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Mathematical Analysis A Modern Approach to Advanced Calculus by Tom Apostol 1973
Mathematical Analysis A Modern Approach to Advanced Calculus by Tom Apostol 1973
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Mathematical Analysis A Modern Approach to Advanced Calculus Apostol.
PREFACE
A glance at the Table of Contents will reveal that most of the topics which usually fall under the heading "Advanced Calculus' are treated in this book: The author's aim has been to provide a development of the subject matter which in honest, rigorous, up-to-date, and, at the same time, not too pedantie. Most of the "hard" theorems which are either omitted or treated rather skimpily in many texts on advanced calculus are proved here with great care. Some of them are ordinarily considered too difficult for a standard course in advanced calculus but too elementary for a course in real or complex function theory. With the inclusion of these theoretna, the book helps to fill the gap between elementary calculus and advanced courses in analysis. More important than this, it introduces the reader to some of the abstract thinking that pervades modern mathematics.
Some of the features to be found in the book include:
A chapter on abstract set theory which contains a precise formulation of the function concept.
Point set topology in n-dimensional Euclidean space.
A satisfactory treatment of differentials.
An elementary discussion of connectedness.
An extensive treatment of the Riemann-Stieltjes integral, including complex integration and a discussion of the winding number.
A development of Jordan content and outer Lebesgue measure.
A proof of Green's theorem for plane regions bounded by arbitrary rectifiable Jordan curves.
A careful treatment of surfaces and surface integrals.
A thorough discussion of the interchange of limit processes.
A chapter on Fourier series and Fourier integrals, including the Fourier integral theorem, the convolution theorem for Fourier trans. forms, and the complex inversion formula for Laplace transforms.
An introduction to the theory of functions of a complex variable.
There is ample material here for a year's course, but many parts can easily be omitted without disturbing the continuity of the presentation.
Nearly 500 exercises are included; many of these are designed to illustrate the general theory or to show where it can break down.
For various reasons, one of which is simply lack of space, there is a minimum of emphasis on applications and physical motivation in this book.
It is a fairly easy matter for a lecturer to give a leisurely heuristic discussion
Bs5.240223.e
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