User talk:Orthografer

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Hello Orthografer, and welcome to Wikipedia! Thank you for your contributions. I hope you like the place and decide to stay. Here are a few good links for newcomers:

I hope you enjoy editing here and being a Wikipedian! Please sign your name on talk pages using four tildes (~~~~); this will automatically produce your name and the date. If you have any questions, check out Wikipedia:Where to ask a question or ask me on my talk page. Again, welcome!  - UtherSRG (talk) 19:59, 9 December 2005 (UTC)

[edit] Re:Stein manifold

I replied on talk:Stein manifold. Cheers, Oleg Alexandrov (talk) 16:24, 13 December 2005 (UTC)

[edit] Re: Indices in Christoffel symbol article

>> Hi, I noticed that you changed the symbols from \Gamma^m_{i\ell} to \Gamma^m {}_{i\ell} in the above article. Recently another user changed it back and I've been asking why either edit happened. I'm very curious - can you explain your end? Orthografer 15:34, 21 August 2006 (UTC) <<

Even though the Christoffel symbol is not really a tensor, notationwise it behaves very much like one. For tensors, there is a notation in which each index is assigned its own "column", in which it is either a superscript or a subscript. In this notation (that I'm using), \Gamma^m {}_{i\ell} is explicitly different from Γiml or Γilm, and if the m index is lowered, then the symbol becomes simply Γmil.
In the alternative notation, there are no "columns": the superscript is "independent" of the subscripts, and the indices might be arranged in a "Delta" (Δ) configuration, with the two subscripts forming the base and the superscript forming the top of the triangle (otherwise the superscript is flushed to the left, just like the subscripts). If the superscript (m) is lowered, the result is a Christoffel symbol of the first kind: Γil,m.
There is even a third notation, in which the Christoffel symbol of the second kind is \left\{ {}_i {}^m {}_l \right\}, with the indices in a Delta configuration, and the Christoffel symbol of the first kind is [m,il].
Which notation is used depends on the author. For example, Bernard F. Schutz (from Cambridge Uni) in his book A first course in general relativity uses the first notation (the one which I was using); Donald H. Menzel (from Harvard Uni) in his book Mathematical Physics uses the alternative notation; whereas Borisenko and Tarapov (from Russia) in their book Vector and Tensor Analysis with Applications use the third notation. Schutz always uses Christoffel symbols of the second kind, never of the first kind, and commas in the subscript row are reserved for differentiation (instead of for Christoffel symbols of the first kind) (see Covariant differentiation#Notation).
When I made the edits I happened to prefer the Cantabrigian notation, and now I notice that Wikipedia has kept it. The bracketed notation used by those two Russians I don't like: I definitely prefer the use of Gamma, whether it have its indices separated in columns or "independent" (whichever of these two alternatives is used, should not be a big deal, really). However, the Russian book is antiquated (copyrighted 1968) and checking the "Christoffel symbol" article in other languages (French, Spanish, German, Dutch, Russian, Chinese) one sees that they all use Gamma, though some use indices separated by columns (German, French, Russian) whereas some do not (Spanish, Dutch, Chinese). The Dutch version mentions the difference between Christoffel symbols of the first kind and of the second kind. The Spanish version ignores Christoffel symbols of the first kind, which so does the English, by the way (but so do the German, French, and Russian: just like Schutz). The Dutch version does not use a Gamma for the Christoffel symbol of the first kind: instead it uses brackets: [m,il]. Go figure... --AugPi 22:49, 11 November 2006 (UTC)
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