Richard Courant Fritz John,Introduction to,Calculus and Analysis. With the assistance of,Albert A Blank and Alan Solomon. With 120 illustrations,Springer Verlag,New York Berlin Heidelberg. London Paris Tokyo Hong Kong,Richard Courant 1888 1972 Fritz John. Courant Institute of Mathematical Sciences,New York University. New York NY 10012, Originally published in 1974 by Interscience Publishers a division of John Wiley and Sons Inc. Mathematical Subject Qassification 26xx 26 01,Printed on acid free paper. Copyright 1989 Springer Verlag New York Inc, Softcover reprint of the hardcover 1st edition 1989. All rights reserved This work may not be translated or copied in whole or in part without the. written permission of the publisher Springer Verlag New York Inc 175 Fifth Avenue New. York NY 10010 USA except for brief excerpts in connection with reviews or scholarly. analysis Use in connection with any form of information storage and retrieval electronic. adaptation computer software or by similar or dissimilar methodol0ltY now known or. hereafter developed is forbidden, The use of general descriptive names trade names trademarks etc in this publication even if. the former are not especially identified is not to be taken as a sign that such names as. understood by the Trade Marks and Merchandise Act may accordingly be used freely by. ISBN 13 978 1 4613 8960 6 e ISBN 13 978 1 4613 8958 3. DOl 10 1007 978 1 4613 8958 3, Richard Courant s Differential and Integral Calculus Vols I and. II 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 II 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 II which deals. with functions of several variables would have to be revised more. drastically than Volume I 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 basic importance 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. vi 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 fields in two. dimensions, Extensive additions were made to the fundamental properties of. multiple integrals in Chapters 4 and 5 Here one is 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 l. 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,September 1973. lin contrast to Volume I these have been incorporated completely into the text. their solutions can be found at the end of the volume. Chapter 1 Functions of Several Variables,and Their Derivatives. 1 1 Points and Points Sets in the,Plane and in Space 1. a Sequences of points Conver,gence 1 b Sets of points in the. plane 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 11,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 26,a Definition Geometrical,representation 26 b Examples. 32 c Continuity and the,existence of partial derivatives 34. 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 Directional. derivatives 43 c Geometric,interpretation of differentiability. The tangent plane 46 d The total,differential of a function 49 e. Application to the calculus of,1 6 Functions of Functions Com. pound Functions and the,Introduction of New In,dependent Variables 53. a Compound functions The chain,rule 53 b Examples 59 c. Change of independent variables 60,1 7 The Mean Value Theorem and. Taylor s Theorem for F qnctions,of Several Variables 64. a Preliminary remarks about,approximation by polynomials 64. 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 71,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 82. a Linear differential forms 82,Contents ix,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 95,a Integration of total differentials. 95 b Necessary conditions for,line integrals to depend only on. the end points 96 c Insufficiency,of the integrability conditions 98. d Simply connected sets 102,e The fundamental theorem 104. 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 3 Basic Notions of the Theory of. Point Sets 113,a Sets and sub sets 113 b Union,and intersection of sets 115 c Ap. plications to sets of points in the,A 4 Homogeneous functions 119. x Contents,Chapter 2 Vectors Matrices Linear,Transformations. 2 1 Operations with Vectors 122,a Definition of vectors 122. b Geometric representation of vectors,124 c Length of vectors Angles. between directions 127 d Scalar,products of vectors 131 e Equa. tion of hyperplanes in vector form,133 f Linear dependence of vec. tors and systems of linear equations,2 2 Matrices and Linear Transforma. 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 153,2 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 djmensional. 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,Contents xi,higher dimensions 190 d Orienta. tion of parallelepipeds in n dimen,sional space 195 e Orientation of. planes and hyperplanes 200,f Change of volume of parallele. pipeds in linear transformations 201,2 5 Vector Notions in Analysis 204. a Vector fields 204 h 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. 3 1 Implicit Functions 218,a General remarks 218 h 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. 