Reviews (10)

I couldn't use it.
This book has everything I found to be correct in a text book but when I tried to teach with it I found it to have a lot of mathematics and not much of physical ideas.
Landau's Classical Field Theory book is much better but it is for advanced level, so I still didn't find a proper book.
Now I will try Schutz one.

A great first book on general relativity
I like this book because it has the best elementary introduction to the mathematics of general relativity.It starts out with simple multivariable calculus and geometric notions about vectors.It then explains the ideas of the natural basis and the dual basis, first in a plane and then on a manifold, with very helpful figures.With too many other books it is possible in a first exposure to completely miss the point of these ideas, which really are pretty simple when you come right down to it.It is true that the physical motivation and meaning of general relativity are not treated in that much depth, but these can be picked up from other sources.In my view it is the mathematics that is the most intimidating thing about general relativity -- the physical ideas are exhilirating and natural by comparison!

Great for independant study
As a person who did postgrad physics and maths over 5 years ago and has been out of the field for way too long, I found that this was a great introduction to GR, a subject I never got to do at university.It introduces the maths (tensors, manifolds and geodesics) in the earlier chapters and relies heavily on them in the introduction to GR.

The book has great solutions, or at least very helpful hints, to the problems that are given throughout the book.Though at times I was stuck with some, it generally it required me to only look at the first step of the solution to be able to solve the problem.

Reviews (8)

As an Outline...
This is not a book to learn tensor calculus from.It is an outline only, no greath depth or insight is presented.This book works perfectly as a supplement to a course in tensor calculus, or as a quick reference for the various techniques and concepts involved, provided one is already somewhat familiar with the material.It would be possible to learn the basics of tensor calculus from this book with some effort, and reflection on the implications of the concepts dealt with, however as a complete course in the subject it is insufficient, and I believe intentionally so.
The more modern aspects of tensor analysis on manifolds are largely ignored in this treatment, but also intentionally so, an approach which I found useful practically.
The book does not aim to be an all-inclusive course in the applications of tensor concepts to all areas of mathmatics, but rather a quick-reference guide supplementing more complete treatments, and as such, is largely successful.

Disappointed.
This book is substandard for the Schaum's Outline Series. The descriptions for the techniques are much too brief, and as a result, it's hard to follow what's going on. The summaries are so thin that it's even difficult to learn how to do the mechanics of tensor operations, a real deficit for an outline book! I have a pretty good background in advanced math, but I don't think I could learn tensor analysis from this book. I was especially disappointed because I have had good luck with other books in the Schaum's series. I'm planning on looking for a more traditional book with more discussion and background of the different techniques.

Reviews (5)

Hard to follow
It's an old fashioned text, confusing and hard to follow.

okay book
this book dosen't take things from basics but goes to do high level calculus.

Reviews (19)

Not really helpful
I tried to use this book as a supplement text for my multivariable calculus class. I found it to be very useless even as a supplement, let alone a main source to rely on. The main problem with this book is the fact that it is very short (Can I say way overpriced? Good thing I got it for free) and it doesn't explain the concepts properly. It is theoretical, but also in my opinion too far out there. I have nothing against theory, in fact I think its great to have theory in a math book. But in this case the material presented very tersely and unclearly. In my opinion books should explain concepts nicely and clearly with a proper use of examples. I do not wish to spend hours trying to understand what an author tried to say, especially when a concept is a really easy one. Another annoying thing about this book is the notation. Author uses "modern" notation for partial derivatives, but for some reason not many other people use it. It is found mostly in the 1950s era math books. This archaic approach to math is devastating to a student. Avoid at all costs.

Must be written by Spivak's evil twin
Spivak's other books are quite good, but don't let that fool you into getting this one. This is a horribly dry and terse text of the type which is convenient for authors and lecturers but hopeless to learn from. The object of Bourbakian worship is of course "the modern Stokes' Theorem", but, Spivak says in his preface, "Yet the proof of this theorem is, in the mathematician's sense, an utter triviality - a straight-forward computation. On the other hand, even the statement of this triviality cannot be understood without a horde of difficult definitions from Chapter 4. There are good reasons why the theorems should all be easy and the definitions hard." Perhaps these "good reasons" are that lazy authors can throw together unhelpful books where everything is "left to the reader".

Reviews (4)

Fun
Iploughed through this book years ago. Ijust noticed that a couple of reviews were only posted this year.
I thought I would do the same.

This was a great read by the way.

I suspect that everyone who picked up this book at some point was looking fora way to circumvent Spivak's terse exposition. I don't blame them.

