Book Blogging: Geometrical Methods of Mathematical Physics

Another book, because reading one book at a time is not nearly enough.

Next on our menu: Geometrical Methods of Mathematical Physics, by Bernard Schutz, Cambridge University Press 1980. (Available from Amazon) for $32.

First, let me say a few words about the physical artifact. Most CUP paperbacks I own are excellently bound, with pages sewn in signatures. That is not the case for my copy of Schutz. After a bit of hard use the glued binding has cracked, and the first sixty-six pages have split from the rest and partially separated from the spine. I also have the impression that the printing is slightly muddy, or too small, or otherwise unsuited to a text very liberally festooned with subscripts and superscripts, some of which are themselves further decorated. There is a hardback version for about five times the cost.

So why bother? The book is a compact, lucid, and relatively elementary presentation of a bunch of geometric ideas which have become central to modern physics. After an introductory chapter, there are three mathematical chapters, devoted respectively to differentiable manifolds and tensors, Lie derivatives and Lie groups, and differential forms. The final two chapters are (5) Applications in physics and (6) Connections on Riemannian manifolds and gauge theories.

One strength of the book is a modest sprinkling (120 or so) of exercises which are embedded in the text, for which the author has provided hints and partial solutions in an appendix. These exercises form an integral part of the text and pretty much need to be done to fully understand the material. Schutz does a nice job of dividing up the material into bite size chunks, about 117 sections averaging a couple of pages each. Each chapter has a tersely annotated bibliography.

I have already started reading the book, but since I neglected to do the exercises in my start, I am now retracing, just in case anyone else wants to read along. I will post occasional comments and welcome questions or comments from others.

Prerequisites (from the back cover blurb): “

The reader is assumed to have some familiarity with advanced calculus, linear algebra, and a little elementary operator theory. The advanced physics undergraduate should therefore find the presentation quite accessible.”

I will post later on the first chapter, a quick review of “Some Basic Mathematics.”