# Alternatives to Chapters 8-10 of Rudin's PMA

I have been hearing that the chapters on multi-varaible analysis in Rudin's PMA are almost nothing like previous insightful chapters in the single-variable analysis, and I verified myself too (very difficult to read). I would like to receive your suggestions on alternative books to learn the analysis in R^n, manifolds, and differential forms. Do I need to read a book in differential geometry/topology?

Unfortunately, I forgot most of concepts from the multivariable calculus...

• I do not like when people complain about these particular chapters because it is "word on the street" that these are the bad chapters -- reading the book with this preconceived notion only worsens the situation. I will tell you this: chapters 8 and 9 are beautifully and efficiently written. The exercises in chapter 9 are on the computational side, aiding in intuition in higher-dimensional thinking. By nature of the subject, differential forms are tough to read (half the battle is notation). You may run into the same problem in other books, but check Spivak's Calc on Manifolds or doCarmo. – user369210 Jul 22 '16 at 15:08
• I do not like when people use this forum to scold others, because it usually stems from a motivation of self-assertion. Rudin's writing is extremely Bourbakist and dry. He omits the most important ingredient: guiding the student how to think so as to have discovered this math on their own. Meanwhile, according to Sir Michael Atiyah (who, unlike Rudin, got the Fields Medal), "The mathematical problems that have been solved or techniques that have arisen out of physics in the past have been the lifeblood of mathematics." – avs Jul 22 '16 at 15:55
• @avs right, because offering a second opinion is scolding. If you dive into a book or movie having heard that it is second-rate, you're going to come out thinking it's second-rate. That's called psychological bias. My point is that a re-read with the understanding that the math is beautifully written would help OP mentally. Anyhow, thanks for the condescending words. – user369210 Jul 22 '16 at 22:09
• No, because criticizing someone's behavior is scolding. Psychological bias is not an inevitability. The Asker did try reading, despite the hearsay. And, at any rate, instead of "I don't like when people" (criticizing behavior), writing "Hearsay can create psychological bias, possibly leading to rejecting a book that's actually good" would have been factual, without getting personal. Far too many people on this forum already try to self-assert by rather asocial means. – avs Jul 23 '16 at 1:29

Zorich, "Mathematical Analysis". Intuitive explanations, physical connections; in short, a human touch'' of a caring teacher.