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                                                          Tom m apostol calculus volume 2 pdf
   Loading ... Loading ... Loading ... Tài liệu hạn chế xem trước, để xem đầy đủ mời bạn chọn Tải xuống Ngày đăng: 12/06/2014, 16:22 [...]... Nonhomogeneous linear systems with constant coefficients 7.17 7.18 7.19 7 .20 7 .21 7 .22 7 .23 7 .24 21 3 21 5 Exercises 21 7 The general linear system Y’(t) = P(t) Y(t) + Q(t) 22 0
   A power-series method for solving homogeneous linear systems 22 1 Exercises 22 2 Proof of the existence theorem by the method of successive approximations The method of successive approximations applied to first-order nonlinear systems... systems 22 7 22 9 Proof of an existence-uniqueness theorem for first-
   order nonlinear systems 23 0 Exercises 23 2 Successive approximations and fixed points of operators 23 3 Normed linear spaces *7 .25 *7 .26 k7 .27 Contraction operators k7 .28 Fixed-point theorem for contraction operators A7 .29 Applications of the fixed-point theorem 23 4 23 5 23 7 PART 2 NONLINEAR ANALYSIS
   8 DIFFERENTIAL CALCULUS OF SCALAR AND VECTOR... equality of mixed partial derivatives Miscellaneous exercises 25 1 25 8 25 9 26 1 26 2 26 3 26 6 26 8 26 9 27 1 27 2 27 3 27 5 27 7 28 1 9 APPLICATIONS OF THE DIFFERENTIAL CALCULUS 9.1 9 .2 Partial differential equations A first-order partial
   differential equation with constant coefficients 9.3 Exercises 9.4 The one-dimensional wave equation 9.5 Exercises 9.6 Derivatives of functions defined implicitly 9.7 Worked examples... 12. 10 12. 11 12. 12 12. 13 12. 14 12. 15 * 12. 16 * 12. 17 12. 18 12. 19 12. 20 12. 21 Exercises The theorem of Stokes The curl and
   divergence of a vector field Exercises Further properties of the curl and divergence Exercises Reconstruction of a vector field from its curl Exercises Extensions of Stokes’ theorem The divergence theorem (Gauss’ theorem:) Applications of the divergence theorem Exercises 417 420 423 ... TOM M. APOSTOL, Emeritus
   Professor at the California Institute of Technology, is the author of several highly regarded texts on calculus, analysis, and number theory, and is Director of Project MATHEMATICS!, a series of computer-animated mathematics videotapes. 1.Tom IN. Apostol CALCULUS VOLUME II Mlul ti Variable Calculus and Linear
   Algebra, with Applications toDifFeren tial Equations and ProbabilitySECOND EDITIONJohn Wiley & SonsNew York London Sydney Toronto2. CONSULTINGEDITORGeorge Springer, Indiana UniversityCOPYRIGHT 0 1969 BY XEROX CORPORATION .All rights reserved. No part of the material covered by this
   copyrightmay 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.1098765432 3. ToJane and Stephen 4. PREFACE This book is a continuation
   of the authors Calculus, Volume I, Second Edition. Thepresent volume has been written with the same underlying philosophy that prevailed in thefirst. Sound training in technique is combined with a strong theoretical development.Every effort has been made to convey the spirit of modern mathematics without
   undueemphasis on formalization. As in Volume I, historical remarks are included to give thestudent a sense of participation in the evolution of ideas. The second volume is divided into three parts, entitled Linear Analysis, NonlinearAna!ysis, and Special Topics. The last two chapters of Volume I have been repeated as
   thefirst two chapters of Volume II so that all the material on linear algebra will be completein one volume. Part 1 contains an introduction to linear algebra, including linear transformations,matrices, determinants, eigenvalues, and quadratic forms. Applications are given toanalysis, in particular to the study of linear
   differential equations. Systems of differentialequations are treated with the help of matrix calculus. Existence and uniqueness theoremsare proved by Picards method of successive approximations, which is also cast in thelanguage of contraction operators. Part 2 discusses the calculus of functions of several variables.
