: Analysis 2 Textbook recommendations please?

[SMHPhysics]

Hello,

My father will be visiting the UK soon. I live in a developing country where not many books are available, so he'll be bringing them here. Now, I have already compiled a list of quite a few books. I am particularly looking forward to Munkres' Topology and Sutherland's Metric Spaces and Topology. However I need a real analysis book, at a level higher that Spivak's Calculus textbook, which I have completed. My 'mathematical maturity' is at the level of Spivak's Calculus, Shilov's linear algebra, Tennenbaum's differential equations, and parts of Apostol's Analysis, etc. I have already had cookbook James Stewart style non rigorous calculus course. I know its not much but I've turned 18 not long ago, and I haven't started college yet (begins fall 2013 I hope).

I would prefer something at the level of Spivak's calculus on manifolds, but its very terse I've noticed from Amazon. What about Munkres' Analysis on Manifolds? Ideally one of these two should do, but I would like to know which would be more appropriate, given that I will be mostly using them for self-study? Also is Pugh's Analysis similar to either of these? The book for me should cover multivariable analysis, with a proof of Stoke's theorem, manifolds etc. Throw in any other suggestions you like! :)

Professor Mathwonk, if you're here Sir, I'm really looking forward to your post :)

Thank You!

PS: PLEASE DO NOT HIJACK THIS THREAD!

 

[Site]

Spivak's Calculus is not quite at the level of introductory real analysis, so moving to a second course in real analysis would be quite a jump. If I were you, I would first learn single variable analysis. Rudin's Principles of Mathematical Analysis essentially boils Spivak's Calculus down to 170 short pages and examines everything from a more general and comprehensive point of view. It should definitely be accessible to you.

After that you will certainly be much more prepared to tackle Spivak's Calculus on Manifolds. Munkres' Analysis book will take you to the same place but with much more words.

 

[ahsanxr]

I own/have access to all three books you mentioned but unfortunately since we only recently got to differential forms in my analysis class, I haven't read them in detail. You are right in that Spivak seems terse although people say if you work out the exercises you'll be golden. As for Pugh, I like his style but keep in mind that he only devotes one out of the 6 chapters in his book to multivariable calculus so his whole treatment is around 50-60 pages long. I would say Munkres would be the most detailed and user-friendly as he motivates things quite well and breaks long up proofs into steps and stuff like that. So if you had to choose one, I'd go for Munkres.

 

[alissca123]

Besides Munkres I reccomend:

-Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach by Hubbard
https://www.amazon.com/dp/0130414085/?tag=pfamazon01-20

- Advanced Calculus of Several Variables by Edwards
https://www.amazon.com/dp/0486683362/?tag=pfamazon01-20

- Functions of Several Variables by Fleming
https://www.amazon.com/dp/0387902066/?tag=pfamazon01-20

 

[rezeew]

Hello,

Thank you for reaching out for recommendations on real analysis textbooks. Based on your current level of mathematical maturity and interests, I would highly recommend Munkres' Analysis on Manifolds. It covers multivariable analysis and includes a proof of Stoke's theorem, making it a great choice for your self-study goals. Additionally, it is written in a clear and concise manner, which should help with understanding the material.

If you are interested in a more challenging and rigorous approach, I would also suggest Spivak's Calculus on Manifolds. However, as you mentioned, it may be quite terse and may require additional resources for a deeper understanding.

Pugh's Analysis is also a good option, but it may not cover all the topics you are looking for. It is more focused on real analysis and may not include as much on multivariable analysis and manifolds.

Other suggestions I have for you are "Real Analysis" by Royden and "Principles of Mathematical Analysis" by Rudin. Both of these texts are highly regarded in the mathematical community and cover a wide range of topics in real analysis.

I hope this helps guide your decision and I wish you all the best in your self-study journey. Happy learning!