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Preface Richard Courant's Differential and Integral Calculus, Vols. I and Il, has been tremendously successful in introducing several gener- ations of mathematicians to higher mathematics. Throughout, those volumes presented the important lesson that meaningful mathematics is created from a union of intuitive imagination and deductive reason ing. In preparing this revision the authors have endeavored to main tain the healthy balance between these two modes of thinking which characterized the original work. Although Richard Courant did not live to see the publication of this revision of Volume I, all major changes had been agreed upon and drafted by the authors before Dr. Courant's death in January 1972 From the outset. the authors realized that volume ll which deals with functions of several variables, would have to be revised more drastically than Volume L. In particular, it seemed desirable to treat the fundamental theorems on integration in higher dimensions with the same degree of rigor and generality applied to integration in one dimension. In addition, there were a number of new concepts and topics of basicimportance, which, in the opinion of the authors, belong to an introduction to analysis Only minor changes were made in the short chapters(6, 7, and 8) dealing, respectively with Differential Equations, Calculus of Vari ations, and Functions of a Complex Variable. In the core of the book Chapters 1-5, we retained as much as possible the original scheme of two roughly parallel developments of each subject at different levels: an informal introduction based on more intuitive arguments together with a discussion of applications laying the groundwork for the subsequent rigorous proofs The material from linear algebra contained in the original Chapter 1 seemed inadequate as a foundation for the expanded calculus struc- ture. Thus, this chapter(now Chapter 2) was completely rewritten and now presents all the required properties of nth order determinants and matrices, multilinear forms. Gram determinants and linear manifolds Preface The new Chapter 1 contains all the fundamental properties of linear differential forms and their integrals. These prepare the reader for the introduction to higher-order exterior differential forms added to Chapter 3. Also found now in Chapter 3 are a new proof of the implicit function theorem by successive approximations and a discus sion of numbers of critical points and of indices of vector fieldsin two dimensions Extensive additions were made to the fundamental properties of multiple integrals in Chapters 4 and 5. Here oneis faced with a familiar difficulty: integrals over a manifold M, defined easily enough by subdividing M into convenient pieces, must be shown to be inde- pendent of the particular subdivision. This is resolved by the sys tematic use of the family of Jordan measurable sets with its finite intersection property and of partitions of unity. In order to minimize topological complications, only manifolds imbedded smoothly into Euclidean space are considered. The notion of orientation"of a manifold is studied in the detail needed for the discussion of integrals of exterior differential forms and of their additivity properties. On this basis, proofs are given for the divergence theorem and for Stokes's theorem in n dimensions. To the section on Fourier integrals in Chapter 4 there has been added a discussion of Parseval's identity and of multiple Fourier integrals. Invaluable in the preparation of this book was the continued generous help extended by two friends of the authors, Professors Albert A. Blank of Carnegie-Mellon University, and Alan Solomon of the University of the Negev. Almost every page bears the imprint of their criticisms, corrections, and suggestions. In addition, they prepared the problems and exercises for this volume. I Thanks are due also to our colleagues, Professors K.O. Friedrichs and Donald Ludwig for constructive and valuable suggestions, and to John Wiley and Sons and their editorial staff for their continuing encouragement and assistance FRITZ JOHN New York September 1973 1In contrast to Volume I, these have been incorporated completely into the text their solutions can be found at the end of the volume Contents Chapter 1 Functions of several va ariables and their derivatives 1.1 Points and points Sets in the Plane and in Space a. Sequences of points. Conver- gence 1 b. Sets of points in the lane, 3 c. The boundary of a set Closed and open sets, 6 d. closure as set of limit points,9 e. Points and sets of points in space, 9 1.2 Functions of Several Independent Variables a. Functions and their domains. 11 b. The simplest types of func. tions, 12 C. Geometrical representa tion of functions. 13 1.3 Continuity 17 a. Definition, 17 b. The concept of limit of a function of several vari- ables. 19 C. The order to which a function vanishes. 22 1.4 The Partial Derivatives of a Function a. Definition. Geometrical representation, 26 b. Examples 32 c. Continuity and the existence of partial derivatives, 84 viii Contents d, Change of the order of differentiation. 36 1.5 The Differential of a Function and Its Geometrical Meaning 40 a. The concept of differentia bility, 40 b. Directiona. derivatives, 43 C. Geometric interpretation of differentiability, The tangent plane, 46 d. The total differential of a function, 49 application to the calculus of errors. 