User talk:Orthografer

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Re:Stein manifold

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

Re: Indices in Christoffel symbol article

>> Hi, I noticed that you changed the symbols from $\Gamma _{i\ell }^{m}$ 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 $\Gamma _{i}{}^{m}{}_{l}$ or $\Gamma _{il}{}^{m}$, and if the m index is lowered, then the symbol becomes simply $\Gamma _{mil}$.

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: $\Gamma _{il,m}$.

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)

Coriolis

You just added a bit of a cryptic concern at Talk:Coriolis effect. The argument there has been dead since April. The article does not state that the Coriolis effect is important in tornadoes, only in cyclones, which agrees with Tornado#rotation. So where is the contradiction? Rracecarr 16:10, 19 August 2007 (UTC)

The second sentence in the cyclone article it says that "cyclone" is a general term that applies to tornadoes. Orthografer 23:15, 19 August 2007 (UTC)

Thanks!

Hi Orthografer—thanks for the barnstar, and glad you've enjoyed the edits! I studied manifolds and surgery for some time (and algebraic surfaces before that), and I've been writing up what I know; I've a few more general manifold contributions, and lots to write on surgery theory. Good luck with your research, and look forward to continuing to contribute! Nbarth (talk) 00:41, 17 November 2007 (UTC)

Differential Structures

Hi there, I really don't understand how talk pages work around here, so it looked plausible that I'll have to notify you in person of some content I added to the talk page of Differential structure. Here it is, I hope you can help (& I promise to read up on how things are done in Wikipedia once I find the time...).

Are we sure we don't? Because it looks to me like we do need it. In particular, having constructed the equivalence class of C^k-compatible atlases, one would first need to equip it with a partial ordering (trivial: otherwise maximality means nothing; we take the partial ordering to be the usual set inclusion), then show that a maximal atlas in that class exists, and finally show that it is unique (trivial: if it is not, there are at least two such maximal atlases, so their union is a member of the class and properly contains both, which contradicts their maximality). The second step (existence), though, can be carried out through Zorn's lemma (since every chain has, indeed, an upper bound: the union of its elements) & Zorn's lemma is equivalent to the Axiom of Choice. Am I making a mistake somewhere? Athenray (talk) 22:56, 3 November 2008 (UTC)

I also can't see this when I look at the article's talk page w/o having logged in - what a mess... Anyway, thanks! Athenray (talk) 11:21, 4 November 2008 (UTC)

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