Applied Engineering Mathematics

Undergraduate engineering students need good mathematics skills. This textbook supports this need by placing a strong emphasis on visualization and the methods and tools needed across the whole of engineering. The visual approach is emphasized, and excessive proofs and derivations are avoided. The visual images explain and teach the mathematical methods. The book's website provides dynamic and interactive codes in Mathematica to accompany the examples for the reader to explore on their own with Mathematica or the free Computational Document Format player, and it provides access for instructors to a solutions manual. Strongly emphasizes a visual approach to engineering mathematics Written for years 2 to 4 of an engineering degree course Website offers support with dynamic and interactive Mathematica code and instructor's solutions manual Brian Vick is an associate professor at Virginia Tech in the United States and is a longtime teacher and researcher. His style has been developed from teaching a variety of engineering and mathematical courses in the areas of heat transfer, thermodynamics, engineering design, computer programming, numerical analysis, and system dynamics at both undergraduate and graduate levels. eResource material is available for this title at (http://www.crcpress.com/9780367432768) www.crcpress.com/9780367432768 . Undergraduate engineering students need good mathematics skills, and this textbook supports this need, with a strong emphasis on visualization and the methods and tools needed across the whole of engineering. Cover 1 Half Title 2 Title Page 4 Copyright Page 5 Table of Contents 6 Preface 12 About the Author 14 Chapter 1 Overview 16 1.1 Objectives 16 1.2 Educational Philosophy 17 1.3 Physical Processes 18 1.4 Mathematical Models 18 1.4.1 Algebraic Equations 19 1.4.2 Ordinary Differential Equations 19 1.4.3 Partial Differential Equations 19 1.5 Solution Methods 20 1.6 Software 22 Chapter 2 Physical Processes 24 2.1 Physical Phenomena 24 2.2 Fundamental Principles 25 2.3 Conservation Laws 26 2.3.1 Conservation of Mass: Continuity 26 2.3.2 Conservation of Momentum: Newton’s Second Law 27 2.3.3 Conservation of Energy: First Law of Thermodynamics 28 2.4 Rate Equations 29 2.4.1 Heat Conduction: Fourier’s Law 29 2.4.2 Heat Convection: Newton’s Law of Cooling 30 2.4.3 Thermal Radiation 30 2.4.4 Viscous Fluid Shear: Newton’s Viscosity Law 31 2.4.5 Binary Mass Diffusion: Fick’s Law 32 2.4.6 Electrical Conduction: Ohm’s Law 32 2.4.7 Stress-Strain: Hooke’s Law 32 2.5 Diffusion Analogies 32 Chapter 3 Modeling of Physical Processes 34 3.1 Cause and Effect 34 3.1.1 General Physical Process 34 3.1.2 Thermal Processes 35 3.1.3 Mechanical Processes 35 3.2 Mathematical Modeling 35 3.3 Complete Mathematical Model 37 3.3.1 Mechanical Vibrations 38 3.3.2 Heat Conduction 39 3.4 Dimensionless Formulation 41 3.4.1 General Procedure 41 3.4.2 Mechanical Vibrations 42 3.4.3 Steady Heat Conduction 44 3.5 Inverse and Parameter Estimation Problems 46 3.5.1 Direct Problem 46 3.5.2 Inverse Problem 46 3.5.3 Parameter Estimation Problem 46 3.6 Mathematical Classification of Physical Problems 47 Problems 47 Chapter 4 Calculus 52 4.1 Derivatives 53 4.1.1 Basic Concept of a Derivative 53 4.1.2 Velocity from Displacement 53 4.1.3 Derivative of tn 54 4.1.4 Chain Rule 55 4.1.5 Product Rule 55 4.1.6 Partial Derivatives 56 4.2 Numerical Differentiation: Taylor Series 57 4.2.1 Taylor Series Expansion 57 4.2.2 First Derivatives Using Taylor Series 58 4.2.3 Second Derivatives Using Taylor Series 59 4.3 Integrals 60 4.3.1 Basic Concept of an Integral 60 4.3.2 Geometric Interpretation of an Integral: Area Under a Curve 61 4.3.3 Mean Value Theorem 62 4.3.4 Integration by Parts 63 4.3.5 Leibniz Rule: Derivatives of Integrals 63 4.4 Summary of Derivatives and Integrals 65 4.5 The Step, PULSE, and Delta Functions 67 4.5.1 The Step Function 67 4.5.2 The Unit Pulse Function 67 4.5.3 The Delta Function 69 4.6 Numerical Integration 70 4.6.1 Trapezoid Rule 71 4.6.2 Trapezoid Rule for Unequal Segments 73 4.6.3 Simpson’s Rule 75 4.6.4 Simpson’s 3/8 Rule 