Mathematical Methods in Engineering

Description

This text focuses on a variety of topics in mathematics in common usage in graduate engineering programs including vector calculus, linear and nonlinear ordinary differential equations, approximation methods, vector spaces, linear algebra, integral equations and dynamical systems. The book is designed for engineering graduate students who wonder how much of their basic mathematics will be of use in practice. Following development of the underlying analysis, the book takes students through a large number of examples that have been worked in detail. Students can choose to go through each step or to skip ahead if they so desire. After seeing all the intermediate steps, they will be in a better position to know what is expected of them when solving assignments, examination problems, and when on the job. Chapters conclude with exercises for the student that reinforce the chapter content and help connect the subject matter to a variety of engineering problems. Students have grown up with computer-based tools including numerical calculations and computer graphics; the worked-out examples as well as the end-of-chapter exercises often use computers for numerical and symbolic computations and for graphical display of the results. Contents......Page 8 Preface......Page 14 1.1.1 One Independent Variable......Page 18 1.1.2 Many Independent Variables......Page 22 1.1.3 Many Dependent Variables......Page 23 1.2 Inverse Function Theorem......Page 28 1.3 Functional Dependence......Page 30 1.4 Leibniz Rule......Page 35 1.5 Optimization......Page 37 1.5.1 Unconstrained Optimization......Page 38 1.5.2 Calculus of Variations......Page 39 1.5.3 Constrained Optimization: Lagrange Multipliers......Page 45 1.6 Non-Cartesian Coordinate Transformations......Page 48 1.6.1 Jacobian Matrices and Metric Tensors......Page 51 1.6.2 Covariance and Contravariance......Page 60 1.6.3 Differentiation and Christoffel Symbols......Page 66 1.6.4 Summary of Identities......Page 70 1.6.5 Nonorthogonal Coordinates: Alternate Approach......Page 71 1.6.6 Orthogonal Curvilinear Coordinates......Page 74 Exercises......Page 76 2.1.1 Cartesian Index Notation......Page 81 2.1.2 Direction Cosines......Page 84 2.1.4 Vectors......Page 89 2.1.5 Tensors......Page 90 2.2.1 Definitions and Properties......Page 98 2.2.3 Cross Product......Page 99 2.2.5 Identities......Page 100 2.3.2 Differential Geometry of Curves......Page 101 2.4 Line Integrals......Page 109 2.6 Differential Operators......Page 111 2.6.1 Gradient......Page 112 2.6.3 Curl......Page 115 2.6.5 Identities......Page 116 2.7.1 Trajectory......Page 117 2.7.3 Gaussian......Page 120 2.8.1 Green’s Theorem......Page 121 2.8.2 Divergence Theorem......Page 122 2.8.4 Stokes’ Theorem......Page 125 Exercises......Page 127 3.1 Paradigm Problem......Page 132 3.2 Separation of Variables......Page 134 3.3 Homogeneous Equations......Page 135 3.4 Exact Equations......Page 137 3.5 Integrating Factors......Page 139 3.6 General Linear Solution......Page 140 3.7 Bernoulli Equation......Page 142 3.8 Riccati Equation......Page 143 3.9.2 Independent Variable x Absent......Page 146 3.10 Factorable Equations......Page 148 3.11 Uniqueness and Singular Solutions......Page 149 3.12 Clairaut Equation......Page 151 3.13 Picard Iteration......Page 153 3.14 Solution by Taylor Series......Page 156 3.15 Delay Differential Equations......Page 157 Exercises......Page 158 4.1 Linearity and Linear Independence......Page 163 4.2.1 Constant Coefficients......Page 166 4.2.2 Variable Coefficients......Page 171 4.3.1 Undetermined Coefficients......Page 173 4.3.2 Variation of Parameters......Page 175 4.3.3 Green’s Functions......Page 177 4.3.4 Operator D......Page 183 4.4 Sturm-Liouville Analysis......Page 186 4.4.1 General Formulation......Page 187 4.4.2 Adjoint of Differential Operators......Page 188 4.4.3 Linear Oscillator......Page 192 4.4.4 Legendre Differential Equation......Page 196 4.4.5 Chebyshev Equation......Page 199 4.4.6 Hermite Equation......Page 202 4.4.7 Laguerre Equation......Page 205 4.4.8 Bessel Differential Equation......Page 206 4.5 Fourier Series Representation......Page 210 4.6 Fredholm Alternative......Page 217 4.7 Discrete and Continuous Spectra......Page 218 4.8 Resonance......Page 219 4.9 Linear Difference Equations......Page 224 Exercises......Page 228 5 Approximation Methods......Page 236 5.1.1 Taylor Series......Page 237 5.1.2 Pade´ Approximants......Page 239 5.2.1 Functional Equations......Page 241 5.2.2 First-Order Differential Equations......Page 243 5.2.3 Second-Order Differential Equations......Page 247 5.2.4 Higher-Order Differential Equations......Page 254 5.3 Taylor Series Solution......Page 