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chapter one vector algebra 1 Introduction I I-1 Definitions 3 1-2 Equality of Vectors and Null Vectors 4 1-3 Vector Operations 5 1-4 Expansion of Vectors 9 1-5 Vector Identities 14 -86 Problems and Applications 15 chapter two matrix and tensor algebra 17 2-1 Definitions 17 2-2 Equality of Matrices and Null Matrices 18 2-3 Matrix Operations 19 2-4 Determinants 23 2-5 Special Matrices 25 2-6 Systems of Linear Equations 30 2-7 Linear Operators 33 2-8 Eigenvalue Problems 37 2-9 Diagonalization of Matrices 40 2-10 Special Properties of Hermitian Matrices 46 2-11 Tensor Algebra 47 2-12 Tensor Operations 48 2-13 Transformation Properties of Tensors 50 2-14 Special Tensors 53 2-15 Problems and Applications 55 chapter three vector calculus 59 3-1 Ordinary Vector Differentiation 59 3-2 Partial Vector Differentiation 64 3-3 Vector Operations in Cylindrical and Spherical Coordinate Systems 68 3-4 Differential Vector Identities 74 3-5 Vector Integration over a Closed Surface 76 3-6 The Divergence Theorem 80 3-7 The Gradient Theorem 82 3-8 The Curl Theorem 82 3-9 Vector Integration over a Closed Curve 83 3-_0 The Two-dimensional Divergence Theorem 87 3-11 The Two-dimensional Gradient Theorem 87 3-12 The Two-dimensional Curl Theorem 88 3-13 Mnemonic Operators 92 3-14 Kinematics of Infinitesimal Volume, Surface, and Line Elements 93 3-15 Kinematics of a Volume Integral 96 3-1.6 Kinematics of a Surface Integral 97 3-17 Kinematics of a Line Integral 99 3-18 Solid Angle 100 3-19 Decomposition of a Vector Field into Solenoidal and Irrotational Parts 102 3-20 Integral Theorems for Discontinuous and Unbounded Functions 103 3-21 Problems and Applications 115 chapter four functions of a complex variable 127 4-1 Introduction 127 4-2 Definitions 127 4-3 Complex Algebra 129 4-4 Domain of Convergence 130 4-5 IAnalytic Functions 131 4-6 Cauchy's Approach 133 4-7 Cauchy's Integral Theorem 134 4-8 Cauchy's integral Representation of an Analytic Function 136 4-9 Taylor's Series 139 4-10 Cauchy's Inequalities 140 4-11 Entire Functions 140 4-12 Riemann's Theory of Functions of a Complex Variable 141 4-13 Physical Interpretation 142 4-14 Functions Defined on Curved Surfaces 145 4-15 Laurent's Series 152 4-16 Singularities of an Analytic Function 154 4-17 Multivalued Functions 155 4-18 Residues 158 4-19 Residue at Infinity 161 4-20 Generalized Residue Theorem of Cauchy 162 4-21 Problems and Applications 167 chapter five integral transforms 173 5-1 Introdution 173 5-2 Orthogonal Functions 174 5-3 Dirac's Notation 175 5-4 Analogy between Expansion in Orthogonal Functions and Expansion in Orthogonal Vectors 177 5-5 Linear Independence of Functions 179 5-6 Mean-square Convergence of an Expansion in Orthogonal Functions 180 5-7 In tgration and Differentiation of Orthogonal Expansions 185 5-8 Pointwise Convergence of an Orthogonal Expansion 185 5-9 Gibbs's bhenorenon 186 5-10 The inite Sine Transform 187 5411 The Finite Cosine Transform 190 5-12 Properties of Finite Fourier Transforms 191 5-13 Connection with Classical Theory of Fourier Series 192 5-14 Applications of Finite Fourier Transforms 194 5i-5 Infinite-range Fourier Transforms 206 5-16 Condiions for the Applicability of the Fourier Transformation 210 5-17 Fourier Sin and Cosine Transforms 211 5-18 Fourier Transforms in n Dimensions 213 5-19 Properties of Fourier Transforms 214 5-20 Physical Interpretation of the Fourier Transform 216 5-21 Applications of the Infinite-range Fourier Transform 218 5-22 The L,avlace Transform 223 5-23 Properties of Laplace Transforms 226 5-24 Application of the Laplace Transform 228 5-25 Problems and Applications 232 chapter six linear differential equations 239 6-1 Introduction 239 6-2 Linear Differential Equations with Constant Coefficients 240 6-3 The Theory of the Seismograph 246 6-4 Linear Differential Equations with Variable Coeffcients 252 6-5 The Special Functions of Mathematical Physics 255 6-6 The Gamma Function 256 6-7 The Beta Function 259 6-8 The Bessel Functions 261 6-9 The Neumann Functions 264 6 -0 Bessel Funetions of Arbitrary Order 267 6-11 The Hankel Functions 269 "6-12 The Hyperbolic Bessel Functions 270 6-13 The Associated Legendre Functions 272 6-14 Representation of Associated Legendre Functions in Terms of Legendre Polynomials 275 6-15 Spherical Harmonies 276 6-16 Spherical Bessel Functions 279 6-17 Hermite Polynomials 281 6-18 General Properties of Linear Second-order Differential Equations with Variable Coefficients 287 6-19 Evaluation of the Wronskian 291 6-20 General Solution of a Homogeneous Equation Using Abels Formula 292 6-21 Solution of an Inhomogeneous Equation Using Abel's Formula 293 6-22 Green's Function 295 6-23 Use of the Green's Function g(xjx') 296 6-24 The Sturm-Liouville Problem 299 6-25 Solution of Ordinary Differential Equations with Variable Coefficients by Transform Methods 303 6-26 Problems and Applications 306 chapter seven partial differential equations 317 7-1 Introduction 317 7-2 The Role of the Laplacian 317 7-3 Laplace's Equation 318 7-4 Poisson's Equation 318 7-5 The Diffusion Equation 319 7-6 The Wave Equation 321 7-7 A Few General Remarks 322 7-8 Solution of Potential Problems in Two Dimensions 323 7-9 Separation of Variables 333 7-10 The Solution of Laplace's Equation in a Half Space 338 7-11 Laplace's Equation in Polar Coordinates 343 7-12 Construction of a Green's Function in Polar Coordinates 344 7-13 The Exterior Dirichlet Problem for a Circle 352 7-14 Laplace's Equation in Cylindrical Coordinates 354 7-15 Construction of the Green's Function 356 7-16 An Alternative Method of Solving Boundary-value Problems 360 -17 Laplace's Equation in Spherical Coordinates 363 7-18 Construction of the Green's Function 365 7-19 Solution of the Interior and Exterior Dirichlet Problems for a Grounded Conducting Sphere 368 "7-20 The One-dimensional Wave Equation 371 7-21 The Two-dimensional Wave Equation 377 7-22 The Helmholtz Equation in Cylindrical Coordinates 382 7-23 The Helmholtz Equation in Rectangular Cartesian Coordinates 392. 7-24 The Helmholtz Equation in Spherical Coordinates 400 7-25 Interpretation of the Integral Solution of Helmholtz's Equation 403 7-26 The Sommerfeld Radiation Condition 405 7-27 Time-dependent Problems 409 7-28 Poisson's Solution of the Wave Equation 413 7-29 The Diffusion Equation 420 7-30 General Solution of the Diffusion Equation 422 7-31 Construction of the Infinite-medium Green's Function for the Diffusion Equation 423 7-32 Problems and Applications 427

Library of Congress subject headings for this publication: Mathematical physics