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Fundamentals of mathematical physics
Author
Title
Fundamentals of mathematical physics / Edgar A. Kraut.
ISBN
9781621986119
162198611X
0486458091
9780486458090
Edition
Dover ed.
Published
Mineola, N.Y. : Dover Publications, 2007.
Physical Description
1 online resource (xiii, 466 pages) : illustrations
Local Notes
Access is available to the Yale community.
Notes
Electronic reproduction. [Place of publication not identified] : HathiTrust Digital Library, 2011.
Access and use
Access restricted by licensing agreement.
Variant and related titles
Knovel. OCLC KB.
Other formats
Print version:
Format
Books / Online
Language
English
Added to Catalog
April 01, 2024
Bibliography
Includes bibliographical references and index.
Contents
Machine derived contents note: 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 Introduction 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.
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