Mathematical Methods for Engineering and Science

This book introduces undergraduate students of engineering and science to applied mathematics essential to the study of many problems. Topics are differential equations, power series, Laplace transforms, matrices and determinants, vector analysis, partial differential equations, complex variables, and numerical methods. Approximately, 160 examples and 1000 homework problems aid students in their study. This book presents mathematical topics using derivations rather than theorems and proofs. This textbook is uniquely qualified to apply mathematics to physical applications (spring-mass systems, electrical circuits, conduction, diffusion, etc.), in a manner that is efficient and understandable. This book is written to support a mathematics course after differential equations, to permit several topics to be covered in one semester, and to make the material comprehensible to undergraduates. An Instructor Solutions Manual, and also a Student Solutions Manual that provides solutions to select problems, is available. Contents 6 Preface 12 1 Ordinary Differential Equations 14 1.1 INTRODUCTION 14 1.2 DEFINITIONS 15 1.3 DIFFERENTIAL EQUATIONS OF FIRST ORDER 16 1.3.1 Separable Equations 16 1.3.2 Exact Equations 19 1.3.3 Integrating Factors 21 1.4 PHYSICAL APPLICATIONS 22 1.4.1 Simple Electrical Circuits 22 1.4.2 The Rate Equation 24 1.4.3 Fluid Flow 26 1.4.4 Dynamics 27 1.5 LINEAR DIFFERENTIAL EQUATIONS 32 1.6 HOMOGENEOUS SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS 34 1.7 SPRING–MASS SYSTEM—FREE MOTION 38 1.7.1 Undamped Motion 39 1.7.2 Damped Motion 42 1.7.3 The Electrical Circuit Analog 47 1.8 NONHOMOGENEOUS SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS 49 1.9 SPRING–MASS SYSTEM—FORCED MOTION 52 1.9.1 Resonance 55 1.9.2 Near Resonance 56 1.9.3 Forced Oscillations with Damping 58 1.10 PERIODIC INPUT FUNCTIONS—FOURIER SERIES 62 1.10.1 Even and Odd Functions 66 1.10.2 Half-Range Expansions 70 1.10.3 Forced Oscillations 72 1.11 THE CAUCHY EQUATION 74 1.12 VARIATION OF PARAMETERS 77 1.13 MISCELLANEOUS INFORMATION 79 PROBLEMS 80 2 Power-Series Methods 89 2.1 POWER SERIES 89 2.2 LINEAR DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS 93 2.3 LEGENDRE’S EQUATION 102 2.4 THE METHOD OF FROBENIUS 106 2.4.1 Distinct Roots Not Differing by an Integer 107 2.4.2 Double Roots 109 2.4.3 Roots Differing by an Integer 113 2.5 BESSEL’S EQUATION 116 PROBLEMS 126 3 Laplace Transforms 130 3.1 INTRODUCTION 130 3.2 THE LAPLACE TRANSFORM 131 3.3 LAPLACE TRANSFORMS OF DERIVATIVES AND INTEGRALS 141 3.4 DERIVATIVES AND INTEGRALS OF LAPLACE TRANSFORMS 145 3.5 LAPLACE TRANSFORMS OF PERIODIC FUNCTIONS 148 3.6 INVERSE TRANSFORMS—PARTIAL FRACTIONS 152 3.6.1 Unrepeated Linear Factor (s - a) 152 3.6.2 Repeated Linear Factor (s − a)m 152 3.7 SOLUTION OF DIFFERENTIAL EQUATIONS 157 PROBLEMS 165 4 Matrices and Determinants 172 4.1 INTRODUCTION 172 4.2 MATRICES 173 4.3 ADDITION OF MATRICES 174 4.4 THE TRANSPOSE AND SOME SPECIAL MATRICES 176 4.5 MATRIX MULTIPLICATION—DEFINITION 180 4.6 MATRIX MULTIPLICATION—ADDITIONAL PROPERTIES 182 4.7 DETERMINANTS 184 4.8 THE ADJOINT AND THE INVERSE MATRICES 190 4.9 SOLUTION OF SIMULTANEOUS LINEAR ALGEBRAIC EQUATIONS 195 4.9.1 Nonhomogeneous Sets of Linear Algebraic Equations 195 4.9.2 Homogeneous Sets of Linear Algebraic Equations 197 