The Geometry of Physics: An Introduction

Some texts are designed to increase understanding, others to aid in practical computation, making them as much references as pedagogic tools. The latter are especially suited for self study. In this new edition, Frankel does something amazing-- instead of completely reorganizing an already stellar text, he "ties it all together" with a new "example" introduction-- a 34 page (roman numeral numbered!) "preface" illustrating Cartan's exterior differential forms with a "metal torsion" example application to Cauchy's stress tensor. Don't mistakely think that this means Frankel limits this text to the differential geometry of engineering mechanics and materials-- he covers a vast field of physics all the way from classic to quantum, sans string but with numerous gauge applications, in 750 packed pages, most containing fully worked out calculations for the aforementioned reference value. It seems today that all publishers just parrot "for grad students or advanced undergrads with a year of calculus and some linear algebra." Is this to sell more books? Not sure, but I wouldn't tackle this for self study or even calculative reference without "advanced" calculus (in my definition, analysis) PLUS a good course in analytic geometry first. Although this is packed with AG, it does not start by teaching AG-- the geometry knowledge is assumed, and we're then treated to an astonishing adventure of detailed APPLICATIONS of geometry to nearly every aspect of physics, including numerous cutting edge and intractable problems. There also are NUMEROUS engineering applications examples, blending physics, engineering and geometry in a way no other text even attempts. I've long felt that some pundits who tease the Greeks for seeing everything as geometric would someday eat their words. Well.. wow. This volume clearly demonstrates how much what comes round goes round. OK, looking at physical spheres is not the same as spacetime curvature spheres, let alone "field" geometry that isn't even a physical geometry, but the geometry of a vector bundle! So, to be honest, if you see the word "introduction" in this text's title, and think you'll be guided through the UNDERSTANDING of Lie algebras, matrix calculus, Yang-Mills and other gauges via geometry-- be careful. "Introduction" as I read it after reading this text means intro to the APPLICATION and CALCULATION techniques available to someone already well grounded in analytic geometry. Don't get me wrong, the author is simply amazing, as were the very successful first two editions, in carefully explaining many neglected applications of AG to physics, but this book would be 3,000 pages if we actually expected it to "introduce" every notation. So, it does blast right off assuming a good base in analytic geometry, and a fair base in physics. One really cool dimension for self-study-- the author is obviously first a mathematician, and within that a geometer, so the pedagogic artifact from older days of showing NUMEROUS diagrams and illustrations made its way into this fine text. Even if I weigh the overall presentation as more computational that didactic, the illustrations themselves bely that evaluation-- each one gives one of those "aha" moments. The author also does take the time to explain the WHY of certain formula elements so you really "get" them. For example, if we're given an element where x^(exp polynomial) = b*(expression), the author WILL digress enough to remind us that b is acting as a proportionality constant. I find this really helpful as a way to generalize the lesson learned, otherwise we're just rote memorizing or referring back to a process we're not really getting! Highly recommended with the caveats mentioned about brushing up on your analytic/ differential geometry. Library Picks reviews only for the benefit of Amazon shoppers and has nothing to do with Amazon, the authors, manufacturers or publishers of the items we review. We always buy the items we review for the sake of objectivity, and although we search for gems, are not shy about trashing an item if it's a waste of time or money for Amazon shoppers. If the reviewer identifies herself, her job or her field, it is only as a point of reference to help you gauge the background and any biases.

Highly recommended. It's well written, covers a lot of material, and is suitable for self study . I wish that the Kindle ebook were a PDF ebook for which the equations are always properly sized in proportion to the text (available from ebooks and vitalsource at much greater cost). I also recommend the classic Differential Forms by Harley Flanders. It's much shorter but still covers some important material such as the converse to the Poincare Lemma, the Frobenius theorem, and an elegant concise derivation of the Riemann tensor-valued curvature two-forms, the Bianchi identity, and the Einstein tensor. But much that is discussed in Frankel's book is entirely missing from Flander's book, such as Frankel's elegant treatment of the Lie derivative, even though it is somewhat sloppy in its treatment of time-dependent "flows" on a manifold M, which can be made onto true flows by considering an extended manifold RxM. Usually when he says that the Lie derivative with respect to the flow vector field X on RxM of a particular time-dependent differential form with no dt^ terms is zero, what he really means is that it's zero after discarding any dt^ terms.

