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I've been learning real analysis from this book:
Elementary Analysis, K.A. Ross

I really liked the style of this book. It is quite old, and sometimes very difficult, but I guess I liked the way it lays out analysis very precise and rigorously. Although, I'm a first-year math student and I don't have much to compare it with.

We just finished this book at my university, and now we are studying: "Analysis from $ℝ$ to $ℝ^n$: multivariable functions" as it is called. The "book" we use is written by the university, and to be fair, I just don't understand it. I think that if I had attended all the lectures then it would have worked out otherwise. But now I'm quite sure that some definitions/theorems/tricks to solve the questions are just missing in the text. I get completely frustrated by this.

Would anybody know a good sequel to Elementary Analysis by Ross?

These are the names of the chapters of the book we use:
1. Normed vector spaces and limits
1.1 Normed vector spaces
1.2 Limits of functions and continuity
1.3 Operator norm
2. Differentiation of functions from $ℝ^n$ to $ℝ^m$
3. The chain rule
4. Geometric interpretation of derivatives
5. Mean value theorem: Higher partial derivatives
6. Taylor series in use
7. Extremes
8. Higher total derivatives: Taylor series

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    $\begingroup$ Why the downvote ? $\endgroup$ – Kasper Apr 4 '13 at 22:14
  • $\begingroup$ I suggest you to change the tags for reference request $\endgroup$ – Sebastian Valencia Apr 4 '13 at 22:21
  • $\begingroup$ @Kasper another nice treatment of the topics you list is found in Edwards' Advanced Calculus, it is almost as cheap as your choice. $\endgroup$ – James S. Cook Jul 11 '13 at 23:07
  • $\begingroup$ I won't presume to recommend my textbook, but you might find some of my lectures helpful. $\endgroup$ – Ted Shifrin Oct 11 '18 at 17:51
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multivariable analysis, as well known as analysis on manifolds, uses ideas of vector calculus, and linear algebra, as well ideas taught to you in your former analysis course, if you want to learn it well, off course you should be familiar with multivariable calculus, may be at Hubbard, Loomis level, and for linear algebra a review of abstract topics would suffice, if you have those prerequisites, you would be able to read great books on analysis on manifolds, the best books i know on these topics are Spivak's Calculus on manifolds, Loomis Advanced calculus and Munkres Analysis on manifolds, one of those would suffice for cover your course material.

http://www.amazon.com/Analysis-Manifolds-Advanced-Books-Classics/dp/0201315963

http://www.amazon.com/gp/product/0805390219/ref=pd_lpo_k2_dp_sr_2?pf_rd_p=1532170842&pf_rd_s=lpo-top-stripe-1&pf_rd_t=201&pf_rd_i=0201315963&pf_rd_m=ATVPDKIKX0DER&pf_rd_r=1Z59F991ZNZ9CVRASJC3

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I think I will go for this one:
Multidimensional Real Analysis I: Differentiation (Cambridge Studies in Advanced Mathematics)
It looks very complete.

For dutch people:
http://www.staff.science.uu.nl/~ban00101/lecnotes/inlan2011.pdf http://www.staff.science.uu.nl/~ban00101/funcr2012/fr_2012.pdf
Are two very good readers.

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