..andthenBrowder came out with his analysis text. So with advanced calculus in view, these (more or less)recent publications make the subject even more accessible to undergraduates.

..and nowSpivak doesn't look so hard, all of a sudden.

Munkres presentation is certainly original.Motivating examples are bountiful, and the figures are excellent.

The perfect prequel to Boothby.

Enjoy.

A good introduction but not the best
One thing i like about this book is the way Munkres presents the counterexamples : why theorem 5.11 wont work if we ease one statement from the hypothesis. Also, the material is accessible and the exercises hard -- both of which, IMHO, are important benchmark for a good math text.

However, compared to his classic textbook of topology, Munkres did not perform as well in connecting with the readers. The text is very hard to read, and is not suitable for self study. This is useful only as a class text, or as a reference for those who already knew (or passed) the subject.

Amasterpiece yet accessible on this topic
This book covers a natural extention to my course on analysis in R^n--only content similar to first one sixth of the book got treated at the end of the course. Having read first half (just before manifold) in a continuous fashion (span of nearly a week for 4 hours-ish p.d.), I find this one exceptionally clearly-written, (unlike some point in Spivak's Calculus on Manifold), and in content it serves as a detailed amplification on Spivak's (Sp seems to try to keep the proofs elegant and concise more than possible, making a couple of important theorems render indigestible).

Other noticeable features are:

1) Mistake-free.

2) Proofs are truncated into stages with explicit objectives in each, making them well-structured on paper and easy to recall in future, and in this way techniques in proofs become highlighted into some elementary theorems (to get most job done) so that the scope of applications are much widened.

3) Motivations scattered throughout the book for integrity.

4) Examples given illustrate as counterexample of how theorem fails with some condition changed or missing.

5) The level of presentation is uniform throughout the book: strictly speaking, only a good single-variable analysis course (Rudin will do, and also helpful to refer to the overlapping topics) and some motivation are needed, all essential concepts of linear algebra, topology are introduced afresh and uniquely and in the favorable context: either indispensible in later proofs (can act as a practice of it) or results proven motivate its introduction and properties, though some knowledge beforehand can help you to appreciate more, and focus on mainbody.

6) Each proof is not necessarily the shortest in methods, you may say, but looks most natural and appropriate at this level. Actually, most time it's quite concise whilst, in main theorems, all details are laid out without undue omission. (In contrast, some authors waffle lavishly between substance, but say bare minimum (sometimes unjustified) when it comes to proofs.) Length is also due to partition of proof into stages, which is way clearer in mind than a gluster of dense but appearingly short arguments. And richness and details of proofs themselves are good for getting hang of techniques.

All in all, Munkres is clearly a master, while reading it, you just feel it cannot get any better. Clarity, style, and organisation put the book far above its peers, and an undeniably outstanding first course in multivariable analysis and manifold alike.

Reviews (6)

Worthy
This text covers many topics and is quite approachable fora beginner but requires very careful reading.The author tends to express important definitions in the flow of the text inconveniencing references to said definitions from later sections. This book does not require and makes little appeal to notions from topology which renders the text more accessible but short changes the reader of important insights.

The definition of tangents and tangent space are introduced rather awkwardly; the set of continuously differentiable functions (C-infinity) is dumped on the readers lap without preliminary discussion.These issues are addressed by "Tensor analysis on manifolds" by Bishop and Goldberg though the latter book is _very_ terse.

I recommend the subject text but suggest that one simultaneously read "Tensor analysis on manifolds" by Bishop and Goldberg.I also recommend that the beginner/self studier consider reading either "Differential Geometry" by Kreyszig or possibly "Differential Forms With Applications to the Physical Sciences" by Harley Flanders before reading either of the said texts.Kreyszig does a much better job of actually writing a readable text (the notation used is a bit old though).Ideally the Bishop and Goldberg texts should be combined and rewritten in the Kreyszig style.

Not for everyone, flawed in basic ways
I disagree with reviewers who found this book useful for self-study. I would not recommend it for individuals first learning this material. The book is frankly contradictory in places, and frustratingly repetitive in others. In the early chapters it assumes concepts not yet explained, and introduces terminology and symbols that are nowhere defined.

If you already know quite a bit, you may find this approach enlightening. But if you're just beginning to master these concepts, I suggest you look elsewhere.

I also suggest that much tighter editing would do this book a world of good. Go with Kreiszig, or Lovelock and Rund instead.

