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Links to all things tensors and indices!!!

Numerous articles have been searched out and linked to here, so people have access quicker and other editors who havn't participated at wikiproject maths talk can still stumble upon here quicker.... =) F = q(E+v×B) ⇄ ∑_ic_i 21:54, 11 April 2012 (UTC)

Names

absolute tensor calculus was the name used by Ricci. Schouten formalized the changes and extensions that Einstein needed for the ART, so Schouten (1924) Der Ricci-Kalkül should be considered a final summarization of the tensor calculus in gravitational physics and differential geometry before WWII and the Manhattan project.--LutzL (talk) 12:10, 12 April 2012 (UTC)

Thats fine, thanks for that, though you could have edited the lead to incorperate your statement. I'll do it now. =) F = q(E+v×B) ⇄ ∑_ic_i 13:00, 12 April 2012 (UTC)

Btw, the tensor article says absolute differential calculus rather than absolute tensor calculus. "Differential" has been added to the article, so whichever it is, if its wrong in the lead please just change it.

Forgot to ask, what is "ART", in relation to Einstien/anyone? F = q(E+v×B) ⇄ ∑_ic_i 13:05, 12 April 2012 (UTC)

ART = Allgemeine RelativitätsTheorie^(LutzL is German).TR 14:44, 12 April 2012 (UTC)

ok. Thanks. F = q(E+v×B) ⇄ ∑_ic_i 14:51, 12 April 2012 (UTC)

Raising and lowering indices

The statement "Any index of a tensor can be raised and lowered using a nonsingular metric tensor" is problematic. It depends on what convention one uses when defining the tensor in question. In particular, there are at least two situations where this may cause a problem:

• In defining the Levi-Civita symbol, the most natural course is to use the same symbol, say ${\displaystyle \varepsilon _{ijk}}$, with both upper and lower indices since the actual value of the symbol is the same in both cases. However, following the raising-lowering convention above would require the use of two different symbols for the same thing.
• As one can see at Alternatives to general relativity and Bimetric theory, it is sometimes necessary to consider situations where there are two (or more) metric tensors. In this case, blindly trying to follow the convention would result in a contradiction since it would force the two metrics to be equal.

${\displaystyle g^{\alpha \beta }A_{\beta }=A^{\alpha }=\gamma ^{\alpha \beta }A_{\beta }\,.}$

Consequently, I believe that this part of the article should be qualified. JRSpriggs (talk) 04:56, 4 May 2012 (UTC)

It should probably be phrased more clearly that this operation changes the tensor. (Which makes both points you make irrelevant.) Also note that the Levi-Civita symbol is not a tensor.TR 05:48, 4 May 2012 (UTC)

I had debated wording it as "Any index of a true tensor...", but decided against it as the meaning would not be adequately clear. From the above, the situation is even more problematic than I had been aware of. Anyone want to try clarifying this in a way that is in keeping with the summarizing nature of the article? (Simply saying "not always" leaves one wondering what is meant.) — Quondum 08:49, 4 May 2012 (UTC)

To TimothyRias: The Levi-Civita symbol is a tensor density (in more than one way) as is explained at Levi-Civita symbol#Tensor density. The Ricci calculus extends to tensor densities and certain non-tensors such as the Christoffel symbols. JRSpriggs (talk) 11:58, 4 May 2012 (UTC)

But a tensor density is not a tensor, hence there is no real problem here.TR 13:05, 4 May 2012 (UTC)

How about (something like)

"By contracting an index with a metric tensor, the type of a tensor can be changed converting a lower index to an upper index or vice versa."

This stresses the fact the raising or lowering an index changes the tensor even though the same symbol is continued to be used. TR 13:05, 4 May 2012 (UTC)

I like that – it's a major improvement. At the risk of sacrificing brevity, one might add "The base symbol in many cases is retained, and repositioning an index is often taken to imply this operation when there is no ambiguity."

On a different but related point: the article is missing a very important operation that would contextualize this better, namely coordinate basis transformations (which may be tied to coordinate transformations on a differentiable manifold)). — Quondum 13:30, 4 May 2012 (UTC)

That is covered at some length in our tensor article. As I understand it, the main function of this article is to summarize the notational convention for tensor index notation.TR 14:56, 4 May 2012 (UTC)

To Quondum: Your rewording of that part of the article looks good to me. Thank you.

To TimothyRias: And thanks to you for suggesting part of the new wording. JRSpriggs (talk) 15:10, 4 May 2012 (UTC)

Recently busy so haven’t touched the article for a while... thanks to all of you =) for the good work and fixing yet more of my errors =(. Much appreciated...

