The Geometry of Physics: An Introduction [Paperback]

This book is definitely a must for the mathematically minded physicist. Self-contained, logically structured throughout, absolutely consistent mathematical notation (which nevertheless does not slide into over-sophistication). It is as if Frankel somehow knew about the anger of readers who are never satisfied with the mathematical presentation within similar textbooks. The covered material is the right collection of things that are 'needed' nowadays and missing topics can easily be added by reading sections in Nakahara (which is the best supplementary text). In comparison to Nakahara, Frankel is much more rigorous and precise. For instance the notion of 'tensor product' and its relation to the wedge-product of p-forms is not properly handled in Nakahara, also, Nakahara usually does not motivate the mathematical need of a new construction. Probably only a pure mathematician may find inconsitencies or unsatisfactory conclusions in Frankel's book.
I do not agree to the previous review that Frankel is not suited for self study. On the contrary, Frankel is THE book for self study, it's a pleasure to go through it page by page. Only real requirement: you must like the field. So if you have a sort of a 'feeling' for the strange beauty of topology and manifolds, then this is the book for you. The nice thing about it is that it nevertheless provides 'practical' knowledge, ie. the reader really learns how to use the mathematical concepts 'practically' with paper and pencil. Frankel is right when he claims in his preface that this volume provides a 'working knowledge' of the mathematical tools. Proofs are given almost throughout and only in cases where they encourage mathematical thinking, otherwise the reader is referred to the original literature. Frankel clearly explains why and when 'classical' theoretical physics notation may lead to errors and misinterpretations in comparison to the modern language of geometry where these problems cannot occur.
You will see that Frankel liked writing this book and teaching you, the reader --quite a seldom luxury I would say.
Congratulations to Frankel for this excellent textbook of mathematical physics, I can only hope that it will set a standard worldwide. I definitely recommend it without restriction to readers and librarians.

Having taken a course out of Frankel (over the first 7 chapters) and now having used it in my senior project (topology of circuit analysis) I have to say that I love this book more by the day.

Beforewarned it is not an easy text and you may have to read a section or a chapter over a hundred times. I have found that the material is dense and deep but in a way that welcomes effort. It is weak as far as rigor goes, but rigor can sometimes get in the way of understanding. Use this book alongside mathematics texts in topology, differential geometry and linear algebra and there is much to gain.

For an undergraduate in mathematical physics (which I am) I have come to love this book I highly recommend it to a serious student.

This is a valuable reference for students pursuing a support or who want to get themselves deeper in the mathemathical part connected with QFT and GR. I particularly appreciated the first chapter about Manifolds and vector fields, the part about algebraic topology (chapter 13: chains, homology groups and De Rahm's theorem, Betti numbers) and the part about homotopy groups. On the other hand the first part about tensors, exterior forms, integration of differential forms and the Lie derivative seems to me a bit uneven compared to the one I've mentioned above. For this section I'd recommend: Aldrovandi - Pereira, "Introduction to geometrical Physics", or V.I. Arnold, "Classical Mechanics" (first part) which is not complete if compared to the other two books (this is a book about the symplectic formulation of CM and not strictly a matemathical book) but things that are contained are exposed in a beautiful way. Another valuable book is Nakahara (a classic one), but I still have to finish reading it so I'll leave a comment about it in the next. The level of T. Frankel is at last yr undergrad - 1st yr graduate.