52 1.6 Functions of Functions(Com pound Functions)and the Introduction of New I dependent variables a Compound functions. The chain rule, 53 b. Examples, 59 C Change of independent variables. 60 1. The Mean value Theorem and Taylors Theorem for Functions of Several variables 64 a. Preliminary remarks about approximation by polynomials, 6 b. The mean value theorem, 66 c. Taylor's theorem for several in dependent variables, 68 1.8 Integrals of a Function Depend- ing on a Parameter a. Examples and definitions. 71 b. Continuity and differentiability of an integral with respect to the parameter, 74 C. Interchange of integrations Smoothing of functions. 80 1.9 Differentials and Line Integrals 89 a. Linear differential forms. 82 b. Line integrals of linear dif ferential forms, 85 C. Dependence of line integrals on endpoints, 92 1.10 The Fundamental Theorem on Integrability of Linear Differential forms a. Integration of total differentials 95 b. Necessary conditions for line integrals to depend only on of the integrability conditions, a the end points, 96 c. Insuficiency d. Simply connected sets, 1 e. The fundamental theorem, 104 APPENDIX A 1. The Principle of the Point of Ac cumulation in Several Dimen- sions and Its Applications 107 a. The principle of the point of accumulation, 107 b. Cauchy,'s convergence test Compactness 108 C. The Heine-Borel covering theorem, 109 d. An application of the Heine- borel theorem to closed sets contains in open sets. 110 A. 2. Basic Properties of Continuous Functions 112 A.8. Basic Notions of the Theory of Point sets 118 a. Sets and sub-sets. 113 b. Union and intersection of sets, 115 C. ap plications to sets of points in the plane. 117 A. 4. Homogeneous functions 119 x Contents Chapter 2 Vectors, Matrices, Linear anstormations 2.1 Operations with Vectors 122 a. Definition of vectors, 122 b. Geometric representation of vectors, 124 C. Length of vectors Angles between directions, 12 d. Scalar products of vectors, 131 e. Equa tion of hyperplanes in vector form, 138 f. Linear dependence of vec. tors and systems of linear equations 186 2.2 Matrices and Linear Transforma tions 148 a. Change of base. Linear spaces 143 b. Matrices, 146 C. Opera tions with matrices, 150 d. Square matrices. The reciprocal of a mat- rix. Orthogonal matrices. 158 2. 3 Determinants 159 a. Determinants of second and third order 159 b. Linear and multi linear forms of vectors. 163 C. Al ternating multilinear forms. Defini- tion of determinants. 166 d. Prin. cipal properties of determinants, 171 e. Application of determinants to systems of linear equations. 175 2.4 Geometrical Interpretation of Determinants 180 a. Vector products and volumes of parallelepipeds in three-dimensional space, 180 b. Expansion of a deter- minant with respect to a column Vector products in higher dimen sions, 187 C. Areas of parallelograms and volumes of parallelepipeds in higher dimensions, 190 d. Orienta- tion of parallelepipeds in n-dimen- sional space 195 e. Orientation of planes and hyperplanes, 200 I. Change of volume of parallele- Bipeds in linear transformations, 201 2.5 Vector Notions in Analysis 204 a. Vector fields. 204 b. Gradient of a scalar, 205 C. Divergence and curl of a vector field. 208 d Families of vectors. Application to the theory of curves in space and to motion of particles, 211 Chapter 3 Developments and applications of the Differential calculus 31 Implicit Functions 918 a. General remarks, 218 b. Geo. metrical interpretation, 219 c. The implicit function theorem, 221 d. Proof of the implicit function theorem, 225 e. The implicit func- tion theorem for more than two independent variables, 228 8.2 Curves and Surfaces in Implicit Form 280 a. Plane curves in implicit form 230 b. Singular points of curves 286 C. Implicit representation of surfaces. 238 3. 3 Systems of Functions, Transfor- mations, and Mappings 241 8. General remarks, 241 b. Cur- vilinear coordinates 246 C: Exten- sion to more than two independent variables. 249 d. Differentiation formulae for the inverse functions stii Contents 252 e. Symbolic product of mappings 957 f. General theorem on the inversion of transformations and of systems of implicit functions Decomposition into primitive map- pings, 261 g. Alternate construc tion of the inverse mapping by the method of successive approxima tions, 266 h. Dependent functions 268 i. Concluding remarks, 27b 3.4 Applications 278 a. Elements of the theory of sur faces. 278 b. Conformal transfor- mation in general, 289 8.5 Families of Curves. Families of Surfaces, and Their Envelopes 290 a. General remarks. 290 b. En velopes of one-parameter families of curves, 292 c. Examples, 296 d. Endevelopes of families of surfaces. 308 a. Definition of alternating due s 3.6 Alternating Differential Form 307 ferential forms. 80 b. Sums and products of differential forms, 310 c. Exterior derivatives of differ- ential forms. 312 d. Exterior differential forms in arbitrar coordinates. 316 8.7 Maxima and Minima 825 a. Necessary conditions, 325 b, Examples, 827 c, Maxima and minima with subsidiary conditions 380 d. proof of the method of unde termined multipliers in the simplest case. 834 e. Generalization of the method of undetermined multipliers 337f。 xamples,840

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