Download PDF Mathematical Analysis: A Modern Approach to Advanced Calculus, by Tom M. Apostol
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Mathematical Analysis: A Modern Approach to Advanced Calculus, by Tom M. Apostol
Download PDF Mathematical Analysis: A Modern Approach to Advanced Calculus, by Tom M. Apostol
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Product details
Hardcover
Publisher: Addison-Wesley; 3rd Printing edition (1960)
ASIN: B000JESU36
Package Dimensions:
9.2 x 6.3 x 1.2 inches
Shipping Weight: 2.1 pounds
Average Customer Review:
5.0 out of 5 stars
2 customer reviews
Amazon Best Sellers Rank:
#1,600,523 in Books (See Top 100 in Books)
Apostol's Mathematical Analysis is a classic introduction to analysis. The book truely is an advanced calculus book, which makes it a great place to start developing the fundamental concepts of analysis. The material covered will look familiar to the calculus student, and it is covered in a clear and concise enough manner to be easily accessible.This book is available in a paperback reprint edition.
While thumbing through analysis books in a local college library, looking for a proof or justification of the method of Lagrange multipliers, I fortuitously stumbled on this baby. Abraham, Marsden, and Ratiu were breezing through this method without justification-just an exercise left to the reader. I found the proof in a chapter of Apostol's book on applications of partial differentiation. It follows from the implicit and/or inverse function theorems-more challenging than the elegant AMR examples-but comprehensible and rigorous now via Apostol. This was the 1st edition but judging from the other reviews of the 2nd edition the content is pretty much the same.Apostol's approach is more like the method of classical analysis rather than Rudin's formal metric topological approach. Apostol doesn't even define an abstract metric, just sticks to the usual Euclidean. He mentions topology on one page. He doesn't do formal measure theory nor full Lebesgue integration but only enough to justify certain integrals or operations. Stokes' Theorem is done in 3 dimensions via Green's Theorem, the classical approach. Notions of orientability and boundary are not presented here as in Rudin's approach via differential forms. Nevertheless with careful reading they can be deduced.What can be learned here? Pretty much most of Rudin's techniques in their classical setting as well as many examples done which are given as exercises in Rudin. For example Apostol constructs a countable base a couple chapters in by assigning rational coordinates to the centers of open ball neighborhoods and rational radii. He doesn't call it a countable base. This notion appears in Rudin's exercises 22 through 25 in chapter 2 where countable dense subsets are discussed (separability) leading to a proof that a metric space is compact if and only if every infinite sequence has a limit point. There are hard to find things like the Lagrange multipliers. Of course Apostol does not have Dedekind's Construction of the reals from the rationals to which the reader is referred to Rudin (by Apostol). If you don't have Rudin get this. It's worth having both if you can.
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