※ 引述《po953 (josh)》之銘言：
: 請問常聽說說的Apostol是哪一本書
: 高微的書嗎
: 唸過微積分再來唸這一本ok嗎
Apostol, Mathematical analysis.
底下抄 MathSciNet.
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2nd edition:
From the author's preface: "The aim has been to provide a
development of analysis at the advanced calculus level that
is honest, rigorous, up to date and, at the same time, not
too pedantic. "The second edition differs from the first
[1957; MR0087718 (19,398e)] in many respects. Point set topology
is developed in the setting of general metric spaces as well
as in Euclidean $n$-space, and two new chapters have been
added on Lebesgue integration. The material on line integrals,
vector analysis and surface integrals has been deleted. The
order of some chapters has been rearranged, many sections have
been completely rewritten, and several new exercises have been
added.
"The development of Lebesgue integration follows the Riesz-Nagy
approach which focuses directly on functions and their integrals
and does not depend on measure theory. The treatment here is
simplified, spread out and somewhat rearranged for presentation
at the undergraduate level."
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第一版:
This is a book on advanced calculus, in that many topics
frequently omitted are treated here in detail. On the other
hand, the book is easily adaptable to more modest needs.
Table of contents:
1. The real and complex number systems.
2. Some basic notions of set theory.
3. Elements of point set theory.
4. The limit concept and continuity.
5. Differentiation of functions of one real variable.
6. Differentiation of functions of several variables.
7. Applications of partial differentiation.
8. Functions of bounded variation, rectifiable curves and connected sets.
9. Theory of Riemann-Stieltjes integration.
10. Multiple integrals and line integrals.
11. Vector analysis.
12. Infinite series and infinite products.
13. Sequences of functions.
14. Improper Riemann-Stieltjes integrals.
15. Fourier series and Fourier integrals.
16. Cauchy's theorem and the residue calculus.
The wealth of material is not indicated by this outline alone;
e.g., Chapter 6 includes a careful discussion of the differential,
and Chapter 9 contains a complete proof for Lebesgue's necessary
and sufficient condition for Riemann integrability. Lists of
references to other texts accompany most chapters. There are
nearly 500 problems, which include counterexamples, and generalizations
of results, as well as straightforward exercises. The presentation
is simple and clear. An outstanding feature is the unusually
careful motivation provided for each new concept (but with no
trace of condescension); likewise, well-chosen comments elucidate
the wherefore of proofs. Occasional diagrams, such as to illustrate
a suitable smaller neighborhood within a given one, are another
welcome feature.
Reviewed by L. Gillman
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