Talk:Raising and lowering indices

From Wikipedia, the free encyclopedia
Jump to: navigation, search
WikiProject Mathematics (Rated C-class, High-importance)
WikiProject Mathematics
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
C Class
High Importance
 Field: Geometry

g_{ij}g^{ij}[edit]

I have corrected the article which was stating that the product of g_{ij}g^{ij} was \operatorname{Tr}(g)=N. Since g^{ij} is the inverse of g_{ij} their product is the identity matrix and thus one obtains the trace of the identity which is, correctly, N in a N-dimensional manifold. -- CristianCantoro (talk) 16:01, 30 July 2010 (UTC)

My own reversion[edit]

Quondum: just in case you're wondering why I reverted myself, its because I copied + pasted + edited the markup separately, then pasted back into the edit panel + saved then realized you had edited. Your tweaks were better than mine so I undid my paste. Apologies for that. F = q(E+v×B) ⇄ ∑ici 13:18, 31 March 2012 (UTC)

Considering that recently I made made only a very minor change to this article (substituting "tensor field" for "tensor"), I guess you must be refering to slightly earlier edits by someone else. Keep up the good work. — Quondum 13:57, 31 March 2012 (UTC)
I thought you did more at a first glance (don't ask...). I'll revert my own change back. Thanks for your encouragement. =) F = q(E+v×B) ⇄ ∑ici 14:52, 31 March 2012 (UTC)

Dual tensor[edit]

The new section (written entirely by me) will be split off to a new article: Dual tensor with redirects Dual of a tensor and Tensor dual. It doesn't really fit too well here (at first I thought it would); just because it involves raising indices, so is this done everywhere else, rather it's another operation worthy of it's own article.

It's astonishing that there is no article on this, for instance in classical electromagnetism and special relativity, where to link the sentence

"There is another way of merging the electric and magnetic fields into an antisymmetric tensor, by replacing E/c B and B → − E/c, to get the dual tensor Gμν."

?? And no - this is not the same thing as a dual space in the context of vector spaces of covectors. F = q(E+v×B) ⇄ ∑ici 15:12, 10 June 2012 (UTC)

But it is almost certainly the same thing as the Hodge dual, and should be called that. Though I prefer the (very) closely related Clifford dual, which is defined by Lounesto and is better-behaved. — Quondum 15:37, 10 June 2012 (UTC)
(ec)PS: Beware, both these duals are (almost certainly) only defined on the same subspace of the tensor algebra as the wedge product is defined; i.e. not on general tensors. So to refer to the Hodge dual of a tensor may be misleading. — Quondum 15:42, 10 June 2012 (UTC)
That article is so abstract, and less followable... Its on anyone who would like to merge. F = q(E+v×B) ⇄ ∑ici 15:40, 10 June 2012 (UTC)
I think that Ricci calculus should deal with it in a section on Levi-Civita symbol and Hodge dual. These are significant aspects of the Ricci calculus not yet covered. Put it in there and we can streamline it later. — Quondum 15:50, 10 June 2012 (UTC)
When Ricci calculus was first started, I did want to write about it there, but thought it was off-topic so left it. Lets go with your plan though, its better than nothing. =) F = q(E+v×B) ⇄ ∑ici 15:54, 10 June 2012 (UTC)
It has been transplanted, and all redirects and double redirects eliminated. F = q(E+v×B) ⇄ ∑ici 16:00, 10 June 2012 (UTC)