   Differential calculus isunified and simplified with the aid of linear algebra. It includes chain rules for scalar andvector fields, and applications to partial differential equations and extremum problems.Integral calculus includes line integrals, multiple integrals, and surface integrals, withapplications to vector analysis. Here the
   treatment is along more or less classical lines anddoes not include a formal development of differential forms. The special topics treated in Part 3 are Probability and Numerical Analysis. The materialon probability is divided into two chapters, one dealing with finite or countably infinitesample 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- andtwo-dimensional random variables. The last chapter contains an introduction to numerical analysis, the chief emphasisbeing on different kinds of polynomial approximation. Here
   again the ideas are unifiedby the notation and terminology of linear algebra. The book concludes with a treatment ofapproximate integration formulas, such as Simpsons rule, and a discussion of Eulerssummation formula. 5. I11 Preface There is ample material in this volume for a full years course meeting three or four
   timesper week. It presupposes a knowledge of one-variable calculus as covered in most first-yearcalculus courses. The author has taught this material in a course with two lectures and tworecitation periods per week, allowing about ten weeks for each part and omitting thestarred sections. This second volume has been
   planned so that many chapters can be omitted for a varietyof shorter courses. For example, the last chapter of each part can be skipped withoutdisrupting 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
   instructorcan choose topics to suit his needs and preferences by consulting the diagram on the nextpage 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 ofCalifornia, Los Angeles, each of whom suggested a number of improvements. Thanks arealso 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 manyways in which she has contributed. In grateful acknowledgement I happily dedicate thisbook to her. T. M. A.Pasadena, CaliforniaSeptember 16, 1968 6. Logical Interdependence of the Chaptersix 1LINEARSPACES INTRODUCTION TRANSFORMATIONS TO
   NUMERICALAND MATRICES ANALYSISDETERMINANTS H 101113I6 LINEARDIFFERENTIAL IDIFFERENTIAL CALCULUS OF SCALAR AND I ILINE INTEGRALSSET FUNCTIONS AND ELEMENTARYPROBABILITY EQUATIONS 4VECTOR FIELDS EIGENVALUES,IANDI 7 SYSTEMS OF
   EIGENVECTORSDIFFERENTIAL I- EQUATIONS OPERATORS ACTING ON EUCLIDEAN1 P R OBABILITI E S 1 1SPACES9 APPLICATIONS OFDIFFERENTIAL INTEGRALSCALCULUS 7. CONTENTSPART 1. LINEAR ANALYSIS 1. LINEAR SPACES1.1 Introduction31.2The definition of a linear space
   31.3Examples of linear spaces41.4Elementary consequences of the axioms61.5Exercises71.6Subspaces of a linear space81.7Dependent and independent sets in a linear space91.8Bases and dimension121.9Components 131.10 Exercises131.11Inner products, Euclidean spaces. Norms 141.12 Orthogonality in a
   Euclidean space 181.13 Exercises201.14Construction of orthogonal sets. The Gram-Schmidt process 221.15 Orthogonal complements. Projections261.16 Best approximation of elements in a Euclidean space by elements in a finite- dimensional subspace 281.17 Exercises302. LINEAR TRANSFORMATIONS AND
   MATRICES2.1 Linear transformations312.2 Null space and range322.3 Nullity and rank34 xi 8. xiiContents2.4 Exercises352.5 Algebraic operations on linear transformations 362.6 Inverses 382.7 One-to-one linear transformations412.8 Exercises422.9 Linear transformations with prescribed values442.10 Matrix
   representations of linear transformations452.11 Construction of a matrix representation in diagonal form482.12 Exercises 502.13 Linear spaces of matrices 512.14 Tsomorphism between linear transformations and matrices 522.15 Multiplication of matrices542.16 Exercises 572.17 Systems of linear equations 582.18