Excellent book
This is a very modern, very concise, and very efficient book. By using vector bundles the curvature forms on semi-Riemannian manifolds are introduced. Definitions are given clearly and intuitively. Without spending tons of pages on digression to minimal surfaces, Hopf-Rinow thm, Gauss-Bonnet thm, etc., the book builds enough machinery to describe the gauge field theory in the last chapter. Most other differential geometry books either throw in too many applications to waste reader's time or give vague definitions (too bluntly abstract or not self-contained) to confuse the reader.

All exercise problems are interesting and important. Hints are given to some of them.

Isbn: 0828403163
Sales Rank: 1045279
Subjects:  1. Differential Geometry    2. Geometry    3. Science/Mathematics   

Reviews (13)

Best DG book out there
This book is rather expensive, but when compared to the other books available, it is not a waste of your money. It has plenty of exercises, many of them with answer or hints in the back of the book, and its exposition is broad, very clear and concise.

It is hard to tell being a math student, but I think anyone with a solid knowledge in multi-variable calculus (Apostol's book would be perfect) or, better yet, who has taken multi-variable analysis course would find this book accessible. One of the advantages of this book is that it is self-contained, so even though it uses, for example, the inverse function theorem (which is something unavoidable for a DG book), it has an appendix on differentiability and continuity which covers this.

The exercises range from easy to very hard, but because of the exposition and of the way the exercises are stated (the tougher ones are many times itemized so that they drive you to the answer) it is rare to find a problem that the reader will not be able to solve upon a little thinking.

The greatest advantage of this book is its clear and well-written exposition. It has very few errors, mostly typographical. It covers a lot of topics and its notation is extremely simple and suggestive, which, believe me, is of great help in a DG book. In short, if you want or have to learn differential geometry, save your time and get this book. As another reader very intelligently put it, there is a reason why this is a classic.

There is a reason why it is a classic.
Before talking about the book itself, let me tell you that I am a mathematician, and when I took a differential geometry course and used do Carmo's book, I already knew I wanted to be a mathematician. So, is this a book for mathematicians? Well, yes, but not exclusively. It is certainly written from a mathematician's point-of-view, and it assumes some maturity on the part of the reader. For instance, the exercises contain very little in the way of drill, and are used to enhance the theory (as pointed out by another reviewer). It seems to me that the author believes that mature readers can provide their own `drill' exercises. So, you won't find many exercises asking for you to find pricipal curvatures for this or that surface, and that other as well; exercises in this book have a theoretical flavor to them. This, of course, makes for some hard exercises, and I do remember spending a lot of time over them, often working together with other students taking the same course. The upside is that we learned the material, and thoroughly. Also, the author provided plenty, plenty of examples. The figures are very well drawn and really allow you to see what is going on - even though these days, with powerful computer packages like Maple, Mathematica, Matlab, and others, any student can provide his/her own pictures. But just because now we can use computers, I wouldn't say the text shows signs of age. It is jus as clear now in its exposition of topics and concepts as it was many years ago. So, even though there are many good alternatives in the market, if I had to teach a course now on this subject, or even better, if I were a student now taking this subject, I would certainly have this book at the top of my list of possible textbooks.

Reviews (11)

Good
I used this text for a course after taking an undergraduate GR course based on Shutz.I found Shutz to be a much clearer and pedagogical text, and don't think I would have learned GR as easily if I had started with Wald.I think one requires greater mathematical preparation than I possess to fully appreciate the discussions involving topology in the second chapter and appendix.Oddly, however, this text becomes clearer as the reader advances through it:later chapters were more straightforward and still concise.

Valuable
A valuable reference for GR. If one has to learn something on GR for the first time, then this is probably not the best book to start with (even if the first part on GR, chapters 1-6, is quite clear). On the other hand the book contains a very good treatment of Energy in GR, Killing fields, and ADM Energy-momentum. This is in brief a great buy, if one does not feel fine facing Hawking-Ellis.

Reviews (14)

Revolutionize the way calculus is taught
This is the textbook used for the math 223/224 Theoretical Calculus and Linear Algebra sequence in Cornell University. The book is designed for prospective math students. Although the book mainly follows a rigorous development of the theories of multi-dimensional calculus, the mathematical machinery used in developing the theories is immensely broad, especially in linear algebra. The book covers most of the standard topics in a first semester linear algebra course and touches on many other areas of mathematics such as, real and complex analysis, set theory, differential geometry, integration theory, measure theory, numerical analysis, probability theory, topology, etc. The highlight of the book is its introduction of differential forms to generalize the fundamental theorems of vector calculus. The author is not the first one who follows this path. There are many other books written before this one that have similar approach, such as Calculus On Manifolds by Spivak, which was written 40 years ago and was too old to suit modern students.
The author tries hard to retain rigor and present to the readers as many examples and applications as possible. Often he tries to cover a broad range of mathematics and digresses a little. The book more or less touches on most of the areas of undergraduate mathematics curriculum and does not go into depth. It sometimes gives me the impression that the book is almost like a survey of undergradute math. The book is also not error-free. There are many typos and some technical errors. If you buy this book, make sure to get the errata from the author's website.

