Tom IN. Apostol CALCULUS VOLUME II Mlul ti Variable Calculus and Linear Algebra, with Applications to DifFeren tial Equations and Probability SECOND EDITION John Wiley & Sons New York London Sydney Toronto CONSULTING EDITOR George Springer, Indiana University COPYRIGHT 0 1969 BY XEROX CORPORATION. All rights reserved. No part of the material covered by this copyright may be produced in any form, or by any means of reproduction. Previous edition copyright 0 1962 by Xerox Corporation. Librar of Congress Catalog Card Number: 67-14605 ISBN 0 471 00007 8 Printed in the United States of America. 1 0 9 8 7 6 5 4 3 2 To Jane and Stephen PREFACE This book is a continuation of the author’s Calculus, Volume I, Second Edition. The present volume has been written with the same underlying philosophy that prevailed in the first. Sound training in technique is combined with a strong theoretical development. Every effort has been made to convey the spirit of modern mathematics without undue emphasis on formalization. As in Volume I, historical remarks are included to give the student a sense of participation in the evolution of ideas. The second volume is divided into three parts, entitled Linear Analysis, Nonlinear Ana!ysis, and Special Topics. The last two chapters of Volume I have been repeated as the first two chapters of Volume II so that all the material on linear algebra will be complete in one volume. Part 1 contains an introduction to linear algebra, including linear transformations, matrices, determinants, eigenvalues, and quadratic forms. Applications are given to analysis, in particular to the study of linear differential equations. Systems of differential equations are treated with the help of matrix calculus. Existence and uniqueness theorems are proved by Picard’s method of successive approximations, which is also cast in the language of contraction operators. Part 2 discusses the calculus of functions of several variables. Differential calculus is unified and simplified with the aid of linear algebra. It includes chain rules for scalar and vector fields, and applications to partial differential equations and extremum problems. Integral calculus includes line integrals, multiple integrals, and surface integrals, with applications to vector analysis. Here the treatment is along more or less classical lines and does not include a formal development of differential forms. The special topics treated in Part 3 are Probability and Numerical Analysis. The material on probability is divided into two chapters, one dealing with finite or countably infinite sample spaces; the other with uncountable sample spaces, random variables, and dis- tribution functions. The use of the calculus is illustrated in the study of both one- and two-dimensional random variables. The last chapter contains an introduction to numerical analysis, the chief emphasis being on different kinds of polynomial approximation. Here again the ideas are unified by the notation and terminology of linear algebra. The book concludes with a treatment of approximate integration formulas, such as Simpson’s rule, and a discussion of Euler’s summation formula. Preface \‘I11 There is ample material in this volume for a full year’s course meeting three or four times per week. It presupposes a knowledge of one-variable calculus as covered in most first-year calculus courses. The author has taught this material in a course with two lectures and two recitation periods per week, allowing about ten weeks for each part and omitting the starred sections. This second volume has been planned so that many chapters can be omitted for a variety of shorter courses. For example, the last chapter of each part can be skipped without disrupting the continuity of the presentation. Part 1 by itself provides material for a com- bined course in linear algebra and ordinary differential equations. The individual instructor can choose topics to suit his needs and preferences by consulting the diagram on the next page which shows the logical interdependence of the chapters. Once again I acknowledge with pleasure the assistance of many friends and colleagues. In preparing the second edition I received valuable help from Professors Herbert S. Zuckerman of the University of Washington, and Basil Gordon of the University of California, Los Angeles, each of whom suggested a number of improvements. Thanks are also due to the staff of Blaisdell Publishing Company for their assistance and cooperation. As before, it gives me special pleasure to express my gratitude to my wife for the many ways in which she has contributed. In grateful acknowledgement I happily dedicate this book to her. T. M. A. Pasadena, California September 16, 1968 Logical Interdependence of the Chapters ix 1 LINEAR SPACES INTRODUCTION TRANSFORMATIONS TO NUMERICAL AND MATRICES ANALYSIS DETERMINANTS I I I 10 1 1 13 6 H DIFFERENTIAL LINE SET FUNCTIONS LINEAR I\ CALCULUS OF INTEGRALS AND ELEMENTARY DIFFERENTIAL SCALAR AND PROBABILITY EQUATIONS 4 VECTOR FIELDS EIGENVALUES , I AND I 7 EIGENVECTORS SYSTEMS OF I- DIFFERENTIAL EQUATIONS OPERATORS ACTING ON EUCLIDEAN 1 1 PROBABILITIES 1 SPACES 9 ’ APPLICATIONS OF DIFFERENTIAL INTEGRALS CALCULUS CONTENTS PART 1. LINEAR ANALYSIS 1. LINEAR SPACES 1.1 Introduction 3 1.2 The definition of a linear space 3 1.3 Examples of linear spaces 4 1.4 Elementary consequences of the axioms 6 1.5 Exercises 7 1.6 Subspaces of a linear space 8 1.7 Dependent and independent sets in a linear space 9 1.8 Bases and dimension 12 1.9 Components 13 1.10 Exercises 13 1.11 Inner products, Euclidean spaces. Norms 14 1.12 Orthogonality in a Euclidean space 18 1.13 Exercises 20 1.14 Construction of orthogonal sets. The Gram-Schmidt process 22 1.15 Orthogonal complements. Projections 26 1.16 Best approximation of elements in a Euclidean space by elements in a finite- dimensional subspace 28 1.17 Exercises 30 2. LINEAR TRANSFORMATIONS AND MATRICES 2.1 Linear transformations 31 2.2 Null space and range 32 2.3 Nullity and rank 34 xi xii Contents 2.4 Exercises 35 2.5 Algebraic operations on linear transformations 36 2.6 Inverses 38 2.7 One-to-one linear transformations 41 2.8 Exercises 42 2.9 Linear transformations with prescribed values 44 2.10 Matrix representations of linear transformations 45 2.11 Construction of a matrix representation in diagonal form 48 2.12 Exercises 50 2.13 Linear spaces of matrices 51 2.14 Tsomorphism between linear transformations and matrices 52 2.15 Multiplication of matrices 54 2.16 Exercises 57 2.17 Systems of linear equations 58 2.18 Computation techniques 61 2.19 Inverses of square matrices 65 2.20 Exercises 67 2.21 Miscellaneous exercises on matrices 68 3. DETERMINANTS 3.1 Introduction 71 3.2 Motivation for the choice of axioms for a determinant function 72 3.3 A set of axioms for a determinant function 73 3.4 Computation of determinants 76 3.5 The uniqueness theorem 79 3.6 Exercises 79 3.7 The product formula for determinants 81 3.8 The determinant of the inverse of a nonsingular matrix 83 3.9 Determinants and independence of vectors 83 3.10 The determinant of a block-diagonal matrix 84 3.11 Exercises 85 3.12 Expansion formulas for determinants. Minors and cofactors 86 3.13 Existence of the determinant function 90 3.14 The determinant of a transpose 91 3.15 The cofactor matrix 92 3.16 Cramer’s rule 93 3.17 Exercises 94