Rating: 
Summary: The perfect first book in differential geometry
Review: 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.
Rating: 
Summary: Bad book.
Review: Frankel's book is provbably the most confusing book I have ever looked into. As other readers noted, it is probably because of his approach not to define things properly. The book's style is extremely wordy, unnecessary wordy that is. The result - total confusion. Mr. Frankel probably thinks the readers are nearly morons, so he tries to re-express some (really simple) notions with words that supposedly will make things lucid. Well, he fails.
Alternative book by Nakahara is way better.I also recommend "Analysis, Manifolds and Physics" by Yvonne Cgiqyet-Bruhat, et al
2 stars for effort.
Rating: 
Summary: Dissapointing!
Review: Having gone through the first 3 chapters of this book, I must say I am really dissapointed. The author is supposedly trying to avoid the mathematical rigor to the account of explaining things in a physical way. Well, he almost completely fails in that and, the worst, confuses the reader. For example, he devotes 15 pages to smoothly introduce the reader to the concept of a manifold, promising a more rigorous definition to be given later on. When the time comes, he uses two "brand new" concepts, namely Haussdorf space and countable base, for the meaning of which, the reader has to look up in other books!(however cited in bibliography) If one wants to understand what all these things (manifolds, diff. forms, Lie deriv, etc.) are about, the best thing to do is buy a mathematical book. After all, it is hardly possible to satisfactorily describe abstract mathematical concepts if you avoid using mathematical language! For the interested, I am NOT a mathematician.
Rating:
Summary: over and over and over again
Review: 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.
Rating: 
Summary: There are better...
Review: I have used this book in an independent study in Geometry of Differential Forms. It did not take me too long to start looking for other references. There is something about its content that makes it diffucult to follow. May be it's too wordy. There are several misprints in notation. After I few weeks of study, I turned to Morita's Geometry of Differential Forms. The mathematical presentation is much clear and it's only 300 pages. I really like Frankel's book mainly for its application to physics. But with respect to the math, I recommend Morita's and Thirring monographs.
Rating:
Summary: the geometry of physics
Review: 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.
Rating:
Summary: I don't like it
Review: This a brilliant and beautiful work, evincing a profound understanding of modern physics. It can be read with profit by everyone from undergraduates to professionals in the field.
Rating:
Summary: A Must
Review: 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.
Rating:
Summary: Be careful with this one
Review: This book is has its ups and downs. It is very comprehensive, for sure. It covers a lot of topics, and in many ways it is self-contained. My biggest gripes: it is sometimes lacking in rigor, it is too heavy for my tastes on its attempts to relate to physics, and it has sometimes screwy notation (or at least, I should say, different from similar books I've read). I felt like it was sometimes wordy, going through lengthy and unnecessary explanations, making it sometimes difficult to extract the substance from the text. I would very much recommend this book for people who have heavy physics backgrounds but weaker math backgrounds. It is certainly written for physicists, and not mathematicians. But then, that's how it advertizes itself. People may want to consider alternatives like Nakahara's "Geometry, Topology, and Physics", which covers mostly the same subjects as Frankel but also other relevant topics like Complex Manifolds, and (though I have only glanced at it) appears more mathematical than Frankel. Also Nash and Sen's "Topology and Geometry for Physicists", which is similar to Nakahara in style, though covers fewer topics (but probably cheaper).
Rating:
Summary: Be careful with this one
Review: This book is has its ups and downs. It is very comprehensive, for sure. It covers a lot of topics, and in many ways it is self-contained. My biggest gripes: it is sometimes lacking in rigor, it is too heavy for my tastes on its attempts to relate to physics, and it has sometimes screwy notation (or at least, I should say, different from similar books I've read). I felt like it was sometimes wordy, going through lengthy and unnecessary explanations, making it sometimes difficult to extract the substance from the text. I would very much recommend this book for people who have heavy physics backgrounds but weaker math backgrounds. It is certainly written for physicists, and not mathematicians. But then, that's how it advertizes itself. People may want to consider alternatives like Nakahara's "Geometry, Topology, and Physics", which covers mostly the same subjects as Frankel but also other relevant topics like Complex Manifolds, and (though I have only glanced at it) appears more mathematical than Frankel. Also Nash and Sen's "Topology and Geometry for Physicists", which is similar to Nakahara in style, though covers fewer topics (but probably cheaper).
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