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Overview
The three-part presentation begins with an exploration of systems with a finite number of degrees of freedom (described by matrices). In part two, the concepts developed for discrete systems in previous chapters are extended to continuous systems. New concepts useful in the treatment of continuous systems are also introduced. The final part examines approximation methods — including perturbation theory, variational methods, and numerical methods — relevant to addressing most of the problems of nature that confront applied physicists. Two Appendixes include background and supplementary material. 1960 edition.
Product Details
| ISBN-13: | 9780486793320 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 02/08/2014 |
| Series: | Dover Books on Physics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 320 |
| File size: | 24 MB |
| Note: | This product may take a few minutes to download. |
About the Author
American theoretical physicist Gerald Goertzel (1919–2002) worked on the Manhattan Project, formed Sage Industries, and worked for many years at IBM, where he was active in design automation, data compression, and digital printing. He developed the Goertzel algorithm, a data signal processing technique.
Nunzio Tralli (1917–79) taught at St. John's University and was the author of several books on physics, including Classical Electromagnetic Theory and Atomic Theory: An Introduction to Wave Mechanics, which he co-wrote with Frank R. Pomilla.
Table of Contents
Preface v
References xiii
Part 1 Systems With A Finite Number Of Degrees Of Freedom
Chapter 1 Formulation of the Problem and Development of Notation 3
1.1 Introduction 3
1.2 Standardization of Notation 5
1.3 Matrices 7
1.4 Elementary Arithmetic Operations with Matrices 9
1.5 The Row-Column Rule 12
1.6 Warnings 13
1.7 Some Properties of Determinants 14
1.8 Inverses 16
1.9 Linear Independence 17
Chapter 2 Solution for Diagonalizable Matrices 21
2.1 Solution by Taylor Series 21
2.2 Eigenvalues and Eigencolumns 22
2.3 Superposition 23
2.4 Completeness 24
2.5 Diagonalization of Nondegenerate Matrices 27
2.6 Outline of Computation Procedure with Examples 29
2.7 Change of Variable 32
2.8 The Steady-state Solution 34
2.9 The Inhomogeneous Problem 35
Chapter 3 The Evaluation of a Function of a Matrix for an Arbitrary Matrix 38
3.1 Introduction 38
3.2 The Cauchy-integral Formula 38
3.3 Application to Matrices 39
3.4 Evaluation of f(A) with Illustrations 40
3.5 The Inversion Formula 43
3.6 Laplace Transforms 44
3.7 Inhomogeneous Equations 46
3.8 The Convolution Theorem 47
Chapter 4 Vector Spaces and Linear Operators 50
4.1 Introduction 50
4.2 Base Vectors and Basis 52
4.3 Change of Basis 55
4.4 Linear Operators 57
4.5 The Representation of Linear Operators by Matrices 57
4.6 The Operator in the Dual Space 58
14.7 Effect of Change of Basis on the Representation of an Operator 59
4.8 The Spectral Representation of an Operator 60
4.9 The Formation of a Basis by Eigenvectors of a Linear Operator 61
4.10 The Diagonalization of Normal Matrices 64
Chapters 5 The Dirac Notation 67
5.1 Introduction 67
5.2 The Change of Basis 68
5.3 Linear Operators in the Dirac Notation 69
5.4 Eigenvectors and Eigenvalues 70
5.5 The Spectral Representation of an Operator 70
5.6 Theorems on Hermitian Operators 70
Chapter 6 Periodic Structures 73
6.1 Motivation 73
6.2 The RC Line 73
6.3 Diagonalizing M 75
6.4 The Loaded String 78
6.5 Difference Operators 79
Part 2 Systems With An Infinite Number Of Degrees Of Freedom
Chapter 7 The Transition to Continuous Systems 85
7.1 Introduction 85
7.2 The RC Line?Change of Notation 85
7.3 The RC Line?Transition to the Continuous Case 87
7.4 Solution of the Discrete Problem 87
7.5 Solution in the Limit (Continuous Problem) 89
7.6 The Fourier Transform 92
Chapter 8 Operators in Costinuous Systems 93
8.1 Introduction 96
8.2 Operators on Functions 97
8.3 The Dirac δ Function 98
8.4 Coordinate Transformations 100
8.5 Adjoints 103
8.6 Orthogonality of Eigenfunctions 104
8.7 Functions of Operators 105
8.8 Three-dimensional Continuous Systems 107
