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Rating: 
Summary: great book!
Review: After a hiatus of many, many years I wanted to resume the study of math that I pursued as an undergraduate, and I wanted to start at the heart of analysis: advanced calculus. I found myself bouncing around among several books because:
1. they were really books about measure theory and real analysis with only a cursory review of advanced calc
2. they were really freshman calc books with a short intro to advanced calc
3. they were 400 pages longer than I wanted to deal with
4. they were, in the words of one Amazon reviewer, "one damn theorem after another, written by mathematicians for mathematicians"Then I found Guzman, who explains, discusses, illustrates and
gives examples. Guzman gives examples after definitions and theorems, sometimes concrete ones from the physical sciences. He discusses why certain suppositions are neccessary and gives counterexamples.
The proofs are about as clear as any I've seen, and he avoids technical tricks which make proofs shorter but harder to understand. Diagrams give insight into the ways a mathematician
might think about the theorems and proofs. The book includes the complete solutions to all problems, which makes it even better for self study (in some math books the problems are used to introduce new concepts not covered in the main text, but Guzman's problem set sticks pretty close to the material covered). In short: Guzman is a master expositor. When I don't understand something it's clear that the fault lies in my efforts, not in the book - with many texts I feel the opposite is true. (one of the few other books I feel this way about is "Abstract Algebra" by Dummit and Foote). I think that in the future math will be taught using interactive software that will allow the reader to experiment with ideas and
visualizations of the material, but in terms of educational materials written on dead trees, it doesn't get much better than this.
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