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I'm a second year math undergraduate and I'm looking for a book for the three analysis courses I'm taking this year: Differentiation of Multivariable functions, Integration of Multivariable functions and Power Series and Lebesgue Integral.

I have read almost all Spivak's Calculus and some of Apostol's Calculus book and done plenty of the exercises in both books as last year I took a Calculus course that covered most of the topics of single-variable calculus. I've also studied the basics of metric spaces (using Kaplansky's Set Theory and Metric Spaces as a reference) and some topology using Mendelson's introduction to topology (I'll be taking this semester a course on General Topology too).

With this background, which book should I get? I was thinking as the two main options either Calculus II (Apostol) with a supplementary book on Lebesgue integral or Mathematical Analysis (Apostol). Is there any other book suitable for my courses? Are there any Dover books (or similar) related that could be useful as a supplementary material (as I've already found for other topics) ? I appreciate suggestions and comments about the books I've already mentioned.

Edit (i): I've also seen a bit about Spivak's Calculus in Manifolds. Can it be suitable for studying multivariable calculus?

gal127
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    I don't think Apostol II is the best choice for someone going into math, though it may be the treatment of line and surface integrals that's influencing me, and that's not part of your curriculum. I really like Apostol's analysis book, but be aware that it treats Lebesgue integration only in $R^n$, not arbitrary measure spaces. In all likelihood, you'll need a separate book for that. Here are some options for computational practice with partial derivatives in case you use an analysis book that assumes prior exposure: the last chapter of Burkill's First Course in Mathematical Analysis, – Anonymous Sep 20 '21 at 00:39
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    (cont'd) the last chapter of Volume 1 of Zorich. Some rigorous books which don't assume prior exposure to partial derivatives: Fleming's Functions of Several Variables (but it does only Lebesgue, not Riemann, integration in $R^n$), C.H. Edwards' Advanced Calculus of Several Variables. Fleming and Edwards also do differential forms on submanifolds of $R^n$, Stokes' Theorem, etc. For multiple Riemann integrals, you could also look at Zorich II or Burkill's Second Course. – Anonymous Sep 20 '21 at 00:45
  • (cont'd) On balance, though, I would say that if you're willing to put off manifolds/Stokes and abstract Lebesgue integration to a later stage (or look at them in other books), then the best approach might be to use one of the two options for partial derivatives followed by Apostol's analysis book (which I think has enough computational examples for multiple integrals). – Anonymous Sep 20 '21 at 00:48
  • If you can read French, you could read Volumes 3 and 4 of Cours de mathématiques spéciales by Ramis. In comparison with Apostol or Zorich, it goes less far on theory, but it's not far, and the exercises are highly instructive, with a good balance of computational and theoretical problems. – Anonymous Sep 20 '21 at 01:59
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    My opinion is that the treatment in Spivak's Calculus on Manifolds is probably too condensed if it's the only thing you read. There aren't enough examples and exercises. – Anonymous Sep 20 '21 at 16:04
  • I was curious and looked to see what I could find out about these three analysis courses at the University of Seville. I think you could improve your post by linking to the "Programa de la asignatura" or lecture notes for each. From the names alone, I wouldn't have guessed that the course on multiple integrals includes vector analysis limited to R3. That changes your options somewhat. With this information, I would probably recommend Fleming's book, using Apostol's MA as a source for series and an alternative for some of the rest. – Anonymous Sep 20 '21 at 19:43
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    Since your course on Lebesgue integration includes some measure theory, you might prefer the three-volume series by Amann and Escher to Apostol and Fleming, It's translated from German and was described in this answer: (https://math.stackexchange.com/questions/3703006/modern-analysis-books-in-languages-other-than-english-or-russian/3703057#3703057) If you just want a concise introduction to abstract Lebesgue integration, there's Bartle's Elements of Integration and Lebesgue Measure.. – Anonymous Sep 20 '21 at 20:06

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