Table Of Content

Alan Jeffrey University of Newcastle-upon-Tyne San Diego San Francisco New York Boston London Toronto Sydney Tokyo SponsoringEditor BarbaraHolland ProductionEditor JulieBolduc PromotionsManager StephanieStevens CoverDesign MontyLewisDesign TextDesign ThompsonSteeleProductionServices FrontMatterDesign Perspectives Copyeditor KristinLandon Composition TechBooks Printer RRDonnelley&Sons,Inc. Thisbookisprintedonacid-freepaper.(cid:3)∞ Copyright(cid:3)C 2002byHARCOURT/ACADEMICPRESS Allrightsreserved. Nopartofthispublicationmaybereproducedortransmittedinanyformorbyany means,electronicormechanical,includingphotocopy,recording,oranyinformation storageandretrievalsystem,withoutpermissioninwritingfromthepublisher. Requestsforpermissiontomakecopiesofanypartoftheworkshouldbemailedto: PermissionsDepartment,Harcourt,Inc.,6277SeaHarborDrive,Orlando,Florida 32887-6777. Harcourt/AcademicPress AHarcourtScienceandTechnologyCompany 200WheelerRoad,Burlington,Massachusetts01803,USA http://www.harcourt-ap.com AcademicPress AHarcourtScienceandTechnologyCompany 525BStreet,Suite1900,SanDiego,California92101-4495,USA http://www.academicpress.com AcademicPress HarcourtPlace,32JamestownRoad,LondonNW17BY,UK http://www.academicpress.com LibraryofCongressCatalogCardNumber:00-108262 InternationalStandardBookNumber:0-12-382592-X PRINTEDINTHEUNITEDSTATESOFAMERICA 01 02 03 04 05 06 DOC 9 8 7 6 5 4 3 2 1 ToLislandourfamily This Page Intentionally Left Blank C O N T E N T S Preface xv P A R T O N E REVIEW MATERIAL 1 1 CHAPTER Review of Prerequisites 3 1.1 Real Numbers, Mathematical Induction, and Mathematical Conventions 4 1.2 Complex Numbers 10 1.3 The Complex Plane 15 1.4 Modulus and Argument Representation of Complex Numbers 18 1.5 Roots of Complex Numbers 22 1.6 Partial Fractions 27 1.7 Fundamentals of Determinants 31 1.8 Continuity in One or More Variables 35 1.9 Differentiability of Functions of One or More Variables 38 1.10 Tangent Line and Tangent Plane Approximations to Functions 40 1.11 Integrals 41 1.12 Taylor and Maclaurin Theorems 43 1.13 Cylindrical and Spherical Polar Coordinates and Change of Variables in Partial Differentiation 46 1.14 Inverse Functions and the Inverse Function Theorem 49 vii P A R T T W O VECTORS AND MATRICES 53 2 CHAPTER Vectors and Vector Spaces 55 2.1 Vectors, Geometry, and Algebra 56 2.2 The Dot Product (Scalar Product) 70 2.3 The Cross Product (Vector Product) 77 2.4 Linear Dependence and Independence of Vectors and Triple Products 82 2.5 n-Vectors and the Vector Space Rn 88 2.6 Linear Independence, Basis, and Dimension 95 2.7 Gram–Schmidt Orthogonalization Process 101 3 CHAPTER Matrices and Systems of Linear Equations 105 3.1 Matrices 106 3.2 Some Problems That Give Rise to Matrices 120 3.3 Determinants 133 3.4 Elementary Row Operations, Elementary Matrices, and Their Connection with Matrix Multiplication 143 3.5 The Echelon and Row-Reduced Echelon Forms of a Matrix 147 3.6 Row and Column Spaces and Rank 152 3.7 The Solution of Homogeneous Systems of Linear Equations 155 3.8 The Solution of Nonhomogeneous Systems of Linear Equations 158 3.9 The Inverse Matrix 163 3.10 Derivative of a Matrix 171 4 CHAPTER Eigenvalues, Eigenvectors, and Diagonalization 177 4.1 Characteristic Polynomial, Eigenvalues, and Eigenvectors 178 4.2 Diagonalization of Matrices 196 4.3 Special Matrices with Complex Elements 205 4.4 Quadratic Forms 210 4.5 The Matrix Exponential 215 viii P A R T T H R E E ORDINARY DIFFERENTIAL EQUATIONS 225 5 CHAPTER First Order Differential Equations 227 5.1 Background to Ordinary Differential Equations 228 5.2 Some Problems Leading to Ordinary Differential Equations 233 5.3 Direction Fields 240 5.4 Separable Equations 242 5.5 Homogeneous Equations 247 5.6 Exact Equations 250 5.7 Linear First Order Equations 253 5.8 The Bernoulli Equation 259 5.9 The Riccati Equation 262 5.10 Existence and Uniqueness of Solutions 264 6 CHAPTER Second and Higher Order Linear Differential Equations and Systems 269 6.1 Homogeneous Linear Constant Coefficient Second Order Equations 270 6.2 Oscillatory Solutions 280 6.3 Homogeneous Linear Higher Order Constant Coefficient Equations 291 6.4 Undetermined Coefficients: Particular Integrals 302 6.5 Cauchy–Euler Equation 309 6.6 Variation of Parameters and the Green’s Function 311 6.7 Finding a Second Linearly Independent Solution from a Known Solution: The Reduction of Order Method 321 6.8 Reduction to the Standard Form u(cid:4)(cid:4) + f(x)u = 0 324 6.9 Systems of Ordinary Differential Equations: An Introduction 326 6.10 A Matrix Approach to Linear Systems of Differential Equations 333 6.11 Nonhomogeneous Systems 338 6.12 Autonomous Systems of Equations 351 ix