Confusing book
Math and Physics major

This is a good book if you are solid in set theory. The prerequisites do not emphasized set theory enough. It gives a brief overview in the begining but not extensive enough.

Also, although I can accept theorems written in set theory, even the examples are in set theory. In fact, alot of examples are really just corollaries. I take math to apply to physics. There are very few practical examples that apply to the real world.

Also, notation is horrendous. Hubbard tends to make up the notation as he goes along. This is going to make it very confusing in later classes.

Needless to say, there are careless typos.

This is not a well written pratical textbook.

A Must Text
I have used this book to teach gifted high school students about the following topics: the implicit function theorem, manifolds, and differential forms.With the Hubbards' approach, even students without a course in linear algebra actually get it!Not only do they understand the material, but they also become amazingly enthusiastic when they begin to see the unifying effect of understanding differential forms.

This is the only text that I have seen that really makes forms clear.It does so by taking the time to carefully, but rigorously, explain them in a "classical" setting.One of the reasons forms are so difficult to grasp is that while some things, such as the exterior derivative and the work form of a function, can be seen as natural objects (when explained well), the connection between these objects and calculating with forms using coordinates is not so easy to make clear.The Hubbards' do make these ideas clear - even when presenting topics as hard as orientation.

Unfortunately, most of us had to wait till graduate school to see forms - usually, in a more abstract setting. By then, we probably didn't have time to sit, calculate, and make clear connections. This text makes that later transition, for those in math, much easier.It also makes physics easier. The Hubbards' make that point by showing that the electric field shouldn't really be a field, but a two form.Any book that lets one explain that - and much more - to high school students, which I do, should be a part of every multivariable calculus course.

Finally, I should note that this book contains much, much more than manifolds, the implicit function theorem and differential forms.But, even if that were all it contained, it would fully be worth the price.

Reviews (73)

An excellent textbook
I think mathematics is a part of our culture.That's why, as a non-math major, I wandered into a very serious analysis class for mathematics majors.That might have been a disaster for me.Luckily, we used this book as a text, and it saved me.I read the whole book and diligently did all the exercises (of course, back then, it was the first edition, with only 227 pages and 140 exercises; it's somewhat more now).And that is my recommendation today.Read the book carefully and do as many exercises as you can.It certainly isn't easy.But it isn't, um, countably hard either.

The material in the book is self-contained.I guess that in theory, it could be mastered by any bright 14-year old who had learned some high school algebra and geometry.But I would surely recommend having much more mathematical sophistication than that as a prerequisite!

If you haven't learned the language of mathematics before, you'll enjoy the use of terms such as "countable," "real," "rational cuts," "measure," "ring," and "complete." By the end of the book, when the author claims that a proof (involving Cauchy sequences no less) is complete, you'll barely be able to suppress a desire to ask "Does every Cauchy sequence in the proof converge?"

In the first edition of this book, Rudin did mess up a little in his section on "the integral as a limit of sums." His theorem as stated was false.We cruelly dubbed it "Rudin's Last Theorem."Worse, he had used it "to prove some elementary properties of the Stieltjes integral."But that was all straightened out by the second edition.

I especially like the first couple of chapters.They give most readers the confidence to continue.And the final chapter, on Lebesgue integration, is very well written.One note of warning, though.Rudin begins this chapter by saying, "Some of the easier propositions are stated without proof.However the reader who has become familiar with the techniques used in the preceding chapters will certainly find no difficulty in supplying the missing steps."That is an exaggeration.It takes work.After all, this is, um, real mathematics you'll be doing!

I'm thankful that I was assigned this as my textbook.

A masterpiece
I absolutely agree with Professor Jorgensen.

I loved it when I was a student of physics, and I still love it because I tend to consider it as my personal standard in Classical Mathematical Analysis (and not only): sort of a "pacemaker" which sets the qualitative level to go back to just every time one is a little confused about what to do and where to go ;)