However... could I please ask that for simple symbols to type we use wiki markup and not LaTeX, e.x. A and not ${\displaystyle A}$ (and no I'll not italicize the Greek letters, that was only to match any LaTeX formulae). If the symbols are clearer in LaTeX then those can be used of course, e.x. ${\displaystyle \scriptstyle \gamma }$ instead of γ (gamma). Its just for neater appearance... F = q(E+v×B) ⇄ ∑_ic_i 20:49, 5 May 2012 (UTC)

Most of the formulae in the article are amenable to HTML use due to their simplicity; I seen some articles where this is done throughout. It tends to produce cleaner, more readible fonts than either PNG or MathJax but cannot deal with complex formulae (optionally using {{math}} for a serif font). The point is that you, as the original author, get to choose. — Quondum 00:44, 6 May 2012 (UTC)

Well no - it’s not me that gets to choose just because I did start the article... It would look better to use LaTeX for the main displayed formulae, but all I was saying was not for inline letters which are easy like "A, B" etc. The current formatting style of the article is fine as it is. I don't think it’s that essential to use {{math}}, but if anyone would like to then of course feel free... F = q(E+v×B) ⇄ ∑_ic_i 06:53, 6 May 2012 (UTC)

Naming of derivatives

The recently added "The first partial derivative is also called the gradient of a tensor field. If this is followed by a contraction between the tensor field and coordinate variable, the result is the divergence of a tensor field" is extrapolating the standard terminology somewhat. Please feel free to give references by way of contradicting me. The terms "gradient" and "divergence" are, AFAIK, applicable only to a scalar (a (0,0)-tensor) and a vector (a (1,0)-tensor) respectively, and then only make sense when using the covariant derivative. I am doubtful that the term "gradient" is used in this way, since the term "covariant derivative" means exactly this. The concept of divergence (∇⋅) does not naturally extend in this way, because the index of contraction is unspecified. The article Tensor derivative (continuum mechanics) does define these concepts, but this looks like WP:OR, and most certainly does not apply in the general contexts that Ricci calculus covers. I'm trimming this from the article. — Quondum 11:53, 10 June 2012 (UTC)

What I wrote is definitley in Gravitation (book) by J.A.Wheeler et al, p.82 says:

• the gradient of a tensor field S_βγ has components S_βγ,δ.
• the divergence of a tensor field S_βγ has components S_βγ,α - i.e. the gradeint then a contraction.

F = q(E+v×B) ⇄ ∑_ic_i 12:07, 10 June 2012 (UTC)

I think what you're saying about the directional derivative is this (again on p.82):

${\displaystyle {\boldsymbol {\nabla }}{\boldsymbol {\mathsf {S}}}({\mathbf {u}},{\mathbf {v}},{\mathbf {w}},{\boldsymbol {\xi }})=\partial _{\boldsymbol {\xi }}{\boldsymbol {\mathsf {S}}}({\mathbf {u}},{\mathbf {v}},{\mathbf {w}})=\partial _{\boldsymbol {\xi }}(S_{\alpha \beta \gamma }u^{\alpha }v^{\beta }w^{\gamma })=S_{\alpha \beta \gamma ,\delta }u^{\alpha }v^{\beta }w^{\gamma }\xi ^{\delta }}$

which is not quite the same... maybe your'e right about the gradient so apologies. =(

However, the divergence is undoubtably the same from that book, and it does say on p.85 that the gradient of tensor is exactly as I wrote using the comma above... that page says "gradient of tensor N has components N_β,γ to form a new tensor S with components S_βγ." So I'm still inclined that the additional statements to the article were correct... but we can leave them out, its not essential. F = q(E+v×B) ⇄ ∑_ic_i 12:18, 10 June 2012 (UTC)

No, I'm not making the distinction that you're making. I'm taking issue with Wheeler's terminology, and suggesting that it does not make sense in the general context. Does he define the terms generally, or is he defining them in a restricted context (e.g. rectilinear coordinates in flat space)? — Quondum 12:31, 10 June 2012 (UTC)

Its very early in the book, and the first section is in flat spacetime, not curved, so I geuss "general context in flat space time", not in actual generality. I'm not sure about how to handle curved spaces yet, so lets just leave them from the article (or possibly state that those statements are true in flat spacetime only, but I'm not fussed)... F = q(E+v×B) ⇄ ∑_ic_i 12:43, 10 June 2012 (UTC)

