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The prerequisites that Lee lists for his book that are also part of Spivak's book are: differentiation of functions $A\subseteq\mathbb{R}^m\to\mathbb{R}^n$, the inverse function theorem, the implicit function theorem, integration, Fubini's theorem, and the change of variable theorem. So Chapters 1,2, and 3. Chapters 4 and 5 on integration on chains and integration on manifolds, respectively, are not needed and it seems that they are covered in Lee's book. I already know from a different source the meterial of Chapters 1,2, and 3 from Spivak's book. Should I go through Spivak's book before going to Lee's?

Edit: So from what I can tell, Spivak's book is not needed (and the same goes for Analysis on Manifolds by Munkres). So what is the purpose of those books?

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  • $\begingroup$ Any reason why you think you should? It seems you have found out yourself that you don't have to. $\endgroup$ Commented Apr 22 at 17:54
  • $\begingroup$ They exist because they discuss a more analysis take on the topic, which may be more useful for more applied contexts such as differential equations. They also fit really well after one has taken a course in real analysis and most calculus as a undergrad. Meanwhile Lee's book has a geometrical take, discussing abstract manifolds, which is more useful for geometrical contexts, and fits well after a Differential Geometry Course (although not required). This difference mostly depends on how different universities/professors structure their math courses. $\endgroup$ Commented Apr 22 at 23:45

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As this is a soft-question related to books, it will be based on the recollection of my past experiences and from the experiences of people I know.

To start off, a brief opinion on the books:

I think M. Spivak's Calculus on Manifolds is one of the worst mathematics textbooks. It's too dense with superficial information, it has too many rough proofs, ill-defined concepts and ill-stated theorem. When I read it as a undergrad, I would get so confused on the results that I would constantly be forced to check other textbooks, to the point that I was just reading Spivak's textbook because it was a shorter introduction. I also got some wrong ideas about concepts such as chains, which I had trouble fixing later on.

I would only ever recommend this book to an engineer that wants to get a brief taste of Advanced Calculus, without mathematical technicalities. I don't think it's a useful book for a mathematician of physicist. But I've heard good opinions on Spivak's other Calculus textbooks, but I never engaged with them.

On the other hand, I think J. M. Lee's Introduction to Smooth Manifolds is a wonderful book for undergraduate students. Sometimes it takes longer to explain some concepts, but I think all discussions have a role in creating a strong intuition and good motivation for definitions and theorems that follow. I really like how he chooses some definitions, and how the topics are organized. But I would not recommend it to a graduate student that already has a good background on these topics, and wants a more dense and in-depth book. Also, I think his Riemannian Manifolds: An introduction to Curvature is also well-written.

I think most math/physics students which I had contact would agree with my opinions here.

Comments aside, they also have different roles:

J. M. Lee has a more geometrical take on Introduction to Smooth Manifolds. Another popular book with a similar take is Loring W. Tu's An introduction to Manifolds, I much prefer Lee's textbook but some of my colleges preferred Tu's textbook. I also have some good recommendations that are exclusively in Portuguese...

On the other hand M. Spivak has a more analysis take on the topic. There are other textbooks with a similar take, such as W. Rudin's Principles of Mathematical Analysis and Munkres' Analysis on Manifolds. I think Munkres' is a much better alternative then Spivak's. But these kind of books almost do not talk about important topics of topological/smooth manifolds as much, so it depends on what you're interested in.

In conclusion:

I believe you should go straight to Lee's, and not read Spivak's. Maybe look into Tu's if you get confused at some parts, to see how things are done there, or just for a comparison. Maybe read Munkres' if that fits better with your intentions.

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    $\begingroup$ On the other hand Spivaks differential geometry volumes are really good and offer unique perspectives. $\endgroup$ Commented Apr 22 at 3:39
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    $\begingroup$ Oh yes, I forgot about them. They also have a delightfully funny covers! Spivak's writing is really good on those ones. I don't how "Calculus on Manifolds" came out so wrong, it's a mystery. First time I read it was in Portuguese, I thought it was a problem with translation, but then I checked out the English version and it was also pretty bad. It would be okay as lecture notes, but it should not have been published in my opinion (specially in multiple languages) $\endgroup$ Commented Apr 22 at 5:13
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    $\begingroup$ There is a very detailed and balanced review on amazon.com. According to the reviewer, Spivak wrote his book when he was a first-year graduate student at the age of 25. $\endgroup$ Commented Apr 22 at 5:52
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    $\begingroup$ Interesting. I was always under the impression that Spivak's Calculus on Manifolds was considered a classic. (I own a copy, but never went through it.) $\endgroup$ Commented Apr 23 at 0:32
  • $\begingroup$ @user1551 So, let me get this straight. I'm a 22-year-old-ass guy who read some guy whose age is 25 on something he's not so sure about xD. Thanks for the comment. I'll switch the book $\endgroup$ Commented May 6 at 4:07
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The most important preparation on the calculus side for differentiable manifolds is fluency with the Inverse Function Theorem and the Implicit Function Theorem. You can just read parts of Spivak's book if you need to.

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