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Synopsis
Excerpt from Lectures on the Theory of Functions of Real Variables, Vol. 1
The present work is based on lectures which the author is accustomed to give at Yale University on advanced calculus and the theory of functions of real variables. It falls in two volumes, and the following remarks apply only to the first.
The student of mathematics, on entering the graduate school of American universities, often has no inconsiderable knowledge of the methods and processes of the calculus. He knows how to differentiate and integrate complicated expressions, to evaluate indeterminate forms, to find maxima and minima, to differentiate a definite integral with respect to a parameter, etc. But no em phasis has been placed on the conditions under which these pro cesses are valid. Great is his surprise to learn that they do not always lead to correct results. Numerous simple examples, how ever, readily convince him that such is nevertheless the case.
The problem therefore arises to examine more carefully the conditions under which the theorems and processes of the calculus are correct, and to extend as far as possible or useful the limits of their applicability.
In doing this it soon becomes manifest that the style of reason ing which the student has heretofore employed must be abandoned. Examples of curves without tangents, of curves completely filling areas, and other strange configurations so familiar to the analyst of to-day, make it clear that the rough and ready reasoning which rests on geometric intuition must give way to a finer and more delicate analysis. It is necessary for him to learn to think in the e, 8 forms of Cauchy and Weierstrass.
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- PublisherForgotten Books
- Publication date2020
- ISBN 10 1330439066
- ISBN 13 9781330439067
- BindingPaperback
- LanguageEnglish
- Number of pages576
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Lectures on the Theory of Functions of Real Variables, Vol. 1 (Classic Reprint)
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Paperback. Condition: New. Print on Demand. This book explores the theoretical underpinnings of calculus from a firm foundation in the real number system, providing a rigorous treatment of the subject that has laid the groundwork for modern mathematical analysis. The author begins by establishing the framework for the real numbers through the development of rational and irrational numbers, a step largely attributed to the mathematician Georg Cantor. The text details the properties of these number systems and establishes a formal structure for ordering, addition, subtraction, multiplication, and division, providing a new perspective on familiar concepts. The author demonstrates how the expansion of the number system from rational to real numbers allows for the precise definition of limits, continuity, and derivatives, concepts essential to the study of calculus. Through a systematic and logical approach, this book offers a fresh vantage point on the theoretical underpinnings of calculus, inviting readers to revisit foundational concepts and appreciate the intricate connections within the subject. This book is a reproduction of an important historical work, digitally reconstructed using state-of-the-art technology to preserve the original format. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in the book. print-on-demand item. Seller Inventory # 9781330439067_0
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Lectures on the Theory of Functions of Real Variables, Vol 1 Classic Reprint
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PAP. Condition: New. New Book. Shipped from UK. Established seller since 2000. Seller Inventory # LW-9781330439067
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Lectures on the Theory of Functions of Real Variables, Vol 1 Classic Reprint
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PAP. Condition: New. New Book. Shipped from UK. Established seller since 2000. Seller Inventory # LW-9781330439067
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