Talk:Tensor

New illustrations

Do any of the new illustrations have any value at all? The one on dyadic tensors seems a little bit offtopic for understanding the article, and it might be more suitable in the dyadic tensor article (if anywhere). I don't think it adds anything helpful or relevant to the Definitions section, so I have removed it.

The other recently-added graphics are more mystifying. The issues are numerous. First, the index conventions are wrong (what is $A_\rho \mathbf{e}_\rho$ supposed to mean, for instance?) Second, the vector itself should not be changed by a coordinate transformation: in fact, that's the whole point of having a covariant/contravariant transformation law, to ensure that the same vector is described with respect to the two different coordinate systems. Thirdly, there appears to be some confusion here between the basis vectors and the xⁱ coordinates: one of these should be, roughly, "dual" to the other, so it's not correct to have the basis vectors pointing along the "coordinate axes" in this way as far as I can tell. So, given these various issues, it's unclear what meaningful information is intended. Sławomir Biały (talk) 01:35, 27 May 2011 (UTC)

Bad Grammar

I've tried to correct basic English in this article to no good effect, as another editor keeps re-introducing the same grammatical errors. The section "As multidimensional arrays" contains the following sentence:

Just as a scalar is described by a single number and a vector can be described by a list of numbers with respect to a given basis, any tensor can be considered as a multidimensional array of numbers with respect to a basis, which are known as the "scalar components" of the tensor or simply its "components".

Any tensor (singular) can be considered as a multidimensional array of numbers (singular) with respect to a basis, which are known (plural!)... As written, the sentence ties the plural components back to the singular array instead of the plural list of numbers. To tie the scalar components back to the numbers and not the array, introduce a new sentence "These numbers are known as the scalar components..." or else re-write the existing sentence to remove the misprision. Ross Fraser (talk) 02:48, 8 June 2011 (UTC)

Well, there was no grammatical error, since "which" referred to numbers not "array of numbers". The correction you made, resulted in a nonsensical statement, on top of which it introduced a grammatical error in numbers by connecting array and components. None the less your confusion shows that the sentence was somewhat ambiguous. As such, I've corrected it by splitting the sentence in two.TR 07:49, 8 June 2011 (UTC)

Künneth theorem

User:TimothyRias requests citation about how tensors enter into the Künneth theorem. But the Künneth theorem is all about how Cartesian products of topological spaces map to tensor products of graded modules under homology. Right? It couldn't be more about tensors. Am I missing something here? Mgnbar (talk) 12:37, 6 September 2011 (UTC)

1) No matter how clear you think the statement is, it still requires a citation.

2) Tensor product != Tensor. The Kunneth theorem involves tensor products of modules rather than vector spaces, i.e. does not involve tensors in the conventional sense. If you claim that an element of a tensor product of modules is referred to as a "tensor" then that certainly needs a citation.TR 12:48, 6 September 2011 (UTC)

Regarding citation: Would a statement that the Künneth theorem involves tensor products require citation in this article? Or is that obvious to anyone who follows the link to Künneth theorem? In other words, is your request for citation entirely separate from your other issue, that tensor product != tensor?

Regarding the other point: To me, a "vector" is an element of a vector space, and a "tensor" is an element of a tensor space, and there is no reason to restrict to tensor spaces over fields. But I will try to find some time to see whether this terminology is used by people other than me.

I should add that this small disagreement ties into my general dissatisfaction, which I have voiced on other occasions here, about how this article seems to favor the physics point of view over the math point of view --- even going so far as to shunt one of the primary aspects of tensors into a separate article. Mgnbar (talk) 15:02, 6 September 2011 (UTC)

To your first question, yes. (Although the statement that really needs a citation is that tensors play a basic role in algebraic geometry).TR 15:35, 6 September 2011 (UTC)

Okay, now I understand your point better. That tensor products play a fundamental role in algebra topology (not geometry, mind) would be very easy to support with citation. Any introductory algebraic topology book, such as Bredon or Massey, is stuffed with tensor products. Additionally, now that I read the paragraph in dispute more completely, I can see how whoever wrote it (it wasn't me, by the way) attempted to stave off complaints such as yours. It even discusses tensors over rings that aren't fields. So it seems to me that a little citation and clarification is all that's needed. Mgnbar (talk) 16:36, 6 September 2011 (UTC)

Understanding check on tensor type

I have some assertions based on my present understanding of tensors, and if they're wrong, could someone explain why?

• My understanding is that a tensor with all the indices raised is contravariant and with all the indices lowered is covariant, and that the two tensors are dual $T^{ijk}\in V\otimes V\otimes V = (T_{ijk}\in V^* \otimes V^* \otimes V^*)^*.$ Because a space of one-forms is a vector space, the dual to the all-covariant tensor should be the same as looking at the all-covariant tensor as an all-contravariant tensor in the dual space - each covariant index is contravariant relative to the dual space. So, if you represented each contravariant index by a vector, the covariant version would be that same set of vectors in the dual space. And vice versa - the level set representation of each covariant component is the same after taking the dual tensor and observing it in the dual space.

Going on this I imagine raising and lowering indices as looking at pieces of an same object being pushed through a door between the collective vector and dual spaces, and when all the pieces are to one side or the other it looks the same from that side of the door (the two products of all non-dual spaces). ᛭ LokiClock (talk) 22:50, 19 December 2011 (UTC)

• Since the object itself isn't altered by a change of basis, only its matrix representation, lowering a component $T^{ij} \rightarrow T_{j}^{i}$ and then applying a rotation will show the contravariant component moving against the rotation, and the covariant component moving in advance of the rotation (or is it equal?), but when you raise the index again it will be the same as if the rotation was applied with both indices contravariant $R_j^iT^{ij} = g^{jj}R_j^iT_{j}^{i}.$ ᛭ LokiClock (talk) 22:50, 19 December 2011 (UTC)

I've moved this discussion to User_talk:LokiClock#Understanding_check_on_tensor_type. — Quondum^t_c 05:54, 20 December 2011 (UTC)