Topics are great. Notation and typesetting could be improved, but since this is already the 3rd edition, I can only say I don't understand the author/publisher's decision of insisting on using unconventional notations every now and then and such ugly typesetting.

Excellent for both physicists and mathematicians. Essentially a differential geometry textbook and how physics has motivated its development and is inherently connected to it. Starts with the basics of manifolds and continues into highly advanced, specialized topics with numerous applications to physics throughout. Writing is not too terse, but not so clear as to where the author is doing the thinking for you. Numerous exercises sprinkle the text, usually at the end of each section. Some sections are full problem sets, with a few definitions and remarks as you develop and apply the tools (such as hamiltonians) through problems. Packed with information that will keep you occupied for months. Pre-requisites include a firm grasp in both mechanics, electromagnetism, and special relativity, as well as an excellent understanding of linear algebra and multi-dimensional analysis. Dr. Shifrin's youtube videos would be a good start or review for the math pre-reqs. In regards to delivery and quality of book -- on time and superb, respectively.

The Geometry of Physics is written in a very modern style and with a great choice of topics. A mathematician can enjoy this book even though it is mathematical physics. The mathematics is presented in very nice form.

I'm happy with the book. It offers an insightful perspective that I learned to really appreciate. My understanding about physics was greatly enhanced by the combined knowledge of Frankel book and the book "Geometrical methods of mathematical physics" by Schultz. I am really impressed by how much intuition one can gain by developing a geometrical view of modern physics. In fact, I can't emphasize enough how important such knowledge is for a professional physicist.

Newly printed paperback books from Cambridge University Press seem to be printed on demand with low quality ink and papers are no longer the glossy and shiny ones. Very disappointed!

Frankel is an excellent teacher, and did a great job with this book.

本の内容が素晴らしいことは図書館で借りて読んだので知っているのですが、Kindle 版を読むと、（Kindle Cloud 、Kindle アプリ共に）以下の要因により非常に読みづらいです。 ・ページをめくるのに時間がかかる。 ・数式が崩れている。また、本文中の数式が縮小されている。式番号が中途半端な位置に表示される。 ・数式に誤植が増えている。（例えば Vであるべきところが νに置き変わっている。） ・数学記号が正しく表示されていない。例えば集合の二項演算子のいくつかは、下半分が消えています。 ・一画面に数行分しか表示されない。 この本に限らず、物理の専門書をKindle で読むのは、まだ大変そうという印象を持ちました。 (2021.6.27)

Excellent

I rated this book with five stars because it was just perfect FOR ME. I was looking for a modern treatment of mathematical physics from a geometric and topological point of view. I had previously strugled with Schutz's, Nakahara's and Nash and Sen's books on the same subject with only partial success. I finally bought this book and found there everything I wanted to learn on the subject thouroughly explained by a mathematician who sacrifices mathematical rigour for physical relevance. That was EXACTLY what I needed but, please, bear in mind that it might not be what YOU need. Moreover this book is not easy reading. You need to work very hard if you want to fully understand it. This book was MUCH, MUCH better FOR ME than those I mentioned before because I found it either MUCH, MUCH clearer (than Nakahara's and Nash and Sen's) or MUCH, MUCH more thorough (than Schutz's).

This is a very abstract book about geometry in mathematics with some applications in physics. There are some examples how the theory can be used in physics but even these are very abstract. I like the discussion about general relativity. I would say that the center-of-mass of the book is in mathematics rather than in physics even though it is not a standard mathematical book (theorem-proof).

ok