David V. Widder - Advanced Calculus

ADVANCED CALCULUSSecond Editionby David V. WidderProfessor of Mathematics, Emeritus
Harvard UniversityD OVER P UBLICATIONS , I NC. , New YorkCopyright Copyright 1947, 1961, 1989 by David V. Widder. All rights reserved. Library of Congress Cataloging-in-Publication Data Widder, D. V. (David Vernon), 1898 Advanced calculus / by David V. (David Vernon), 1898 Advanced calculus / by David V.

Widder.2nd ed. p. em. Reprint. Originally published: Englewood Cliffs, N.J. (Prentice-Hall mathematics series). (Prentice-Hall mathematics series).

Includes index. ISBN-13: 978-0-486-66103-2 E-ISBN-10: 0-486-66103-2 1. Calculus. I. Title. II.

Series: Prentice-Hall mathematics series. QA303.W48 1989

515-dc20

89-33578

CIP Manufactured in the United States by Courier Corporation
66103213
www.doverpublications.com This book is designed for students who have had a course in elementary calculus covering the work of three or four semesters. However, it is arranged in such a way that it may also be used to advantage by students with somewhat less preparation. The reader is expected to have considerable skill in the manipulations of elementary calculus, but it is not assumed that he will be very familiar with the theoretic side of the subject, Consequently, the book emphasizes first the type of manipulative problem the student has been accustomed to and gradually changes to more theoretic problems. In fact, the same sort of crescendo appears within the chapters themselves. In certain cases a fundamental theorem, whose meaning is easily understood, is stated and used at the beginning of a chapter; its proof is deferred to the end of it.

Believing that clarity of exposition depends largely on precision of statement, the author has taken pains to state exactly what is to be proved in every case. Each section consists of definitions, theorems, proofs, examples, and exercises. An effort has been made to make the statement of each theorem so concise that the student can see at a glance the essential hypotheses and conclusions. Three of the chapters involve the Stieltjes integral and the Laplace transform, topics which do not appear in the traditional course in advanced calculus. The author believes that these subjects have now reached the stage where a knowledge of them must be part of the equipment of every serious student of pure or applied mathematics. The book may be used as a text in various ways.

Certainly, the usual college course of two semesters cannot include so much material. The authors own procedure in his classes has been to present all of any chapter used but to offer different chapters in different years. Another method, which might be particularly useful for the engineering student or for the prospective applied mathematician, would be to use the first two thirds of each chapter. The final third could then be used for reference purposes. It should be observed that the separate chapters are more or less independent. Subject to the fact that the latter half of the book is more difficult than the first, the order of presentation may be greatly varied.

For example, . D.V.W.

In this revision of the text the main features of the first edition are preserved. There follows below a list of the more important changes. The dot-cross notation for vector operations has been substituted for the dash-roof system. This change has necessitated the use of some distinctive designation for vectors, and an arrow over a letter representing a vector is now used. (Bold-faced type, though satisfactory in a text, is not easily transferred to a blackboard.) Certain theorems have been sharpened, where this could be done without too much sacrifice of simplicity. (Bold-faced type, though satisfactory in a text, is not easily transferred to a blackboard.) Certain theorems have been sharpened, where this could be done without too much sacrifice of simplicity.

For example, the class of differentiable functions of several variables has been interpolated between C and C1. The treatment of Stieltjes integrals has been altered somewhat with the purpose of making it more useful to the student who is not very familiar with the basic facts about the Riemann integral. In fact, such a student may, if he wishes, correct his deficiency without studying the Stieltjes integral by concentrating on . The material on series has been augmented by the inclusion of the method of partial summation, of the Schwarz-Hlder inequalities, and of additional results about power series. There has also been added a brief discussion of general infinite products and an elementary derivation of an infinite product for the gamma function. Many new exercises have been added, some of which are intended to be of the easier variety needed for developing initial skills.

Answers to some of the exercises are in a final section. The author wishes to acknowledge here his debt to the many persons who have given him suggestions for improvement of the text. At the risk of unintentional omissions there follows a list of their names: R. D. Accola, L. V.

Ahlfors, Albert A. Bennett, Garrett Birkhoff, B. H. Bissinger, R. E. P. Boas, D. L. Guy, I. Guy, I.

Hirschman, L. H. Loomis, K. O. May, E. Post, D. G. G.

Quillen, A. E. Taylor, H. Whitney, J. L. D.V.W. CONTENTS

We shall be dealing in this chapter with real functions of several real variables, such as u = f(x, y), u = f(x, y, z), etc. Introduction We shall be dealing in this chapter with real functions of several real variables, such as u = f(x, y), u = f(x, y, z), etc.

In these examples the variables x, y, z, are called the independent variables or arguments of the function, u is the dependent variable or the value of the function. Unless otherwise stated, functions will be assumed single-valued; that is, the value is uniquely determined by the arguments. Multiple-valued functions may be studied as combinations of single-valued ones. For example, the equation defines two single-valued functions, A function of two variables clearly represents a surface in the space of the rectangular coordinates x, y, u. In the study of functions of more than two variables, geometrical language is often retained for purposes of analogy, even though geometric intuition then fails.

A partial derivative of a function of several variables is the ordinary derivative with respect to one of the variables when all the rest are held constant.

Various notations are used. The partial derivatives of u = f(x, y, z) are An important advantage of the subscript notation is that it indicates an operation on the function that is independent of the particular letters employed for the arguments. Thus, if

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