Mathematical Methods For Engineers And Scientists 2 Vector Analysis Ordinary Differential Equations And Laplace Transforms

Copyright......Page 4
Preface......Page 6
Table of Contents\r......Page 8
Part I: Vector Analysis\r......Page 14
1: Vectors\r......Page 16
1.2 Vector Operations......Page 17
1.2.3 Addition and Subtraction......Page 18
1.2.4 Dot Product......Page 19
1.2.5 Vector Components......Page 23
1.2.6 Cross Product......Page 26
1.2.7 Triple Products......Page 30
1.3.1 Straight Lines......Page 36
1.3.2 Planes in Space......Page 40
2: Vector Calculus\r......Page 48
2.1.1 Velocity and Acceleration......Page 49
2.1.2 Angular Velocity Vector......Page 50
2.2 Differentiation in Noninertial Reference Systems......Page 55
2.3 Theory of Space Curve......Page 60
2.4.1 The Gradient of a Scalar Function......Page 64
2.4.2 Geometrical Interpretation of Gradient......Page 66
2.4.3 Line Integral of a Gradient Vector......Page 69
2.5 The Divergence of a Vector......Page 74
2.5.1 The Flux of a Vector Field......Page 75
2.5.2 Divergence Theorem......Page 78
2.5.3 Continuity Equation......Page 82
2.6 The Curl of a Vector......Page 83
2.6.1 Stokes’ Theorem......Page 84
2.7 Further Vector Differential Operations......Page 91
2.7.1 Product Rules......Page 92
2.7.2 Second Derivatives......Page 94
2.8.1 Green’s Theorem......Page 98
2.8.2 Other Related Integrals......Page 99
2.9.1 Irrotational Field and Scalar Potential......Page 102
2.9.2 Solenoidal Field and Vector Potential......Page 105
2.10.1 Functions of Relative Coordinates......Page 108
2.10.2 Divergence of R/ |R|2 as a Delta Function\r......Page 111
2.10.3 Helmholtz’s Theorem......Page 114
2.10.4 Poisson’s and Laplace’s Equations......Page 117
2.10.5 Uniqueness Theorem......Page 118
3.1 Cylindrical Coordinates......Page 126
3.1.1 Differential Operations......Page 129
3.1.2 Infinitesimal Elements......Page 133
3.2 Spherical Coordinates......Page 135
3.2.1 Differential Operations......Page 138
3.2.2 Infinitesimal Elements......Page 141
3.3.1 Coordinate Surfaces and Coordinate Curves......Page 143
3.3.2 Differential Operations in Curvilinear Coordinate Systems......Page 146
3.4 Elliptical Coordinates......Page 151
3.4.1 Coordinate Surfaces......Page 152
3.4.2 Relations with Rectangular Coordinates......Page 154
3.5 Multiple Integrals......Page 157
3.5.1 Jacobian for Double Integral......Page 158
3.5.2 Jacobians for Multiple Integrals......Page 160
4: Vector Transformation and Cartesian Tensors\r......Page 168
4.1.1 Transformation of Position Vector......Page 169
4.1.2 Vector Equations......Page 171
4.1.3 Euler Angles......Page 172
4.1.4 Properties of Rotation Matrices......Page 175
4.1.5 Definition of a Scalar and a Vector in Terms of Transformation Properties\r......Page 178
4.2.1 Definition......Page 182
4.2.2 Kronecker and Levi-Civita Tensors......Page 184
4.2.3 Outer Product......Page 187
4.2.4 Contraction......Page 189
4.2.5 Summation Convention......Page 190
4.2.6 Tensor Fields......Page 192
4.2.7 Quotient Rule......Page 195
4.2.8 Symmetry Properties of Tensors......Page 196
4.2.9 Pseudotensors......Page 198
4.3.1 Moment of Inertia Tensor......Page 202
4.3.2 Stress Tensor......Page 203
4.3.3 Strain Tensor and Hooke’s Law......Page 206
Part II: Differential Equations and Laplace Transforms\r......Page 212
5.1 First-Order Differential Equations......Page 214
5.1.1 Equations with Separable Variables......Page 215
5.1.2 Equations Reducible to Separable Type......Page 217
5.1.3 Exact Differential Equations......Page 218
5.1.4 Integrating Factors......Page 220
5.2 First-Order Linear Differential Equations......Page 223
5.2.1 Bernoulli Equation......Page 226
5.3 Linear Differential Equations of Higher Order......Page 227
5.4 Homogeneous Linear Differential Equations with Constant Coefficients\r......Page 229
5.4.1 Characteristic Equation with Distinct Roots......Page 230
5.4.3 Characteristic Equation with Complex Roots......Page 231
5.5.1 Method of Undetermined Coefficients......Page 235
5.5.2 Use of Complex Exponentials......Page 242
5.5.3 Euler–Cauchy Differential Equations......Page 243
5.5.4 Variation of Parameters......Page 245
5.6 Mechanical Vibrations......Page 248
5.6.1 Free Vibration......Page 249
5.6.2 Free Vibration with Viscous Damping......Page 251
5.6.3 Free Vibration with Coulomb Damping......Page 254
5.6.4 Forced Vibration without Damping......Page 257
5.6.5 Forced Vibration with Viscous Damping......Page 260
5.7 Electric Circuits......Page 262
5.7.1 Analog Computation......Page 263
5.7.2 Complex Solution and Impedance......Page 265
5.8.1 The Reduction of a System to a Single Equation......Page 267
5.8.2 Cramer’s Rule for Simultaneous Differential Equations......Page 268
5.8.3 Simultaneous Equations as an Eigenvalue Problem......Page 270
5.8.4 Transformation of an nth Order Equation into a System of n First-Order Equations\r......Page 272
5.8.5 Coupled Oscillators and Normal Modes......Page 274
5.9 Other Methods and Resources for Differential Equations\r......Page 277
6.1.1 Laplace Transform – A Linear Operator......Page 284
6.1.2 Laplace Transforms of Derivatives......Page 287
6.1.3 Substitution: s-Shifting......Page 288
6.1.5 A Short Table of Laplace Transforms......Page 289
6.2.1 Inverse Laplace Transform......Page 291
6.2.2 Solving Differential Equations......Page 301
6.3.1 The Dirac Delta Function......Page 304
6.3.2 The Heaviside Unit Step Function......Page 307
6.4 Differential Equations with Discontinuous Forcing Functions\r......Page 310
6.5.1 The Duhamel Integral......Page 315
6.5.2 The Convolution Theorem......Page 317
6.6.2 Integration of Transforms......Page 320
6.6.3 Scaling......Page 321
6.6.4 Laplace Transforms of Periodic Functions......Page 322
6.6.5 Inverse Laplace Transforms Involving Periodic Functions......Page 324
6.6.6 Laplace Transforms and Gamma Functions......Page 325
6.7 Summary of Operations of Laplace Transforms......Page 326
6.8.1 Evaluating Integrals......Page 329
6.8.2 Differential Equation with Variable Coefficients......Page 332
6.8.3 Integral and Integrodifferential Equations......Page 334
6.9 Inversion by Contour Integration......Page 336
6.10 Computer Algebraic Systems for Laplace Transforms\r......Page 339
References......Page 346
Index......Page 348