Mathematics of physics and engineering

Contents......Page 12
Preface......Page 8
1.1.1 The Postulates of Euclid......Page 16
1.1.2 Relative Position and Position Vectors......Page 18
1.1.3 Euclidean Space as a Linear Space......Page 19
1.2.1 Inner Product......Page 24
1.2.2 Cross Product......Page 32
1.2.3 Scalar Triple Product......Page 38
1.3.1 Vector-Valued Functions of a Scalar Variable......Page 39
1.3.2 The Tangent Vector and Arc Length......Page 42
1.3.3 Frenet's Formulas......Page 45
1.3.4 Velocity and Acceleration......Page 48
2.1.1 Newton's Laws of Motion and Gravitation......Page 54
2.1.2 Parallel Translation of Frames......Page 61
2.1.3 Uniform Rotation of Frames......Page 63
2.1.4 General Accelerating Frames......Page 76
2.2.1 Non-Rigid Systems of Points......Page 81
2.2.2 Rigid Systems of Points......Page 88
2.2.3 Rigid Bodies......Page 94
2.3 The Lagrange-Hamilton Method......Page 99
2.3.1 Lagrange's Equations......Page 100
2.3.2 An Example of Lagrange's Method......Page 105
2.3.3 Hamilton's Equations......Page 108
2.4 Elements of the Theory of Relativity......Page 110
2.4.1 Historical Background......Page 112
2.4.2 The Lorentz Transformation and Special Relativity......Page 114
2.4.3 Einstein's Field Equations and General Relativity......Page 120
3.1.1 Functions Sets and the Gradient......Page 136
3.1.2 Integration and Differentiation......Page 145
3.1.3 Curvilinear Coordinate Systems......Page 156
3.2.1 Green's Theorem......Page 165
3.2.2 The Divergence Theorem of Gauss......Page 166
3.2.3 Stokes's Theorem......Page 170
3.2.4 Laplace's and Poisson's Equations......Page 172
3.3.1 Maxwell's Equations in Vacuum......Page 178
3.3.2 The Electric and Magnetic Dipoles......Page 185
3.3.3 Maxwell's Equations in Material Media......Page 188
4.1.1 Basic Definitions......Page 194
4.1.2 The Complex Plane......Page 198
4.1.3 Applications to Analysis of AC Circuits......Page 202
4.2.1 Continuity and Differentiability......Page 205
4.2.2 Cauchy-Riemann Equations......Page 206
4.2.3 The Integral Theorem and Formula of Cauchy......Page 209
4.2.4 Conformal Mappings......Page 217
4.3.1 Series of Complex Numbers......Page 221
4.3.2 Convergence of Power Series......Page 223
4.3.3 The Exponential Function......Page 228
4.4.1 Laurent Series......Page 230
4.4.2 Residue Integration......Page 237
4.4.3 Power Series and Ordinary Differential Equations......Page 246
5.1 Fourier Series......Page 256
5.1.1 Fourier Coefficients......Page 257
5.1.2 Point-wise and Uniform Convergence......Page 262
5.1.3 Computing the Fourier Series......Page 269
5.2.1 From Sums to Integrals......Page 276
5.2.2 Properties of the Fourier Transform......Page 282
5.2.3 Computing the Fourier Transform......Page 286
5.3.1 Discrete Functions......Page 289
5.3.2 Fast Fourier Transform (FFT)......Page 293
5.4.1 Definition and Properties......Page 296
5.4.2 Applications to System Theory......Page 300
6.1.1 Transport Equation......Page 306
6.1.2 Heat Equation......Page 309
6.1.3 Wave Equation in One Dimension......Page 322
6.2.1 Classification of Equations and Characteristics......Page 331
6.2.2 Variation of Parameters......Page 336
6.2.3 Separation of Variables......Page 340
6.3.1 Telegraph Equation......Page 348
6.3.2 Helmholtz's Equation......Page 353
6.3.3 Wave Equation in Two and Three Dimensions......Page 358
6.3.4 Maxwell's Equations......Page 362
6.3.5 Equations of Fluid Mechanics......Page 368
6.4.1 Schrodinger's Equation......Page 371
6.4.2 Dirac's Equation of Relativistic Quantum Mechanics......Page 388
6.4.3 Introduction to Quantum Computing......Page 394
6.5 Numerical Solution of Partial Differential Equations......Page 405
6.5.1 General Concepts in Numerical Methods......Page 406
6.5.2 One-Dimensional Heat Equation......Page 410
6.5.3 One-Dimensional Wave Equation......Page 413
6.5.4 The Poisson Equation in a Rectangle......Page 416
6.5.5 The Finite Element Method......Page 418
7.1 Geometry and Vectors......Page 424
7.2 Kinematics and Dynamics......Page 428
7.3 Special Relativity......Page 437
7.4 Vector Calculus......Page 441
7.5 Complex Analysis......Page 445
7.6 Fourier Analysis......Page 451
7.7 Partial Differential Equations......Page 455
8.1 Linear Algebra and Matrices......Page 466
8.3 Tensors......Page 470
8.4 Lumped Electric Circuits......Page 478
8.5 Physical Units and Constants......Page 480
Bibliography......Page 482
List of Notations......Page 486
Index......Page 488