76 4.6.5 Gauss Quadrature 77 4.7 Multiple Integrals 79 Problems 80 Chapter 5 Linear Algebra 92 5.1 Introduction 93 5.2 Cause and Effect 93 5.3 Applications 94 5.3.1 Networks 94 5.3.2 Finite Difference Equations 95 5.4 Geometric Interpretations 96 5.4.1 Row Interpretation 96 5.4.2 Column Interpretation 96 5.5 Possibility of Solutions 97 5.6 Characteristics of Square Matrices 97 5.7 Square, Overdetermined, and Underdetermined Systems 100 5.7.1 Overdetermined Systems 100 5.7.2 Underdetermined Systems 100 5.7.3 Square Systems 100 5.8 Row Operations 101 5.9 The Determinant and Cramer’s Rule 102 5.10 Gaussian Elimination 102 5.10.1 Naïve Gaussian Elimination 102 5.10.2 Pivoting 103 5.10.3 Tridiagonal Systems 104 5.11 LU Factorization 104 5.12 Gauss–Seidel Iteration 105 5.13 Matrix Inversion 105 5.14 Least Squares Regression 106 Problems 108 Chapter 6 Nonlinear Algebra: Root Finding 114 6.1 Introduction 114 6.2 Applications 115 6.2.1 Simple Interest 115 6.2.2 Thermodynamic Equations of State 116 6.2.3 Heat Transfer: Thermal Radiation 116 6.2.4 Design of an Electric Circuit 117 6.3 Root Finding Methods 119 6.4 Graphical Method 119 6.5 Bisection Method 120 6.6 False Position Method 120 6.7 Newton–Raphson Method 122 6.8 Secant Method 123 6.9 Roots of Simultaneous Nonlinear Equations 124 Problems 127 Chapter 7 Introduction to Ordinary Differential Equations 134 7.1 Classification of Ordinary Differential Equations 134 7.1.1 Autonomous versus Nonautonomous Systems 135 7.1.2 Initial Value and Boundary Value Problems 135 7.2 First-Order Ordinary Differential Equations 136 7.2.1 First-Order Phase Portraits 136 7.2.2 Nonautonomous Systems 138 7.2.3 First-Order Linear Equations 138 7.2.4 Lumped Thermal Models 139 7.2.5 RC Electrical Circuit 140 7.2.6 First-Order Nonlinear Equations 141 7.2.7 Population Dynamics 141 7.3 Second-Order Initial Value Problems 143 7.3.1 Second-Order Phase Portraits 143 7.3.2 Second-Order Linear Equations 144 7.3.3 Mechanical Vibrations 145 7.3.4 Mechanical and Electrical Circuits 145 7.3.5 Second-Order Nonlinear Equations 146 7.3.6 The Pendulum 146 7.3.7 Predator–Prey Models 147 7.4 Second-Order Boundary Value Problems 148 7.5 Higher-Order Systems 148 Problems 149 Chapter 8 Laplace Transforms 154 8.1 Definition of the Laplace Transform 154 8.2 Laplace Transform Pairs 155 8.3 Properties of the Laplace Transform 155 8.4 The Inverse Laplace Transformation 156 8.4.1 Partial-Fraction Expansion Method 156 8.4.2 Partial-Fraction Expansion for Distinct Poles 157 8.4.3 Partial-Fraction Expansion for Multiple Poles 158 8.5 Solutions of Linear Ordinary Differential Equation 158 8.5.1 General Strategy 158 8.5.2 First-Order Ordinary Differential Equations 159 8.5.3 Second-Order Ordinary Differential Equations 161 8.6 The Transfer Function 161 8.6.1 The Impulse Response 162 8.6.2 First-Order Ordinary Differential Equations 163 Problems 164 Chapter 9 Numerical Solutions of Ordinary Differential Equations 166 9.1 Introduction to Numerical Solutions 167 9.2 Runge–Kutta Methods 168 9.2.1 Euler’s Method 168 9.2.2 Heun’s Method 168 9.2.3 Higher-Order Runge–Kutta Methods 170 9.2.4 Numerical Comparison of Runge–Kutta Schemes 170 9.3 Coupled Systems of First-Order Differential Equations 171 9.4 Second-Order Initial Value Problems 172 9.5 Implicit Schemes 173 9.6 Second-Order Boundary Value Problems: The Shooting Method 175 Problems 176 Chapter 10 First-Order Ordinary Differential Equations 184 10.1 Stability of the Fixed Points 184 10.1.1 RC Electrical Circuit 186 10.1.2 Population Model 186 10.2 Characteristics of Linear Systems 186 10.3 Solution Using Integrating Factors 187 10.4 First-Order Nonlinear Systems and Bifurcations 189 10.4.1 Saddle-Node Bifurcation 190 10.4.2 Transcritical Bifurcation 191 10.4.3 Example of a Transcritical Bifurcation: Laser Threshold 192 10.4.4 Supercritical Pitchfork Bifurcation 194 10.4.5 Subcritical Pitchfork Bifurcation 194 Problems 197 Chapter 11 Second-Order Ordinary Differential Equations 208 11.1 Linear Systems 208 11.2 Classification of Linear Systems 211 11.3 Classical Spring-Mass-Damper 211 11.4 Stability Analysis of the