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 Preface 1 Ordinary Differential Equations 1.1 INTRODUCTION 1.2 DEFINITIONS 1.3 DIFFERENTIAL EQUATIONS OF FIRST ORDER 1.3.1 Separable Equations 1.3.2 Exact Equations 1.3.3 Integrating Factors 1.4 PHYSICAL APPLICATIONS 1.4.1 Simple Electrical Circuits 1.4.2 The Rate Equation 1.4.3 Fluid Flow 1.4.4 Dynamics 1.5 LINEAR DIFFERENTIAL EQUATIONS 1.6 HOMOGENEOUS SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS 1.7 SPRING–MASS SYSTEM—FREE MOTION 1.7.1 Undamped Motion 1.7.2 Damped Motion 1.7.3 The Electrical Circuit Analog 1.8 NONHOMOGENEOUS SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS 1.9 SPRING–MASS SYSTEM—FORCED MOTION 1.9.1 Resonance 1.9.2 Near Resonance 1.9.3 Forced Oscillations with Damping 1.10 PERIODIC INPUT FUNCTIONS—FOURIER SERIES 1.10.1 Even and Odd Functions 1.10.2 Half-Range Expansions 1.10.3 Forced Oscillations 1.11 THE CAUCHY EQUATION 1.12 VARIATION OF PARAMETERS 1.13 MISCELLANEOUS INFORMATION PROBLEMS 2 Power-Series Methods 2.1 POWER SERIES 2.2 LINEAR DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS 2.3 LEGENDRE’S EQUATION 2.4 THE METHOD OF FROBENIUS 2.4.1 Distinct Roots Not Differing by an Integer 2.4.2 Double Roots 2.4.3 Roots Differing by an Integer 2.5 BESSEL’S EQUATION PROBLEMS 3 Laplace Transforms 3.1 INTRODUCTION 3.2 THE LAPLACE TRANSFORM 3.3 LAPLACE TRANSFORMS OF DERIVATIVES AND INTEGRALS 3.4 DERIVATIVES AND INTEGRALS OF LAPLACE TRANSFORMS 3.5 LAPLACE TRANSFORMS OF PERIODIC FUNCTIONS 3.6 INVERSE TRANSFORMS—PARTIAL FRACTIONS 3.6.1 Unrepeated Linear Factor (s - a) 3.6.2 Repeated Linear Factor (s − a)m 3.7 SOLUTION OF DIFFERENTIAL EQUATIONS PROBLEMS 4 Matrices and Determinants 4.1 INTRODUCTION 4.2 MATRICES 4.3 ADDITION OF MATRICES 4.4 THE TRANSPOSE AND SOME SPECIAL MATRICES 4.5 MATRIX MULTIPLICATION—DEFINITION 4.6 MATRIX MULTIPLICATION—ADDITIONAL PROPERTIES 4.7 DETERMINANTS 4.8 THE ADJOINT AND THE INVERSE MATRICES 4.9 SOLUTION OF SIMULTANEOUS LINEAR ALGEBRAIC EQUATIONS 4.9.1 Nonhomogeneous Sets of Linear Algebraic Equations 4.9.2 Homogeneous Sets of Linear Algebraic Equations 4.9.3 Solutions to Sets of Linear Equations by MATLAB 4.10 LEAST-SQUARES FIT AND THE PSEUDO INVERSE 4.11 EIGENVALUES AND EIGENVECTORS 4.12 EIGENVALUE PROBLEMS IN ENGINEERING 4.12.1 Moments of Inertia 4.12.2 Stress 4.12.3 Linear Dynamic Systems and Stability PROBLEMS 5 Vector Analysis 5.1 INTRODUCTION 5.2 VECTOR ALGEBRA 5.2.1 Definitions 5.2.2 Addition and Subtraction 5.2.3 Components of a Vector 5.2.4 Multiplication 5.3 