   Computation techniques612.19 Inverses of square matrices 652.20 Exercises 672.21 Miscellaneous exercises on matrices 683. DETERMINANTS3.1 Introduction 713.2 Motivation for the choice of axioms for a determinant function 723.3 A set of axioms for a determinant function 733.4 Computation of
   determinants763.5The uniqueness theorem793.6Exercises 793.7 The product formula for determinants 813.8 The determinant of the inverse of a nonsingular matrix 833.9Determinants and independence of vectors833.10 The determinant of a block-diagonal matrix843.11 Exercises 853.12 Expansion formulas for
   determinants. Minors and cofactors 863.13 Existence of the determinant function 903.14 The determinant of a transpose913.15 The cofactor matrix 923.16 Cramers rule 933.17 Exercises 94 9. ... Contents XIII4. EIGENVALUES AND EIGENVECTORS 4.1 Linear transformations with diagonal matrix representations 96
   4.2 Eigenvectors and eigenvalues of a linear transformation 97 4.3 Linear independence of eigenvectors corresponding to distinct eigenvalues100 4.4 Exercises *101 4.5 The finite-dimensional case. Characteristic polynomials102 4.6 Calculation of eigenvalues and eigenvectors in the finite-dimensional case 103 4.7
   Trace of a matrix106 4.8 Exercises107 4.9 Matrices representing the same linear transformation. Similar matrices 108 4.10 Exercises 1125. EIGENVALUES OF OPERATORS ACTING ON EUCLIDEAN SPACES5.1 Eigenvalues and inner products1145.2 Hermitian and skew-Hermitian transformations1155.3
   Eigenvalues and eigenvectors of Hermitian and skew-Hermitian operators1175.4 Orthogonality of eigenvectors corresponding to distinct eigenvalues 1175.5 Exercises 1185.6 Existence of an orthonormal set of eigenvectors for Hermitian and skew-Hermitian operators acting on finite-dimensional spaces 1205.7 Matrix
   representations for Hermitian and skew-Hermitian operators 1215.8 Hermitian and skew-Hermitian matrices. The adjoint of a matrix1225.9 Diagonalization of a Hermitian or skew-Hermitian matrix 1225.10 Unitary matrices. Orthogonal matrices1235.11 Exercises1245.12 Quadratic forms1265.13 Reduction of a real
   quadratic form to a diagonal form1285.14 Applications to analytic geometry1305.15 Exercises134A5.16 Eigenvalues of a symmetric transformation obtained as values of its quadratic form 135k5.17 Extremal properties of eigenvalues of a symmetric transformation136k5.18 The finite-dimensional case 1375.19 Unitary
   transformations1385.20 Exercises141 10. xivContents 6. LINEAR DIFFERENTIAL EQUATIONS 6.1 Historical introduction 142 6.2Review of results concerning linear equations of first and second orders 143 6.3 Exercises 144 6.4Linear differential equations of order n 145 6.5 The existence-uniqueness theorem1476.6
   The dimension of the solution space of a homogeneous linear equation 1476.7 The algebra of constant-coefficient operators1486.8 Determination of a basis of solutions for linear equations with constant coefficients by factorization of operators1506.9 Exercises1546.10 The relation between the homogeneous and
   nonhomogeneous equations 1566.11 Determination of a particular solution of the nonhomogeneous equation. The method of variation of parameters 1576.12 Nonsingularity of the Wronskian matrix of n independent solutions of a homogeneous linear equation 1616.13 Special methods for determining a particular
   solution of the nonhomogeneous equation. Reduction to a system of first-order linear equations 1636.14 The annihilator method for determining a particular solution of the nonhomogeneous equation 1636.15 Exercises 1 tom m apostol calculus volume 2 solutions. calculus volume 2 tom m. apostol. tom m apostol
   calculus volume 2 solutions manual
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