3 2 Curves and Surfaces in Implicit,a Plane curves in implicit form. 230 h Singular points of curves,236 c Implicit representation of. surfaces 238,3 3 Systems of Functions Transfor,mations and Mappings 241. a General remarks 241 h Cur,vilinear coordinates 246 c Exten. sion to more than two independent,variables 249 d Differentiation. formulae for the inverse functions,xii Contents,252 e Symbolic product of mappings. 257 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,tion s 266 h Dependent functions. 268 i Concluding remarks 275,3 4 Applications 278,a Elements of the theory of sur. faces 278 b Conformal transfor,mation in general 289. 3 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 303,3 6 Alternating Differential Forms 307. a Definition of alternating dif,ferential forms 307 b Sums and. products of differential forms 310,c Exterior derivatives of differ. ential forms 312 d Exterior,differential forms in arbitrary. coordinates 316,3 7 Maxima and Minima 325,a Necessary conditions 325. b Examples 327 c Maxima and,minima with subsidiary conditions. 330 d Proof of the method of unde,termined multipliers in the simplest. case 334 e Generalization of the,method of undetermined multipliers. 337 f Examples 340,Contents xiii,A l Sufficient Conditions for. Extreme Values 345,A 2 Numbers of Critical Points Re. lated to Indices of a Vector Field 352,A 3 Singular Points of Plane Curves 360. A 4 Singular Points of Surfaces 362,A 5 Connection Between Euler s and. Lagrange s Representation of the,motion of a Fluid 363. A 6 Tangential Representation of a,Closed Curve and the Isoperi. metric Inequality 365,Chapter 4 Multiple Integrals. 4 1 Areas in the Plane 367,a Definition of the Jordan meas. ure of area 367 b A set that does,not have an area 370 c Rules for. operations with areas 372,4 2 Double Integrals 374. a The double integral as a,volume 374 b The general anal. ytic concept of the integral 376,c Examples 379 d Notation. Extensions Fundamental rules 381,e Integral estimates and the mean. value theorem 383,4 3 Integrals over Regions in three. and more Dimensions 385,xiv Contents,4 4 Space Dift erentiation Mass and. Density 386,4 5 Reduction of the Multiple,Integral to Repeated Single. Integrals 388,a Integrals over a rectangle 388,b Change of order of integration. Differentiation under the integral,sign 390 c Reduction of double. in tegrals to single integrals for,more general regions 392 d Ex. tension of the results to regions in,several dimensions 397. 4 6 Transformation of Multiple,Integrals 398,a Transformation of integrals in. the plane 398 b Regions of more,than two dimensions 403. 4 7 Improper Multiple Integrals 406,a Improper integrals of functions. over bounded sets 407 b Proof of,the general convergence theorem. for improper integrals 411,c Integrals over unbounded regions. 4 8 Geometrical Applications 417,a Elementary calculation of. volumes 417 b General remarks,on the calculation of volumes Solids. of revolution Volumes in spherical,coordinates 419 c Area of a curved. surface 421,4 9 Physical Applications 431,a Moments and center of mass. 431 b Moments of inertia 433,c The compound pendulum 436. d Potential of attracting masses 438,Contents xv,4 10 Multiple Integrals in Curvilinear. Coordinates 445,a Resolution of multiple integrals. 445 b Application to areas swept,out by moving curves and volumes. swept out by moving surfaces,Guldin s formula The polar. planimeter 448,4 11 Volumes and Surface Areas in,Any Number of Dimensions 453. a Surface areas and surface in,tegrals in more than three dimen. sions 453 b Area and volume of,the n dimensional sphere 455. c Generalizations Parametric,Representations 459,4 12 Improper Single Integrals as. Functions of a Parameter 462,a Uniform convergence Continu. ous dependence on the parameter,462 b Integration and differentia. tion of improper integrals with re,spect to a parameter 466. c Examples 469 d Evaluation,of Fresnel s integrals 473. 4 13 The Fourier Integral 476,a Introduction 476 b Examples. 479 c Proof of Fourier s integral,theorem 481 d Rate of conver. gence in Fourier s integral theorem,485 e Parseval s identity for. Fourier transforms 488 f The,Fourier transformation for func. tions of several variables 490,4 14 The Eulerian Integrals Gamma. Function 497,a Definition and functional equa,xvi Contents. tion 497 b Convex functions,Proof of Bohr and Mollerup s. theorem 499 c The infinite prod,ucts for the gamma function 503. d The nextensio theorem 507,e The beta function 508. f Differentiation and integration of,fractional order Abel s integral. equation 511,APPENDIX DETAILED ANALYSIS OF,THE PROCESS OF INTEGRATION. A l Area 515,a Subdivisions of the plane and,the corresponding inner and outer. areas 515 b Jordan measurable,sets and their areas 517 c Basic. properties of areas 519,A 2 Integrals of Functions of Several. Variables 524,a Definition of the integral of a,function x y 524 b Integrabili. ty of continuous functions and,integrals over sets 526 c Basic. ruies for multiple integrals 528,d Reduction of multiple integrals. to repeated single integrals 531,A 3 Transformation of Areas and. Integrals 534,a Mappings of sets 534 b Trans,formation of multiple integrals. A 4 Note on the Definition of the,Area of a Curved Surface 540.

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