It sounds rather like the book is taking a pedagogical approach, trying to build on understanding by using familar terminology from vector calculus. I doubt that the terms remain in use in the final developed subject (and hence should not be in this article). And for clarity, even in a flat space, the statements are invalid without the additional restriction on the coordinate system that it must be rectilinear: i.e. at right angles, straight, and with the same scale, as with Cartesian coordinates. — Quondum 13:01, 10 June 2012 (UTC)

I see what you mean, thanks. F = q(E+v×B) ⇄ ∑_ic_i 13:06, 10 June 2012 (UTC)

Notice that the (partial = covariant) divergence of a fully-antisymmetric contravariant tensor density of weight +1 is also a fully-antisymmetric contravariant tensor density of weight +1, whether the order of the tensor is: 1, 2, 3, or 4. JRSpriggs (talk) 16:30, 10 June 2012 (UTC)

Uhh. Maybe tensor densities fit in here in some way; for now tensor densities are beyond my ken, so I'm unable to comment or sensibly use this for the article. And now that you mention it, the term divergence probably does make sense for a fully antisymmetric contravariant tensor . — Quondum 16:48, 10 June 2012 (UTC)

Wedge product

The definition of the wedge product added today is rather curious. Primarily, the factor of p! does not match the definition in exterior algebra, nor in geometric algebra. Or, for that matter, the definition in my Collins Dictionary of Mathematics. Secondarily, is the term "wedge product" in general use in the context of Ricci calculus? — Quondum 12:47, 10 June 2012 (UTC)

Ok - that was from Gravitation also. I added those extra bits becuase if we have all the stuff on commutator coefficients, Lie brackets etc, we should at least state the tensor product components, wedge product components, and that tensor components are multilinear functions of basis vectors which a reader should know first. I have no intension of extensive description. Should we just remove them? F = q(E+v×B) ⇄ ∑_ic_i 12:53, 10 June 2012 (UTC)

There may be merit in showing how the wedge product can be expressed in terms of tensors. My main problem at the moment is that someone would give a definition of a "wedge product" that is close to what I'm familiar with in several closely related fields, but wrong by a factor. The commutator and the Lie bracket have the factor as you gave it, but the the wedge product is not the same thing. — Quondum 13:09, 10 June 2012 (UTC)

The reason for the factor p! is becuase of the antisymm of the components as defined in the article, but I'm mistaken on how the components are summed.

According to the main article on wedge product, for two 2d vectors written in the notation of this article - a = (a, a), b = (b, b):

${\displaystyle {\begin{aligned}{\mathbf {a} }\wedge {\mathbf {b} }&=(a^{1}{\mathbf {e} }_{1}+a^{2}{\mathbf {e} }_{2})\wedge (b^{1}{\mathbf {e} }_{1}+b^{2}{\mathbf {e} }_{2})\\&=a^{1}b^{1}{\mathbf {e} }_{1}\wedge {\mathbf {e} }_{1}+a^{1}b^{2}{\mathbf {e} }_{1}\wedge {\mathbf {e} }_{2}+a^{2}b^{1}{\mathbf {e} }_{2}\wedge {\mathbf {e} }_{1}+a^{2}b^{2}{\mathbf {e} }_{2}\wedge {\mathbf {e} }_{2}\\&=(a^{1}b^{2}-a^{2}b^{1}){\mathbf {e} }_{1}\wedge {\mathbf {e} }_{2}\end{aligned}}}$

and using more notation from this article:

${\displaystyle {\begin{aligned}{\mathbf {a} }\wedge {\mathbf {b} }&=(a^{1}b^{2}-a^{2}b^{1}){\mathbf {e} }_{1}\wedge {\mathbf {e} }_{2}\\&=2a^{}{\mathbf {e} }_{1}\wedge {\mathbf {e} }_{2}\\&=2a^{}({\mathbf {e} }_{1}\otimes {\mathbf {e} }_{2}-{\mathbf {e} }_{2}\otimes {\mathbf {e} }_{1})\\&=2a^{}{\mathbf {e} }_{1}\otimes {\mathbf {e} }_{2}-2a^{}{\mathbf {e} }_{2}\otimes {\mathbf {e} }_{1}\\\end{aligned}}}$

which proves my error. For now, the equations will be corrected to have the basis wedge products as shown here. F = q(E+v×B) ⇄ ∑_ic_i 13:37, 10 June 2012 (UTC)

On second thought maybe there is no point. Lets just leave it at the tensor product (which is correct) and link to the main article. F = q(E+v×B) ⇄ ∑_ic_i 13:45, 10 June 2012 (UTC)

To correct an error in your working:

