Table of contents for Fundamentals of mathematical physics / Edgar A. Kraut.


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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