Customer Reviews

Topology and Geometry for Physicists

5 star:		(1)
4 star:		(2)
3 star:		(2)
2 star:		(1)
1 star:		(0)

 

 

Nash's book commits the sin many mathematical physics textbooks out there commit: "oh, we're writing for dimwit physicists, lets just give them a few scrawny examples and assure them everything else works alright." I'm sorry but writing for physicists is NOT an excuse for writing a sloppy textbook. Would you feel alright not knowing how an integral is defined? Would you use a numerical evaluation software to calculate integrals in serious research without understanding the algorithm it uses? If you do then you're a pretty shoddy physicist. I'm not saying this out of some "macho" sentiment many purist physicists have - I'm simply saying this because I feel the way this book teaches you diff. geometry is wrong - it teaches you to draw pictures and go by the pictures. When the pictures run out, so does your understanding.

This book is supposed to teach differential geometry. However, very little can be learned from it unless one already knows differential geometry: definitions are sometimes not general and sometimes not present at all, theorems are often stated only for special cases and even more often than that not proved at all. Sure, the book offers nice geometrical intuition, but this is not enough. An example: the book "proves" Stoke's theorem around page 40. Now, even a rigorous and condensed book would have problems doing that, considering the amount of "machinery" one needs to build up for it (tensors, differential forms, manifolds and so forth). This means the book makes a mess of it - big time.
There are many fine diff. geometry books out there, some for physicists, some not, which you should check out - Nakahara's text is so much better. For geometrical intuition I suggest picking up Schutz's book. Several books from the GTM (Graduate texts in mathematics series, the yellow ones) are really very accessible, such as Introduction to Topological Manifolds/Smooth Manifolds. Another good one is Allen Hatcher's Algebraic Topology for homotopy, homology and cohomology. For a good and responsible exposition, do yourself a favor and look for something else.

When reading this book one can both admire these authors and feel sympathy with them. They have made an honest effort to explain the concepts of differential geometry and topology in a way that is understandable and appreciated by the physicist reader. But the book falls short in many places, although there are some places where they do a fine job. They have taken on a very difficult project in this book, for it is quite straightforward to expound on the formalism of mathematics, but explaining it in a way that grants insight into its conceptual meaning is another matter altogether. Many physicists complain, with justification, that the way mathematics is presented in textbooks is not sufficient for giving them a deep appreciation of the underlying ideas involved. This, they argue, is what is needed for devising new physical theories and results based on these ideas. Physicists must assimilate very complex mathematical ideas very quickly in order to formulate these theories in a reasonable time frame. This is especially true in high energy physics, which in the last two decades has used mathematics like it has never been used before. Indeed, the mathematical complexity of high energy physics is dizzying, and if progress is going to be made in this field by the students of the 21st century, they are going to need mathematics books and documents that are more than just formal expositions. But, again, writing these kinds of books is very hard to do, and has yet to be done in a book to this date, although there are helpful discussions scattered throughout the mathematical literature.

Some of the concepts that need more in-depth explanation include: the theory of characteristic classes, sheaf theory, the theory of schemes in algebraic geometry, and spectral sequences in algebraic topology. There are of course many others, and some of the ones that the authors do a fairly good job of explaining in this book include: 1. the reason that the continuity of a function is defined in terms of inverses of open sets; 2. The orientability of a manifold; 3. The fundamental group and its relation with the first homology group. 4. The discussion on Morse theory.

This book is written by physicists. Like a book by M. Nakahara it describes basics of diff geometry and topology. Though it stresses physical intuition more than formal definitions. I especially liked discussion of fiber bundles and characteristic classes. Highly recommended.

Unlike many physics students, I grant a lot of leeway to books on mathematics for physicists. I think it's all right for an author to engage in hand-waving arguments if this enhances physical intuition or even to make the occasional statements without proof if this allows more ground to be covered. However, if a proof actually is presented, I expect this proof to be correct. In this book, proofs are sometimes only for special cases of theorems stated more generally and often contain logical errors.

This book shows you the geometric view of some advanced mathematical topics. It can greatly assist your intuition of what is going on in a mathematical setting when reading a true mathematics book. Armed with this book the other advanced text in Topology, Algebraic Geometry and Differential Geometry make more sense from a Physics point of view.

This book shows you the geometric view of some advanced mathematical topics. It can greatly assist your intuition of what is going on in a mathematical setting when reading a true mathematics book. Armed with this book the other advanced text in Topology, Algebraic Geometry and Differential Geometry make more sense from a Physics point of view.