An intro to differential forms

ronaldor9

Which book/books are a good intro into manifolds? Maybe a book that is both oriented towards a physicist but also includes rigor.
How is this book An Introduction to Manifolds by Loring W. Tu
In the preface it says one year of real analysis and a semester of abstract algebra would suffice as a prerequisite. Would it be to ambitious to attempt to learn manifolds without such a background. If not which books should I study/read before tackling this book?

Thanks

n!kofeyn

One of my favorite newfound subjects. I'll give a few suggestions that are geared towards mathematicians and a few geared towards physicists.

Books geared towards mathematicians:
1) Advanced Calculus: A Differential Forms Approach by Harold Edwards
This book is fantastic and is not written like just another textbook. He gives intuitive discussions of the material in the first three chapters and then goes on in chapters 4-6 to prove everything thoroughly. He also has a nice chapter on applications, which goes from complex analysis, the Lebesgue integral, and physics, even proving E=mc². Highly recommend for a first viewing of differential forms.
2) Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach by John Hubbard
I will soon be getting this book. From what I've read of the excerpts, reviews, and table of contents, it looks to be a great book. Hubbard covers all the necessary linear algebra and presents to you calculus on manifolds, while integrating it into vector calculus. I look forward to going through this book. He also has some very nice physical applications, which includes Maxwell's equations.
3) An Introduction to Manifolds by Loring Tu
The more abstract and general of the three books listed here, but it is still accessible to senior undergraduates. This book gives differential forms based upon their general definition, which requires the development of multi-linear and tensor algebra.

Books geared towards physicists:
1) Div, Grad, Curl Are Dead by William Burke
William Burke passed away young, so this book was unfinished by him. I've read there are a lot of mistakes, but it is well worth reading to get Burke's perspective.
2) Differential Forms with Applications to the Physical Sciences by Harley Flanders
A nice amount of applications of differential forms written for physicists and engineers.
3) Tensor Analysis on Manifolds by Bishop and Goldberg
This is more rigorous than the two books above.

There is also A Geometric Approach to Differential Forms by David Bachman. I didn't know which heading to fit it under. :) There is actually a thread here where someone wanted to get a group to go through the book and in which Bachman took part in, until mathwonk ran him off.

n!kofeyn

Sorry. I had missed your edit. If you have had neither analysis nor abstract algebra, then I would say Tu's book will be too much. How much calculus have you had? Are you a math major, physics major, or both? The books by Edwards and Hubbard are geared specifically towards undergraduates.

ronaldor9

Well I'm math/phys major with a concentration on physics and more applied math. My calc/math background is strong (i.e. apostol level book and vector calculus by colley)


Would you suggest I begin with the Hubbard/Edwards book ( I was thinking about this before hand) then after completing them would the book by Tu be a good continuation or still too ambitious?

Based on you suggestions I'm thinking of studying both the Hubbard/Edwards books. After completion would I be able to comprehend the book by Tu or would i still need additional preparation?

In addition, would you suggest I study, concurrently, a book on analysis along side the Hubbard/Edwards?

Thank you very much

Sankaku

Thanks for the recommendation of this book! I had not heard of it before...

n!kofeyn

Edwards' book is older, being at around 40 years old, but it has been updated (sometime in the 90s). It has a nice, warm feel to it, and I have never felt rushed or pressured by the material when reading it. He has some really nice discussions and unique developments, and I really like the end of his book when he discusses applications and math in general. The book isn't too linear, meaning that you can jump around more easily than in most books, which he even encourages in the preface. There is a lot of intuition development.

Hubbard's book is newer (the 4th edition was published just this September), so of course it is probably more modern, but like I said, I haven't received it yet. It does cover a lot more material though and is over 800 pages long (Edwards's book is around 500)! I think Hubbard is probably more comprehensive, and he does have original and unique presentations as well. There is about a 100 page appendix containing the analysis proofs, and I know that he has unique proofs in the text so that some theorems can be accessible to undergraduates. For example, his development of Lebesgue integration (although Edwards has a small section on this as well).

I don't think you can go wrong with either one. Use your library's interlibrary loan if your library doesn't carry them to see which one you like the best. Edwards' book contains the solutions to all exercises in the back of the book, and Hubbard has a separate solutions manual that completely solves the odd numbered problems.

I think you will be better off spending your time learning the above material than analysis. It is very important, but you'll eventually take a course in it, I'm sure. As a physicist, you won't encounter that material as much, and both of the above texts have analysis in them anyway. Be sure to browse the other texts I mentioned as well. They all present something different. Also, there is another book, Advanced Calculus by Loomis and Sternberg, that is well respected. This material is very rich, and so the best way to learn it is to get different perspectives. I learned it from Introduction to Smooth Manifolds by John Lee and also from Loring Tu's book. Now I'm going back to get the more practical and computational understanding that is missing from the abstract texts.
No problem Sankaku! It is a really nice and unique text. The same author also wrote a linear algebra text and a few others.

ronaldor9

Thank you very much n1kofeyn for your very knowledgeable advise

xepma

I can also highly recommend the book by John Baez (gauge fields, knots and topology)
http://www.amazon.com/Gauge-Fields-K...7176842&sr=1-1

It's very nice to read, very well-written and very accessible. It may not touch on all topics that you are looking for (it's aimed at the fiber bundle construction of gauge fields), but if you get a chance to borrow this in your library you should not hesitate to browse through it. The version I read had a lot of (math) errors in it, so be aware of this.

Note that only the last 1/3rd of this book deals with (quantum) gravity, so even if you're not interested in this topic you're not confronted with it untill the end of the book.

dx

Thanks. I'm a fan a Burke and his books, and assumed I would never see this one since he died before completing it.

George Jones

At the second-year level, there is also A Course in Mathematics for Students of Physics 1 by Bamberg and Sternberg,

http://www.amazon.com/Course-Mathema...9&sr=8-1-spell.

n!kofeyn

This book is rather nice, and I plan on going through it myself to pick up the physics. Thanks for mentioning it here. You're right in that it doesn't really delve into the differential form material in depth, but I think once you have learned the material then this is a great way to learn to apply it.

My sort of approach to these things is to learn the math first and then the physics. It may be because I am a mathematician, but I feel that if you have mastered the math, then when learning the physics, the physics is not obscured by the struggle to learn the math. This allows full concentration upon the physics. Like I said above, the material is so rich that there are many different approaches, and they are all interesting.

iamthegelo

The reviews on Amazon kind of deterred me from getting the book. Is it worth it? Any comments?

dx

Bad reviews on amazon tend to be written by frustrated and unprepared students. Read a review by an actual physicist: http://www.ucolick.org/~burke/forms/bamberg.html

n!kofeyn

This webpage of William Burke has some nice one to two sentence reviews of books on differential forms.

Daverz

I'm surprised no one mentioned Burke's Spacetime, Geometry, Cosmology. His Applied Differential Geometry is also fascinating, though it can't really be used as an introductory text as Burke is too elliptical at times (I think he even admits to doing so deliberately.)

n!kofeyn

From my understanding that book is a very watered down approach, which he even used to teach physics to art majors. So it's probably a good surface level book, but it sounds like (again I'm not familiar with the book's actual content) it won't give a good mathematical development. Most of the books above have great motivations, but have the rigorous developments as well.

Daverz

The "Physics for art majors" story is very misleading. These must have been art majors with very strong math backgrounds, or "art majors" was an exaggeration for the sake of the story. The book is not as superficial as that makes it sound, and there's plenty of content that would be of interest even to the sophisticated, as is usually the case with anything Burke wrote. I suggest taking a look if your library has it. It's a very nicely produced book as well.