An intro to differential forms

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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
 
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  • #2
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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) https://www.amazon.com/dp/0817637079/?tag=pfamazon01-20&tag=pfamazon01-20 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) http://matrixeditions.com/UnifiedApproach4th.html" 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) https://www.amazon.com/dp/0387480986/?tag=pfamazon01-20&tag=pfamazon01-20 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) http://count.ucsc.edu/~rmont/papers/Burke_DivGradCurl.pdf" 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) https://www.amazon.com/dp/0486661695/?tag=pfamazon01-20&tag=pfamazon01-20 by Harley Flanders
A nice amount of applications of differential forms written for physicists and engineers.
3) https://www.amazon.com/dp/0486640396/?tag=pfamazon01-20&tag=pfamazon01-20 by Bishop and Goldberg
This is more rigorous than the two books above.

There is also https://www.amazon.com/dp/0817644997/?tag=pfamazon01-20&tag=pfamazon01-20 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.
 
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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.
 
  • #4
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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
 
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1) https://www.amazon.com/dp/0817637079/?tag=pfamazon01-20&tag=pfamazon01-20 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.
Thanks for the recommendation of this book! I had not heard of it before...
 
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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?
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.
Thanks for the recommendation of this book! I had not heard of it before...
No problem Sankaku! It is a really nice and unique text. The same author also wrote a linear algebra text and a few others.
 
  • #7
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Thank you very much n1kofeyn for your very knowledgeable advise
 
  • #8
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I can also highly recommend the book by John Baez (gauge fields, knots and topology)
https://www.amazon.com/dp/9810220340/?tag=pfamazon01-20&tag=pfamazon01-20

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.
 
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  • #11
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I can also highly recommend the book by John Baez (gauge fields, knots and topology)
https://www.amazon.com/dp/9810220340/?tag=pfamazon01-20&tag=pfamazon01-20

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.
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.
 
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This http://www.ucolick.org/~burke/forms/books.html" [Broken] of William Burke has some nice one to two sentence reviews of books on differential forms.
 
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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.)
 
  • #16
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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.)
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.
 
  • #17
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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.
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.
 
  • #18
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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.
I had heard about it before from Burke's website, and I'll definitely try to take a look at it sometime. I thought I had read from Burke's website that he had taught art majors with it, but I remembered wrong. Ends up I read it on an Amazon https://www.amazon.com/review/R3D2RJS9R7KYH3/ref=cm_cr_rdp_perm"&tag=pfamazon01-20 for his Applied Differential Geometry book. The reviewer states that he took 4 courses from Burke, and that it was English and theatre majors that he taught from the book.

I didn't mean to downplay the book, but I just meant to state it probably isn't as mathematically rigorous as say the books by Edwards/Hubbard. Though this makes sense since Burke was a physicist. I myself enjoy books that focus on motivation and intuition. Burke said on his website that its got some neat applications.

In the past few days, I came across the book https://www.amazon.com/dp/0226890481/?tag=pfamazon01-20&tag=pfamazon01-20. It's very short and reads very quickly.
 
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I have just found that an earlier version (2003) of this is available for free on arXiv:

http://arxiv.org/abs/math/0306194
As Sankaku mentioned this is from 2003, I would just like to further mention that this version of the text is three years younger than the published version. David Bachman, the author who was invited to join this https://www.physicsforums.com/showthread.php?t=67268", mentions that this version is old and explained he had made significant changes.

Last spring, I was actually able to find a .pdf of the published version through my university library, so it's a possibility other libraries have an electronic version as well.
 
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As Sankaku mentioned this is from 2003, I would just like to further mention that this version of the text is three years younger than the published version. David Bachman, the author who was invited to join this https://www.physicsforums.com/showthread.php?t=67268", mentions that this version is old and explained he had made significant changes.

Last spring, I was actually able to find a .pdf of the published version through my university library, so it's a possibility other libraries have an electronic version as well.
Yes, my library only has an electronic version. Maybe I'm old-fashioned, but I find e-versions difficult to use.
 
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  • #22
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Yes, my library only has an electronic version. Maybe I'm old-fashioned, but I find e-versions difficult to use.
Definitely! I was able to "rip" the .pdf file from their interface so that I can use it by actually scrolling or printing more than one page at a time. The interfaces these libraries use are despicable and only allow page by page viewing where you have to click to go to each new page. You can also only print a few pages at a time. My library seems to think this is a solution, but nothing's better than a good math book in your hands.
 
  • #23
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Hi everybody, what's the general opinion on Munkres' Analysis on Manifolds?

I'm looking at either this book or the book by Edwards for some self-study over winter break. I'm an applied math major, but I'm not turned off by proofs and mathematical rigor; in fact, I prefer books that emphasize such aspects.
 
  • #25
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Hi everybody, what's the general opinion on Munkres' Analysis on Manifolds?

I'm looking at either this book or the book by Edwards for some self-study over winter break. I'm an applied math major, but I'm not turned off by proofs and mathematical rigor; in fact, I prefer books that emphasize such aspects.
Then go with Edwards's book, at least out of those two. Munkres is preparing the student for the more abstract and technical version of the theory of differential forms, and Edwards' book contains a lot more physical descriptions and applications. Munkres will also require more prerequisites than Edwards.

I haven't looked at it in any real detail, but the Flanders book I mentioned in my first post is supposed to be well known for its nice amount of applications, specifically targeted to engineers and physicists. Although, Edwards is an extremely good introduction to the subject. I received a copy of Hubbard's book today, which is very good as well and comes with lots of proofs and examples.
 

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