ChapterZero

I’m attempting to read way too many books at one time. I had to give up on Solid Shape; it seems that it would be more beneficial to read that book after I already understand the concepts that it attempts to present in a non-mathematical manner. I would rather gain the technical knowledge and then build the intuition, as opposed to culture an incorrect intuition that interferes with learning the technical details.

Anyhow, I found yet another nice book on Differential Geometry: A Comprehensive Introduction to Differential Geometry, by Michael Spivak; I think there are 5 volumes. I have to say, I ordered the first volume from the library without having a clue what to expect. I thought that, since it is comprehensive, it would cover the basic curve and surface theory stuff that we’re doing in class— which, incidentally, is why I was ordering a dg book in the first place— there was only one good one in the library (Elementary Differential Geometry by Springer), and I turned that back in so a friend who needed it more than I (i.e. was taking the class for credit) could check it out. Turns out it doesn’t seem to have that kind of intro dg stuff at all, but it still seems pretty interesting. It is the first book, dg or not, that I’ve seen that has a straight forward, non-intimidating definition of a manifold, offered on page 1:

… a manifold is a metric space with the following property: if , then there is some neighborhood of and some integer such that is homeomorphic to .

No mention of derivatives, or the implicit function theorem! Yeah! It’s nice that he didn’t jump right into that stuff, which I personally find very confusing. Speaking of the implicit function theorem, at UH, we learn about that in the Advanced Multivariable Calculus class, or at least we’re supposed to. When I took it, we spent maybe a month on basic vector calculus: identities, box product, etc. Nothing even as complicated as determinants, which we covered in a day. But we spent maybe two weeks on the implicit function/inverse function theorems. Considering how important it seems to be, I’m suprised and disappointed by that. But whatever— now, I’m getting the idea of finding out when those topics will be covered in that class this semester, and sitting in on those lectures.