PREFACE - Advanced Calculus of Several Variables

Advanced Calculus of Several Variables (1973)

PREFACE

This book has developed from junior–senior level advanced calculus courses that I have taught during the past several years. It was motivated by a desire to provide a modern conceptual treatment of multivariable calculus, emphasizing the interplay of geometry and analysis via linear algebra and the approximation of nonlinear mappings by linear ones, while at the same time giving equal attention to the classical applications and computational methods that are responsible for much of the interest and importance of this subject.

In addition to a satisfactory treatment of the theory of functions of several variables, the reader will (hopefully) find evidence of a healthy devotion to matters of exposition as such—for example, the extensive inclusion of motivational and illustrative material and applications that is intended to make the subject attractive and accessible to a wide range of “typical” science and mathematics students. The many hundreds of carefully chosen examples, problems, and figures are one result of this expository effort.

This book is intended for students who have completed a standard introductory calculus sequence. A slightly faster pace is possible if the students' first course included some elementary multivariable calculus (partial derivatives and multiple integrals). However this is not essential, since the treatment here of multivariable calculus is fully self-contained. We do not review single-variable calculus; with the exception of Taylor's formula in Section II.6 (Section 6 of Chapter II) and the fundamental theorem of calculus in Section IV.1.

Chapter I deals mainly with the linear algebra and geometry of Euclidean n-space Imagen. With students who have taken a typical first course in elementary linear algebra, the first six sections of Chapter I can be omitted; the last two sections of Chapter I deal with limits and continuity for mappings of Euclidean spaces, and with the elementary topology of Imagen that is needed in calculus. The only linear algebra that is actually needed to start Chapter II is a knowledge of the correspondence between linear mappings and matrices. With students having this minimal knowledge of linear algebra, Chapter I might (depending upon the taste of the instructor) best be used as a source for reference as needed.

Chapters II through V are the heart of the book. Chapters II and III treat multivariable differential calculus, while Chapters IV and V treat multivariable integral calculus.

In Chapter II the basic ingredients of single-variable differential calculus are generalized to higher dimensions. We place a slightly greater emphasis than usual on maximum–minimum problems and Lagrange multipliers—experience has shown that this is pedagogically sound from the standpoint of student motivation. In Chapter III we treat the fundamental existence theorems of multivariable calculus by the method of successive approximations. This approach is equally adaptable to theoretical applications and numerical computations.

Chapter IV centers around Sections 4 and 5 which deal with iterated integrals and change of variables, respectively. Section IV.6 is a discussion of improper multiple integrals. Chapter V builds upon the preceding chapters to give a comprehensive treatment, from the viewpoint of differential forms, of the classical material associated with line and surface integrals, Stokes' theorem, and vector analysis. Here, as throughout the book, we are not concerned solely with the development of the theory, but with the development of conceptual understanding and computational facility as well.

Chapter VI presents a modern treatment of some venerable problems of the calculus of variations. The first part of the Chapter generalizes (to normed vector spaces) the differential calculus of Chapter II. The remainder of the Chapter treats variational problems by the basic method of “ordinary calculus”—equate the first derivative to zero, and then solve for the unknown (now a function). The method of Lagrange multipliers is generalized so as to deal in this context with the classical isoperimetric problems.

There is a sense in which the exercise sections may constitute the most important part of this book. Although the mathematician may, in a rapid reading, concentrate mainly on the sequence of definitions, theorems and proofs, this is not the way that a textbook is read by students (nor is it the way a course should be taught). The student's actual course of study may be more nearly defined by the problems than by the textual material. Consequently, those ideas and concepts that are not dealt with by the problems may well remain unlearned by the students. For this reason, a substantial portion of my effort has gone into the approximately 430 problems in the book. These are mainly concrete computational problems, although not all routine ones, and many deal with physical applications. A proper emphasis on these problems, and on the illustrative examples and applications in the text, will give a course taught from this book the appropriate intuitive and conceptual flavor.

I wish to thank the successive classes of students who have responded so enthusiastically to the class notes that have evolved into this book, and who have contributed to it more than they are aware. In addition, I appreciate the excellent typing of Janis Burke, Frances Chung, and Theodora Schultz.