User talk:Shellgirl

Welcome!

Hello, Shellgirl, and welcome to Wikipedia! Thank you for your contributions. I hope you like the place and decide to stay. Here are some pages that you might find helpful:

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And don't forget, the edit summary is your friend. :) – Oleg Alexandrov (talk) 16:34, 4 April 2007 (UTC)[reply]

Best linear approximation

And a warm welcome from me too! I see you have already made some useful edits to several pages. Concerning Derivative, although I am a big fan of the "best linear approximation" point of view, I wasn't convinced when Innerproduct added this paragraph recently, as the first section is already rather overloaded, and linear approximation is discussed in a later paragraph. I would be inclined to delete it, or use it to elaborate the final paragraph of the "Jacobian and differential" section. Let me know what you think, either here, or we can take the discussion over to Talk:derivative.

My other reservation about this is (as you have probably seen for yourself) that Big O notation is something of a mess from a mathematical point of view, since it is heavily geared towards the compute science perspective: if you feel the impulse to improve it at some point, I encourage you to be bold! I hope you have fun here, anyway. Geometry guy 17:22, 4 April 2007 (UTC)[reply]

Jessica replies Hi geometry guy, and thanks for the welcome!

I like the linear approximation definition, since it is a geometric intuition that persists through all levels of sophistication and at the same time is accessible to novices. For the same reason, I don't like the use of little o notation in this paragraph, since this is likely to pull a novice mathematician out of context -- especially because this notation isn't used in most calculus classes, as far as I am aware. I preserved it from an earlier edition, but now I think I'll go back and take it out.

It seems like the best place to make edits is at the edge of my understanding. That is, when I first understand something clearly and well -- perhaps with the help of wikipedia -- and there is some change that I think would have helped out my process of comprehension. But of course, this takes a little more nerve.

I am hesitant to do a lot of editing on big O notation, precisely because I am a mathematician and not a computer scientist. Although I guess there are certainly times when I use big O notation (for example when examining asymptotic behavior of stastical physics models).

David Mumford has an interesting discussion about presenting calculus where he quotes Lancelot Hogben's introduction to the derivative from Mathematics for the Millions, where he points out that "virtually no sentence in English not written in mathematical jargon has an unambiguous interpretation not depending on the use of common sense by the listener." He also points out that once a precise definition is given, "most students become convinced something very complicated must be going on," and that this belief can even be exacerbated by also presenting a simple description of the meaning (i.e., they don't get the precise statement and now additionally they don't get the connection between the precise statement and the intuitive description). His article is at http://www.dam.brown.edu/people/mumford/Papers/CalcReform.pdf.

On a subject up your alley. Suppose you have a connection on the tangent bundle of a smooth complex manifold, not inherited a priori from a metric but just a bundle whose direct sum with the vertical sub-bundle gives the whole tangent bundle of the tangent bundle. If parallel transport by this connection gives complex linear maps on the tangent spaces of the manifold, does it follow that the connection induces a Riemannian metric?

Cheers,

Jessica