4.9.3 Solutions to Sets of Linear Equations by MATLAB 199 4.10 LEAST-SQUARES FIT AND THE PSEUDO INVERSE 205 4.11 EIGENVALUES AND EIGENVECTORS 214 4.12 EIGENVALUE PROBLEMS IN ENGINEERING 222 4.12.1 Moments of Inertia 222 4.12.2 Stress 226 4.12.3 Linear Dynamic Systems and Stability 229 PROBLEMS 232 5 Vector Analysis 241 5.1 INTRODUCTION 241 5.2 VECTOR ALGEBRA 241 5.2.1 Definitions 241 5.2.2 Addition and Subtraction 243 5.2.3 Components of a Vector 243 5.2.4 Multiplication 245 5.3 VECTOR DIFFERENTIATION 254 5.3.1 Ordinary Differentiation 254 5.3.2 Partial Differentiation 259 5.4 THE GRADIENT 262 5.5 CYLINDRICAL AND SPHERICAL COORDINATES 272 5.5.1 Cylindrical Coordinates 272 5.5.2 Spherical Coordinates 276 5.6 INTEGRAL THEOREMS 280 5.6.1 The Divergence Theorem 280 5.6.2 Stokes’s Theorem 283 PROBLEMS 290 6 Partial Differential Equations 295 6.1 INTRODUCTION 295 6.2 WAVE MOTION 297 6.2.1 Vibration of a Stretched, Flexible String 297 6.2.2 The Vibrating Membrane 299 6.2.3 Longitudinal Vibrations of an Elastic Bar 301 6.2.4 Transmission-Line Equations 302 6.3 THE D’ALEMBERT SOLUTION OF THE WAVE EQUATION 305 6.4 SEPARATION OF VARIABLES 309 6.5 DIFFUSION 321 6.6 SOLUTION OF THE DIFFUSION EQUATION 324 6.6.1 A Long, Insulated Rod with Ends at Fixed Temperatures 324 6.6.2 A Long, Totally Insulated Rod 328 6.6.3 Two-Dimensional Heat Conduction in a Long, Rectangular Bar 332 6.7 ELECTRIC POTENTIAL ABOUT A SPHERICAL SURFACE 337 6.8 HEAT TRANSFER IN A CYLINDRICAL BODY 339 6.9 GRAVITATIONAL POTENTIAL 343 PROBLEMS 346 7 Complex Variables 350 7.1 INTRODUCTION 350 7.2 COMPLEX NUMBERS 350 7.3 ELEMENTARY FUNCTIONS 356 7.4 ANALYTIC FUNCTIONS 362 7.5 COMPLEX INTEGRATION 366 7.5.1 Green’s Theorem 366 7.5.2 Cauchy’s Integral Theorem 369 7.5.3 Cauchy’s Integral Formula 373 7.6 SERIES 378 7.7 RESIDUES 386 PROBLEMS 395 8 Numerical Methods 399 8.1 INTRODUCTION 399 8.2 FINITE-DIFFERENCE OPERATORS 401 8.3 THE DIFFERENTIAL OPERATOR RELATED TO THE DIFFERENCE OPERATORS 405 8.4 TRUNCATION ERROR 410 8.5 NUMERICAL INTEGRATION 413 8.6 NUMERICAL INTERPOLATION 420 8.7 ROOTS OF EQUATIONS 422 8.8 INITIAL-VALUE PROBLEMS—ORDINARY DIFFERENTIAL EQUATIONS 425 8.8.1 Taylor’s Method 426 8.8.2 Euler’s Method 426 8.8.3 Adams’ Method 427 8.8.4 Runge–Kutta Methods 427 8.8.5 Direct Method 431 8.9 HIGHER-ORDER EQUATIONS 434 8.10 BOUNDARY-VALUE PROBLEMS—ORDINARY DIFFERENTIAL EQUATIONS 441 8.10.1 Iterative Method 441 8.10.2 Superposition 442 8.10.3 Simultaneous Equations 442 8.11 NUMERICAL STABILITY 445 8.12 NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS 445 8.12.1 The Diffusion Equation 446 8.12.2 The Wave Equation 447 8.12.3 Laplace’s Equation 448 PROBLEMS 456 Bibliography 459 Appendix A 460 Appendix B Introduction to MATLAB 467 B.1 INTRODUCTION 467 B.2 REAL AND COMPLEX NUMBERS 469 B.3 VECTORS AND MATRICES 471 B.4 FORMAT AND SCIENTIFIC NOTATION 477 B.5 PROGRAMMING LOOPS 479 B.6 PLOTTING 481 B.7 STRING ARRAYS 483 B.8 MATLAB FILES, INPUT, AND OUTPUT 484 B.8.1 Setting the Path 484 B.8.2 Script Files, or m-Files 484 B.8.3 Input Files and Output Files 486 B.8.4 Interactive Input and Output 487 B.9 FUNCTIONS 488 B.10 THE WORKSPACE BROWSER 490 B.11 FINAL REMARKS 491 Answers to Selected Problems 492 Index 505 < div="">This book introduces undergraduate students of engineering and science to applied mathematics essential to the study of many problems.