Customer Reviews

The Geometry of Physics: An Introduction

5 star:		(11)
4 star:		(3)
3 star:		(3)
2 star:		(2)
1 star:		(0)

 

 

This book introduces the methods of modern differential geometry and its uses in theoretical physics. The only prerequisites are a good working knowledge of multivariable calculus and linear algebra. The book is very much written for a physics audience(i.e. the book is actually READABLE unlike so many graduate texts in mathematics, and there is an emphasis in actually learning how to CALCULATE things, rather than just staring weary eyed at mathematicians beloved polished proofs that only they can understand) There is an emphasis on physical understanding of the mathematical structures and not too many proofs. Proving things is not a bad thing, but Dr. Frankel seems to know when its most appropriate to do this, and doesn't get too bogged down in the proofs. There is a lot of material in this book (22 chapters) The book is broken into 3 main sections. The first section is on "Manifolds, Tensors, Exterior Forms" Differential forms are not that familiar to physicsts and this is a great place to learn about them. There is very nice section on how to relate Forms to vector Analysis in 3 space that physicists love dearly (see page 94). The second section is on "Geometry and Topology"-mainly Riemannian Geometry and Some Algebraic Topology like DeRham Cohomology, and the third is "Lie Groups, bundles, and Chern Forms". In this third section there is a Chapter on the Dirac equation, and its relation to Spin geometry. The only thing that the book is lacking is that there is no complex algebraic geometry (for aspiring string theorists). It would be nice if some day Dr. Frankel could write a book on this subject, since at this time none exist. I think that even mathematicians could learn a thing or two from this book. Most of differential geometry originated in Physics, not the reverse.

I picked this book for self-study in order to understand differential topology in physics. It is an excellent book for the breath of ideas applicable to many areas of physics and the author has examples from stat mech, thermo, e & m, classical dynamics as well as relativity. I agree with the previous review that it lacks a sense of direction. Occasionally, Frankel uses a concept without explanation only to define it a hundred pages or so later (e.g. the star operator on em fields). For me these problems made the text tough going. I was frequently derailed by complicated notation (without glossary), lack of direction, and deus ex machina concepts dropped without adequate explanation. Some of the confusion derives from use of coordinates which Frankel finds necessary in order to motivate development of coordinate free forms. It seems that the author could have avoided this as did Darling or introduced Clifford algebra early on. I do not recommend this book for independent study without other texts like Flanders, Darling, Misner Thorne Wheeler, etc. to refer. It would be best to have a tutor guide one through it or re-read it after getting sufficient grounding with other texts. This being said it is valuable addition to my library and I still think highly of Frankel's effort.

The other reviews on this page give this book anywhere from 1 to 5 stars, and they are all correct in their own way. The book is inspired, deep and full of physics applications and insights. On the other hand, it skims over mathematical rigor to a large degree and focuses more on defining things, getting a feel for them and moving on to application.

My advice: buy the book for its strengths, and read other books in parallel if you need more rigor. But still, buy it.

Also, things can be confusing on the first two or three reads, but keep at it and you will be glad you did.

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.

Differential geometry can be a very intimidating subject due to its heavy formalism. There are complete books (such as Kobayashi& Nomizu) very good as reference books, and there very few books that show the reader the picture behind the formulas.


This is one such book. It tells you the intuition behind each construction and from this point of view it has many things in common with Arnold's famous book on Math. Methods in Classical Mechanics. But where as Arnold does not pay too much attention to formalism, this book achieves this task as well. It shows the reader how to do those impossible computations as well.

This is definitely the first place to look at if you want to really learn differential geometry. If it seems difficult it is only because the subject is so.

I just finished a class in mathematical physics, and the text we used was Bamberg & Sternberg. I found that books treatment muddled and shortsighted. I mean, most of the linear algebra in the book deals only with 2 dimensional vector spaces. And the book was entirely useless in teaching differential forms...

So i went looking for a better book to learn diferential forms. i didn t like flanders, it was too brief. this is the book for me. Don t expect to find any linear algebra here, but you d better know lin. alg. before you open this book.

it is a challenging book, mathematically speaking, to study on your own (for a senior ugrad phys major, anyway), but it s treatment of forms and tensors is comprehensive, thorough, and detailed. and it shows you all the applications to relativity and electrodynamics, etc... it also builds up all the theory in with a background of differential geometry and topology, which are developed in the first chapter (but wasn t i glad to have already studied those topics beforehand!)

this book prepared me for my mathematical physics class, plus gave me months of other material to study. it is difficult, so i read and reread each chapter.

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.

This book is ideal for a full year advanced undergrad or beginning grad course designed as an intro to theoretical and mathematical physics. Frankel writes clearly and is very careful not to slip anything past the reader - every statement is fully justified by text.

The biggest problem with this book is that in trying to justify everything, Frankel often forgets the big picture while focusing on details. The focusing on details is great, but he isn't explicit in formulating where these details will be used and how they are incorporated into the global structure of the theory.

This means that anyone teaching from this book should not be shy about slashing out extraneous material and coordinating developments in this book with other texts, such as Goeckler & Shuker, or Nakahara. These books act as good surveys of the mathematics behind the physics and better illuminate how all the details tie together to form a coherent theory.

This book can be quite confusing if you start without any background on the idea of manifold or knows nothing about general relativity. However, it does have strong points:

1. The notation is very up-to-date, and is entirely coordinate-independant approach.

2. The author explains in great details of formulation of modern differential geometry, and the details are comparatively lacking in other reference books.

3. The author never hesitate to use graphs and diagrams to illustrate points, and stroke nice balance in between mathematics rigor and physical insight.

Although it appears quite verbose at some point, it is mainly because differential geometry is such a heavy subject. Another book nice to have as companion reading is Goldburg's "Tensor analysis on Manifold", a terse, well-written text book.

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.