255 5.4.1 Polynomial and Transcendental Equations......Page 257 5.4.2 Regular Perturbations......Page 261 5.4.3 Strained Coordinates......Page 264 5.4.4 Multiple Scales......Page 270 5.4.5 Boundary Layers......Page 273 5.4.6 Interior Layers......Page 278 5.4.7 WKBJ Method......Page 280 5.4.8 Solutions of the Type eS(x)......Page 283 5.4.9 Repeated Substitution......Page 284 5.5 Asymptotic Methods for Integrals......Page 285 Exercises......Page 288 6.1 Sets......Page 296 6.2 Integration......Page 297 6.3 Vector Spaces......Page 300 6.3.1 Normed......Page 305 6.3.2 Inner Product......Page 314 6.4 Gram-Schmidt Procedure......Page 321 6.5.1 Nonorthogonal......Page 324 6.5.2 Orthogonal......Page 330 6.5.3 Orthonormal......Page 331 6.5.4 Reciprocal......Page 341 6.7 Operators......Page 347 6.7.1 Linear......Page 349 6.7.2 Adjoint......Page 351 6.7.3 Inverse......Page 354 6.8 Eigenvalues and Eigenvectors......Page 356 6.9 Rayleigh Quotient......Page 367 6.10 Linear Equations......Page 371 6.11 Method of Weighted Residuals......Page 376 6.12 Ritz and Rayleigh-Ritz Methods......Page 388 6.13 Uncertainty Quantification Via Polynomial Chaos......Page 390 Exercises......Page 396 7.1 Paradigm Problem......Page 407 7.2.1 Determinant and Rank......Page 408 7.2.3 Column, Row, and Left and Right Null Spaces......Page 409 7.2.4 Matrix Multiplication......Page 411 7.2.5 Definitions and Properties......Page 413 7.3 Systems of Equations......Page 416 7.3.1 Overconstrained......Page 417 7.3.2 Underconstrained......Page 420 7.3.3 Simultaneously Over- and Underconstrained......Page 422 7.3.4 Square......Page 423 7.3.5 Fredholm Alternative......Page 425 7.4.1 Ordinary......Page 427 7.4.2 Generalized in the Second Sense......Page 431 7.5 Matrices as Linear Mappings......Page 432 7.6 Complex Matrices......Page 433 7.7 Orthogonal and Unitary Matrices......Page 436 7.7.1 Givens Rotation......Page 439 7.7.2 Householder Reflection......Page 440 7.8 Discrete Fourier Transforms......Page 443 7.9.1 L · D · U......Page 449 7.9.2 Cholesky......Page 451 7.9.3 Row Echelon Form......Page 452 7.9.4 Q · U......Page 456 7.9.5 Diagonalization......Page 458 7.9.6 Jordan Canonical Form......Page 464 7.9.7 Schur......Page 466 7.9.8 Singular Value......Page 467 7.9.9 Polar......Page 470 7.10 Projection Matrix......Page 473 7.11 Least Squares......Page 475 7.11.1 Unweighted......Page 476 7.11.2 Weighted......Page 477 7.12 Neumann Series......Page 478 7.13 Matrix Exponential......Page 479 7.14 Quadratic Form......Page 481 7.15 Moore-Penrose Pseudoinverse......Page 484 Exercises......Page 487 8.1 Definitions......Page 497 8.2.1 First Kind......Page 498 8.2.2 Second Kind......Page 499 8.3.1 First Kind......Page 504 8.3.2 Second Kind......Page 506 8.5 Fourier Series Projection......Page 507 Exercises......Page 512 9.1 Iterated Maps......Page 514 9.2.1 Cantor Set......Page 518 9.2.3 Menger Sponge......Page 519 9.3 Introduction to Differential Systems......Page 520 9.3.1 Autonomous Example......Page 521 9.3.2 Nonautonomous Example......Page 525 9.3.3 General Approach......Page 527 9.4 High-Order Scalar Differential Equations......Page 529 9.5.1 Inhomogeneous with Variable Coefficients......Page 531 9.5.2 Homogeneous with Constant Coefficients......Page 532 9.5.3 Inhomogeneous with Constant Coefficients......Page 542 9.6 Nonlinear Systems......Page 545 9.6.1 Definitions......Page 546 9.6.2 Linear Stability......Page 549 9.6.3 Heteroclinic and Homoclinic Trajectories......Page 550 9.6.4 Nonlinear Forced Mass-Spring-Damper......Page 556 9.6.5 Lyapunov Functions......Page 558 9.6.6 Hamiltonian Systems......Page 560 9.7.1 Linear Homogeneous......Page 562 9.7.2 Nonlinear......Page 565 9.8.1 Poincare´ Sphere......Page 566 9.8.2 Projective Space......Page 570 9.9 Bifurcations......Page 571 9.9.1 Pitchfork......Page 572 9.9.2 Transcritical......Page 573 9.9.3 Saddle-Node......Page 574 9.9.4 Hopf......Page 575 9.10 Projection of Partial Differential Equations......Page 576 9.11 Lorenz Equations......Page 579 9.11.1 Linear Stability......Page 580 9.11.2 Nonlinear Stability: Center Manifold Projection......Page 582 9.11.3 Transition to Chaos......Page 586 Exercises......Page 590 A.1.2 Quadratic......Page 602 A.1.3 Cubic......Page 603 A.1.4 Quartic......Page 604 A.2 Cramer’s Rule......Page 606 A.3 Gaussian Elimination......Page 607 A.5 Trigonometric Relations......Page 608 A.7.1 Gamma......Page 610 A.7.3 Sine, Cosine, and Exponential Integral......Page 611 A.7.4 Hypergeometric......Page 612 A.7.6 Dirac δ and Heaviside......Page 613 A.8.1 Euler’s Formula......Page 615 A.8.2 Polar and Cartesian Representations......Page 616 Exercises......Page 617 Bibliography......Page 620 Index......Page 626