,CONTENTS
1 A Review of Vector and Matrix Algebra Using
SubscriptlSummationConventions 1
1.1 Notation, I
1.2 Vector Operations, 5
2 Differential and Integral Operations on Vector and Scalar Fields 18
2.1 Plotting Scalar and Vector Fields, 18
2.2 Integral Operators, 20
2.3 Differential Operations, 23
2.4 Integral Definitions of the Differential Operators, 34
2.5 TheTheorems, 35
3 Curvilinear Coordinate Systems 44
3.1 The Position Vector, 44
3.2 The Cylindrical System, 45
3.3 The Spherical System, 48
3.4 General Curvilinear Systems, 49
3.5 The Gradient, Divergence, and Curl in Cylindrical and Spherical
Systems, 58
,viii CONTENTS
4 Introduction to Tensors 67
4.1 The Conductivity Tensor and Ohm’s Law, 67
4.2 General Tensor Notation and Terminology, 71
4.3 TransformationsBetween Coordinate Systems, 7 1
4.4 Tensor Diagonalization, 78
4.5 Tensor Transformationsin Curvilinear Coordinate Systems, 84
4.6 Pseudo-Objects, 86
5 The Dirac &Function 100
5.1 Examples of Singular Functions in Physics, 100
5.2 Two Definitions of &t), 103
5.3 6-Functions with Complicated Arguments, 108
5.4 Integrals and Derivatives of 6(t), 111
5.5 Singular Density Functions, 114
5.6 The Infinitesimal Electric Dipole, 121
5.7 Riemann Integration and the Dirac &Function, 125
6 Introduction to Complex Variables 135
6.1 A Complex Number Refresher, 135
6.2 Functions of a Complex Variable, 138
6.3 Derivatives of Complex Functions, 140
6.4 The Cauchy Integral Theorem, 144
6.5 Contour Deformation, 146
6.6 The Cauchy Integrd Formula, 147
6.7 Taylor and Laurent Series, 150
6.8 The Complex Taylor Series, 153
6.9 The Complex Laurent Series, 159
6.10 The Residue Theorem, 171
6.1 1 Definite Integrals and Closure, 175
6.12 Conformal Mapping, 189
, CONTENTS ix
7 Fourier Series 219
7.1 The Sine-Cosine Series, 219
7.2 The Exponential Form of Fourier Series, 227
7.3 Convergence of Fourier Series, 231
7.4 The Discrete Fourier Series, 234
8 Fourier Transforms 250
8.1 Fourier Series as To -+ m, 250
8.2 Orthogonality, 253
8.3 Existence of the Fourier Transform, 254
8.4 The Fourier Transform Circuit, 256
8.5 Properties of the Fourier Transform, 258
8.6 Fourier Transforms-Examples, 267
8.7 The Sampling Theorem, 290
9 Laplace Transforms 303
9.1 Limits of the Fourier Transform, 303
9.2 The Modified Fourier Transform, 306
9.3 The Laplace Transform, 313
9.4 Laplace Transform Examples, 314
9.5 Properties of the Laplace Transform, 318
9.6 The Laplace Transform Circuit, 327
9.7 Double-Sided or Bilateral Laplace Transforms, 331
10 Differential Equations 339
10.1 Terminology, 339
10.2 Solutions for First-Order Equations, 342
10.3 Techniques for Second-Order Equations, 347
10.4 The Method of Frobenius, 354
10.5 The Method of Quadrature, 358
10.6 Fourier and Laplace Transform Solutions, 366
10.7 Green’s Function Solutions, 376
1 A Review of Vector and Matrix Algebra Using
SubscriptlSummationConventions 1
1.1 Notation, I
1.2 Vector Operations, 5
2 Differential and Integral Operations on Vector and Scalar Fields 18
2.1 Plotting Scalar and Vector Fields, 18
2.2 Integral Operators, 20
2.3 Differential Operations, 23
2.4 Integral Definitions of the Differential Operators, 34
2.5 TheTheorems, 35
3 Curvilinear Coordinate Systems 44
3.1 The Position Vector, 44
3.2 The Cylindrical System, 45
3.3 The Spherical System, 48
3.4 General Curvilinear Systems, 49
3.5 The Gradient, Divergence, and Curl in Cylindrical and Spherical
Systems, 58
,viii CONTENTS
4 Introduction to Tensors 67
4.1 The Conductivity Tensor and Ohm’s Law, 67
4.2 General Tensor Notation and Terminology, 71
4.3 TransformationsBetween Coordinate Systems, 7 1
4.4 Tensor Diagonalization, 78
4.5 Tensor Transformationsin Curvilinear Coordinate Systems, 84
4.6 Pseudo-Objects, 86
5 The Dirac &Function 100
5.1 Examples of Singular Functions in Physics, 100
5.2 Two Definitions of &t), 103
5.3 6-Functions with Complicated Arguments, 108
5.4 Integrals and Derivatives of 6(t), 111
5.5 Singular Density Functions, 114
5.6 The Infinitesimal Electric Dipole, 121
5.7 Riemann Integration and the Dirac &Function, 125
6 Introduction to Complex Variables 135
6.1 A Complex Number Refresher, 135
6.2 Functions of a Complex Variable, 138
6.3 Derivatives of Complex Functions, 140
6.4 The Cauchy Integral Theorem, 144
6.5 Contour Deformation, 146
6.6 The Cauchy Integrd Formula, 147
6.7 Taylor and Laurent Series, 150
6.8 The Complex Taylor Series, 153
6.9 The Complex Laurent Series, 159
6.10 The Residue Theorem, 171
6.1 1 Definite Integrals and Closure, 175
6.12 Conformal Mapping, 189
, CONTENTS ix
7 Fourier Series 219
7.1 The Sine-Cosine Series, 219
7.2 The Exponential Form of Fourier Series, 227
7.3 Convergence of Fourier Series, 231
7.4 The Discrete Fourier Series, 234
8 Fourier Transforms 250
8.1 Fourier Series as To -+ m, 250
8.2 Orthogonality, 253
8.3 Existence of the Fourier Transform, 254
8.4 The Fourier Transform Circuit, 256
8.5 Properties of the Fourier Transform, 258
8.6 Fourier Transforms-Examples, 267
8.7 The Sampling Theorem, 290
9 Laplace Transforms 303
9.1 Limits of the Fourier Transform, 303
9.2 The Modified Fourier Transform, 306
9.3 The Laplace Transform, 313
9.4 Laplace Transform Examples, 314
9.5 Properties of the Laplace Transform, 318
9.6 The Laplace Transform Circuit, 327
9.7 Double-Sided or Bilateral Laplace Transforms, 331
10 Differential Equations 339
10.1 Terminology, 339
10.2 Solutions for First-Order Equations, 342
10.3 Techniques for Second-Order Equations, 347
10.4 The Method of Frobenius, 354
10.5 The Method of Quadrature, 358
10.6 Fourier and Laplace Transform Solutions, 366
10.7 Green’s Function Solutions, 376