VECTOR DIFFERENTIATION 5.3.1 Ordinary Differentiation 5.3.2 Partial Differentiation 5.4 THE GRADIENT 5.5 CYLINDRICAL AND SPHERICAL COORDINATES 5.5.1 Cylindrical Coordinates 5.5.2 Spherical Coordinates 5.6 INTEGRAL THEOREMS 5.6.1 The Divergence Theorem 5.6.2 Stokes’s Theorem PROBLEMS 6 Partial Differential Equations 6.1 INTRODUCTION 6.2 WAVE MOTION 6.2.1 Vibration of a Stretched, Flexible String 6.2.2 The Vibrating Membrane 6.2.3 Longitudinal Vibrations of an Elastic Bar 6.2.4 Transmission-Line Equations 6.3 THE D’ALEMBERT SOLUTION OF THE WAVE EQUATION 6.4 SEPARATION OF VARIABLES 6.5 DIFFUSION 6.6 SOLUTION OF THE DIFFUSION EQUATION 6.6.1 A Long, Insulated Rod with Ends at Fixed Temperatures 6.6.2 A Long, Totally Insulated Rod 6.6.3 Two-Dimensional Heat Conduction in a Long, Rectangular Bar 6.7 ELECTRIC POTENTIAL ABOUT A SPHERICAL SURFACE 6.8 HEAT TRANSFER IN A CYLINDRICAL BODY 6.9 GRAVITATIONAL POTENTIAL PROBLEMS 7 Complex Variables 7.1 INTRODUCTION 7.2 COMPLEX NUMBERS 7.3 ELEMENTARY FUNCTIONS 7.4 ANALYTIC FUNCTIONS 7.5 COMPLEX INTEGRATION 7.5.1 Green’s Theorem 7.5.2 Cauchy’s Integral Theorem 7.5.3 Cauchy’s Integral Formula 7.6 SERIES 7.7 RESIDUES PROBLEMS 8 Numerical Methods 8.1 INTRODUCTION 8.2 FINITE-DIFFERENCE OPERATORS 8.3 THE DIFFERENTIAL OPERATOR RELATED TO THE DIFFERENCE OPERATORS 8.4 TRUNCATION ERROR 8.5 NUMERICAL INTEGRATION 8.6 NUMERICAL INTERPOLATION 8.7 ROOTS OF EQUATIONS 8.8 INITIAL-VALUE PROBLEMS—ORDINARY DIFFERENTIAL EQUATIONS 8.8.1 Taylor’s Method 8.8.2 Euler’s Method 8.8.3 Adams’ Method 8.8.4 Runge–Kutta Methods 8.8.5 Direct Method 8.9 HIGHER-ORDER EQUATIONS 8.10 BOUNDARY-VALUE PROBLEMS—ORDINARY DIFFERENTIAL EQUATIONS 8.10.1 Iterative Method 8.10.2 Superposition 8.10.3 Simultaneous Equations 8.11 NUMERICAL STABILITY 8.12 NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS 8.12.1 The Diffusion Equation 8.12.2 The Wave Equation 8.12.3 Laplace’s Equation PROBLEMS Bibliography Appendix A Appendix B Introduction to MATLAB B.1 INTRODUCTION B.2 REAL AND COMPLEX NUMBERS B.3 VECTORS AND MATRICES B.4 FORMAT AND SCIENTIFIC NOTATION B.5 PROGRAMMING LOOPS B.6 PLOTTING B.7 STRING ARRAYS B.8 MATLAB FILES, INPUT, AND OUTPUT B.8.1 Setting the Path B.8.2 Script Files, or m-Files B.8.3 Input Files and Output Files B.8.4 Interactive Input and Output B.9 FUNCTIONS B.10 THE WORKSPACE BROWSER B.11 FINAL REMARKS Answers to Selected Problems Index < div="">This book introduces undergraduate students of engineering and science to applied mathematics essential to the study of many problems.