoboi19 (talk) 17:18, 7 January 2024 (UTC)

The reason is that, in the "As multilinear maps" section, we are considering tensors as maps from the vector space in question to R. That is, a tensor is not an element of the vector space in question, but rather of its dual. And this dual-ness shows up as the "opposite" vector space in the section "Using tensor products".

I agree that this might deserve clarification in the article. But I'm not sure what to write. Does anyone want to take a stab at it? Mgnbar (talk) 22:33, 7 January 2024 (UTC)

The bulk of this article should not use the Einstein summation convention

The bulk of this article should not use the Einstein summation convention. The Einstein summation convention should be introduced in a special section near the bottom of the article. 134.16.68.202 (talk) 01:22, 25 January 2024 (UTC)

I tend to agree, although I don't feel strongly. There are a few cases (general transformation rule) where explicitly showing the summations would significantly expand the length of the expression. But it would be a net benefit to novice readers. Mgnbar (talk) 21:07, 25 January 2024 (UTC)

Incorrect definition

The definitions in the "as multilinear maps" and "using tensor products" sections are wrong. Requiring all copies of to be the same vector space would exclude non-square matrices from being tensors (a $2\times 3$ matrix for example is a multilinear map $\mathbb {R} ^{2}\times \mathbb {R} ^{3}\to \mathbb {R}$). 65.93.245.82 (talk) 06:09, 23 April 2024 (UTC)

The definition is not so much incorrect as restrictive. This article views tensors in terms of a single vector space V and its dual V*. In the past, we have had discussions about whether "tensor" simply means "element of a tensor product of vector spaces", which would be broad enough to handle your objection. The consensus was that the term "tensor" is not used that broadly in the literature. Mgnbar (talk) 11:46, 23 April 2024 (UTC)

First image and caption don't match

The image at the top of the article and its caption are mismatched in several fairly confusing ways. First, the unit vectors e are not shown in the picture. Second, the output vector T is not shown in the picture. The reader is left trying to guess which e goes where, and why there are three output vectors when it says a second-order tensor has only one. Also, the σ coefficients aren't mentioned in the caption at all. 2600:8800:1180:25:AD40:E4C1:631E:5E07 (talk) 21:58, 25 April 2024 (UTC)

Surely the $e_{i}$ are represented by the locations at which $T$ is evaluated, the outputs $T(e_{i})$ are shown in a bluish color? --JBL (talk) 23:26, 25 April 2024 (UTC)

The anonymous poster has a point. One can guess where the $e_{i}$ are, but one should not have to. More importantly, $\sigma$ is prominent in the figure and missing from the caption. In fact, is it true that $\sigma =T$? Or is the figure reserving $T$ for the traction vectors that the stress tensor $\sigma$ produces? If that's the case, then is the caption wrong? It's all a bit of a mess. Mgnbar (talk) 00:22, 26 April 2024 (UTC)

It is helpful to compare with an earlier revision revision of the caption, although this doesn't fully resolve the difficulty. Tito Omburo (talk) 13:14, 26 April 2024 (UTC)

It doesn't help that everything is a subscript, preventing use of the Einstein summation convention. I would guess that ${\overrightarrow {\sigma _{ij}}}=\mathbf {T} ({\overrightarrow {e_{i}}})\cdot {\overrightarrow {e_{j}}}{\overrightarrow {e_{j}}}$, the projection of $\mathbf {T} ({\overrightarrow {e_{i}}})$ on ${\overrightarrow {e_{j}}}$. Things would have been clearer had the caption spelled things out, used $\mathbf {T} =T^{ij}{\overrightarrow {e_{i}}}{\overrightarrow {e_{j}}}$ and not introduced an apparently extraneous additional name. I agree with Tito Omburo that the earlier version is clearer, although I find the nomenclature awkward. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:33, 26 April 2024 (UTC) -- Revised 19:04, 26 April 2024 (UTC)

The current caption does describe what/where the e_i are (they are the normals to the faces of the cube). I agree about the sigmas (I mean, I know what must have been intended, but it's not good that it's entirely implicit). --JBL (talk) 18:16, 26 April 2024 (UTC)