${\displaystyle {\mathbf {e} }_{1}\wedge {\mathbf {e} }_{2}=({\mathbf {e} }_{1}\otimes {\mathbf {e} }_{2}-{\mathbf {e} }_{2}\otimes {\mathbf {e} }_{1})/2}$

— Quondum 14:08, 10 June 2012 (UTC)

... I should have known. What you wrote does show up in Wheeler et al, on p.92 (and others, though I seemed to have neglected the boxful of properties on basis differential forms, which I don't understand and shouldn't write about, yet became fascinated and ego-istic. Not good.) =( Thank you for pointing this out, and for your guidance throughout this, also for trimming the intro section on coord. labels and indices which I made too long and wordy (excessive clarification is not clarification...).

On a more general note (and before anyone raises the question) - I'll stop adding material to the article now, anything else will spill out of context completely (unless there are more relevant conventions I'm not aware of then please add them). The recent additions were to add extra clarification not already emphasized in the article, and to transfer understanding to the reader, on pedagogical/mnemonical and technical attributes of the notation. F = q(E+v×B) ⇄ ∑_ic_i 14:38, 10 June 2012 (UTC)

On further thought, while the wedge product does make sense (i.e. is well-defined) on a particular subalgebra of a tensor algebra (in particular the space spanned by all fully antisymmetric tensor products of 0 to n basis covectors, n being the vector space dimension), it does not appear to make any sense for general tensors. On the other hand it is closely related to the Levi-Civita symbol, which is used extensively in the Ricci calculus: so don't take this comment too seriously. It would be interesting to know whether your reference defines the wedge product more generally. — Quondum 16:02, 10 June 2012 (UTC)

On p.92 the wedge product has the usual linearity and associative properties, in addition the commmutation rule:

${\displaystyle {\boldsymbol {\alpha }}\wedge {\boldsymbol {\beta }}=(-1)^{pq}{\boldsymbol {\beta }}\wedge {\boldsymbol {\alpha }}}$

where α is a p-form

${\displaystyle {\boldsymbol {\alpha }}={\frac {1}{p!}}\alpha _{|i_{1}i_{2}\cdots i_{p}|}\mathbf {e} ^{i_{1}}\wedge \mathbf {e} ^{i_{2}}\cdots \mathbf {e} ^{i_{p}}}$

and simalarly β is a q-form.

Indeed

${\displaystyle {\boldsymbol {\alpha }}\wedge {\boldsymbol {\beta }}={\frac {1}{2}}(\alpha _{j}\beta _{k}-\alpha _{k}\beta _{j})\mathbf {e} ^{j}\wedge \mathbf {e} ^{k}}$

F = q(E+v×B) ⇄ ∑_ic_i 16:14, 10 June 2012 (UTC)

Ah, yes. It's starting to come back to me now. If we express differential forms as tensor fields, it should all hang together. This is precisely the subalgebra that I was referring to. — Quondum 16:32, 10 June 2012 (UTC)

Here is something else I found now:

There is the formula for the exterior product of p vectors:

${\displaystyle {\begin{aligned}(\mathbf {u} _{1}\wedge \mathbf {u} _{2}\cdots \mathbf {u} _{p})^{\alpha _{1}\cdots \alpha _{p}}&={\mathcal {E}}_{\mu \cdots \nu }^{\alpha _{1}\cdots \alpha _{p}}(u_{1})^{\mu }\cdots (u_{p})^{\nu }\\&=p!(u_{1}){}^{}\\&={\mathcal {E}}_{1\cdots p}^{\alpha _{1}\cdots \alpha _{p}}\det\end{aligned}}}$

with the permutation tensor

${\displaystyle {\mathcal {E}}_{\mu \cdots \nu }^{\alpha _{1}\cdots \alpha _{p}}=-\epsilon ^{\alpha _{1}\cdots \alpha _{p}\chi }\epsilon _{\mu \cdots \nu \chi }}$

Would this be of any interest/use? F = q(E+v×B) ⇄ ∑_ic_i 16:48, 10 June 2012 (UTC)

I think so – it hangs together with the use of the Levi-Civita symbol, p-forms and the like. I've never been familiar with this area, but I'll certainly like to get to know it well enough to flesh out this part of the article. So let's just hang onto that thought for now. — Quondum 17:28, 10 June 2012 (UTC)

New section on the Levi-Civita symbol and Hodge dual

For starters, see here. A new section has been suggested by Quondum. =) F = q(E+v×B) ⇄ ∑_ic_i 16:03, 10 June 2012 (UTC)

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