8.9 Differential Operators 108
Chapter 9 Hie Lapladan (∇2) in One Dimension 111
9.1 Introduction 111
9.2 The Infinite Domain, ? ∞ <x <+ ∞ 112
9.3 The Semi-infinite Domain, 0 ≤ x < + ∞ 116
9.4 The Finite Domain, 0 ≤ x ≤ L 119
9.5 The Circular Domain 122
9.6 The Method of Images 123
Chapter 10 The Lapiacian (∇2) in Two Dimensions 128
10.1 Introduction 128
10.2 Conduction of Heat in an Infinite Insulated PlateCartesian Coordinates 128
10.3 The Vibrating Rectangular Membrane 130
10.4 Conduction of Heat in an Infinite Insulated Plate; Plane Polar Coordinates 131
10.5 Theorems on Cylindrical Functions (of Integral Order n) 134
10.6 Conduction of Heat in an Infinite Insulated Plate; Plane Polar Coordinates (Concluded) 140
10.7 The Circular Membrane 141
10.8 The Vibrating Circular Ring and Circular Sector 143
Chapter 11 The Lapiacian (∇2) in Three Dimensions 146
11.1 Introduction 146
11.2 The Wave Equation in Three Dimensions 147
11.3 The Eigenvalues of L2 andLx 149
11.4 The Simultaneous Eigenfunctions of L and Lz 152
11.5 Solution of ∇2 φ = 0 154
11.6 Solution of (∇2 + k2)φ 155
11.7 Recurrence Relations for the Spherical Harmonics 158
11.8 Some Expansion Theorems 160
11.9 Solution of the Wave Equation 162
11.10 Heat Conduction in an Infinite Solid 163
Chapter 12 Green's Functions 165
12.1 Definition 165
12.2 The Necessary and Sufficient Condition for a Green's Function 166
12.3 The Operator -α2d2/dx2 + 1 in an Infinite Domain 167
12.4 The Operator ?α2d2/dx2 + 1 in a Finite Domain 169
12.5 The Operator ?α2∇2 + 1 in Spherical Coordinates 171
Chapter 13 Radiation and Scattering Problems 176
13.1 The Outgoing Wave Condition 176
13.2 The Green's Function Solution 177
13.3 The Multipole Expansion 179
13.4 The Radiation Far from the Source 180
13.5 Radiation from an Infinitely Long Cylinder 181
13.6 The Scattering Problem 183
13.7 The Scattering Cross Section 184
13.8 The Method of Partial Waves 185
13.9 The Born Approximation 190
13.30 Gratings 192
Part 3 approximate Methods
Chapter 14 Perturbation of Eigenvalues 201
14.1 Introduction 201
14.2 Formulation of the Problem 202
14.3 A Simple Solution 203
14.4 Nondegenerate Eigenvalues 204
14.5 Change of Notation and an Extension 205
14.6 An Application?The Vibrating String 206
14.7 Degenerate Eigenvalues 207
Chapter 15 Variational Estimates 211
15.1 Introduction 211
15.2 The Rayleigh Variational Principle 212
15.3 A Lower Bound 213
15.4 The Ritz Method 215
15.5 Higher Eigenvalues by the Ritz Method 220
15.6 Example of the Ritz Method 221
Chapter 16 Iteration Procedures 225
16.1 Introduction 225
16.2 Eigenvalue Problems 225
16.3 Inverses by Iteration 229
Chapter 17 Construction of Eigenvalue Problems 233
17.1 Introduction 233
17.2 The Method 233
17.3 Application to the Scattering Problem 234
Chapter 18 Numerical Procedures 236
18.1 Introduction 236
18.2 Simplification of the Model 236
18.3 Difference Equations from the Variational Principle 239
Appendix 1A Determinants 243
Appendix IB Convergence of Matrix Power Series 252
Appendix 1C Remarks on Theory of Functions of Complex Variables 254
1C.I Analytic Functions 254
1C.2 The Cauchy Integral Theorem and Corollary 255
1C.3 Singularities 256
1C.4 Cauchy's Integral Formula 257
1C.5 The Theorem of Residues 259
Appendix 2A Evaluation of Integrals of the Form$$$ 261
Appendix 2B Fourier Transforms, Integrals, and Series 266
2B.1 Introduction 266
2B.2 Transforms 267
2B.3 Infinite One-dimensional Transforms 268
2B.4 Infinite Multidimensional Transforms?Cartesian Coordinates 271
2B.5 Finite One-dimensional Transforms 272
2B.6 The Fourier-Bessel Integral 275
2B.7 The Fourier-Bessel Expansion 276
Appendix 2C The Cylindrical Functions 280
2C.1 Introduction 280
2C.2 The Integral Representation of Jn(p) 280
2C.3 The Integral Representations of the General Cylindrical Functions 281
2C.4 The Integral Representation of the Bessel Function Jv 283
2C.5 The Hankel Functions 284
2C.6 Series Expansions at the Origin 286
2C.7 The Asymptotic Expansions 287
2C.8 The Asymptotic Series of Debye 291
2C.9 The Addition Theorems for Bessel Functions 292
Index 295