Fixed Points 215 11.5 Pendulum 217 11.5.1 Fixed Points: No Forcing, No Damping 219 11.5.2 Fixed Points: General Case 219 11.6 Competition Models 220 11.6.1 Coexistence 221 11.6.2 Extinction 223 11.7 Limit Cycles 223 11.7.1 van der Pol Oscillator 226 11.7.2 Poincare–Bendixson Theorem 226 11.8 Bifurcations 227 11.8.1 Saddle-Node Bifurcation 227 11.8.2 Transcritical Bifurcation 228 11.8.3 Supercritical Pitchfork Bifurcation 229 11.8.4 Subcritical Pitchfork Bifurcation 229 11.8.5 Hopf Bifurcations 229 11.8.6 Supercritical Hopf Bifurcation 230 11.8.7 Subcritical Hopf Bifurcation 232 11.9 Coupled Oscillators 232 Problems: LINEAR SYSTEMS 234 Problems: NONLINEAR SYSTEMS 236 Index 244 applied,engineering;,applied,mathematics;,engineering,mathematics;,conservation,laws;,thermodynamics;,physical,processes;,modeling;,calculus;,linear,algebra;,nonlinear,algebra;,ordinary,differential,equations;,laplace,transforms;,numerical,analysis;,Mathematica applied engineering,applied mathematics,engineering mathematics,conservation laws,thermodynamics,physical processes,modeling,calculus,linear algebra,nonlinear algebra,ordinary differential equations,laplace transforms,numerical analysis,Mathematica Cover Half Title #2,0,-32767Title Page #4,0,-32767Copyright Page #5,0,-32767Table of Contents #6,0,-32767Preface #12,0,-32767About the Author Chapter 1 Overview 1.1 Objectives 1.2 Educational Philosophy 1.3 Physical Processes 1.4 Mathematical Models 1.4.1 Algebraic Equations 1.4.2 Ordinary Differential Equations 1.4.3 Partial Differential Equations 1.5 Solution Methods 1.6 Software Chapter 2 Physical Processes 2.1 Physical Phenomena 2.2 Fundamental Principles 2.3 Conservation Laws 2.3.1 Conservation of Mass: Continuity 2.3.2 Conservation of Momentum: Newton’s Second Law 2.3.3 Conservation of Energy: First Law of Thermodynamics 2.4 Rate Equations 2.4.1 Heat Conduction: Fourier’s Law 2.4.2 Heat Convection: Newton’s Law of Cooling 2.4.3 Thermal Radiation 2.4.4 Viscous Fluid Shear: Newton’s Viscosity Law 2.4.5 Binary Mass Diffusion: Fick’s Law 2.4.6 Electrical Conduction: Ohm’s Law 2.4.7 Stress-Strain: Hooke’s Law 2.5 Diffusion Analogies Chapter 3 Modeling of Physical Processes 3.1 Cause and Effect 3.1.1 General Physical Process 3.1.2 Thermal Processes 3.1.3 Mechanical Processes 3.2 Mathematical Modeling 3.3 Complete Mathematical Model 3.3.1 Mechanical Vibrations 3.3.2 Heat Conduction 3.4 Dimensionless Formulation 3.4.1 General Procedure 3.4.2 Mechanical Vibrations 3.4.3 Steady Heat Conduction 3.5 Inverse and Parameter Estimation Problems 3.5.1 Direct Problem 3.5.2 Inverse Problem 3.5.3 Parameter Estimation Problem 3.6 Mathematical Classification of Physical Problems Problems Chapter 4 Calculus 4.1 Derivatives 4.1.1 Basic Concept of a Derivative 4.1.2 Velocity from Displacement 4.1.3 Derivative of tn 4.1.4 Chain Rule 4.1.5 Product Rule 4.1.6 Partial Derivatives 4.2 Numerical Differentiation: Taylor Series 4.2.1 Taylor Series Expansion 4.2.2 First Derivatives Using Taylor Series 4.2.3 Second Derivatives Using Taylor Series 4.3 Integrals 4.3.1 Basic Concept of an Integral 4.3.2 Geometric Interpretation of an Integral: Area Under a Curve 4.3.3 Mean Value Theorem 4.3.4 Integration by Parts 4.3.5 Leibniz Rule: Derivatives of Integrals 4.4 Summary of Derivatives and Integrals 4.5 The Step, PULSE, and Delta Functions 4.5.1 The Step Function 4.5.2 The Unit Pulse Function 4.5.3 The Delta Function 4.6 Numerical Integration 4.6.1 Trapezoid Rule 4.6.2 Trapezoid Rule for Unequal Segments 4.6.3 Simpson’s Rule 4.6.4 Simpson’s 3/8 Rule 4.6.5 Gauss Quadrature 4.7 Multiple Integrals Problems Chapter 5 Linear Algebra 5.1 Introduction 5.2 Cause and Effect 5.3 Applications 5.3.1 Networks 5.3.2 Finite Difference Equations 5.4 Geometric Interpretations 5.4.1 Row Interpretation 5.4.2 Column Interpretation 5.5 Possibility of Solutions 5.6 Characteristics of Square Matrices 5.7 Square, Overdetermined, and Underdetermined Systems 5.7.1 Overdetermined Systems 5.7.2 Underdetermined Systems 5.7.3 Square Systems 5.8 Row Operations 5.9 The Determinant and Cramer’s Rule 5.10 Gaussian Elimination 5.10.1 Naïve Gaussian Elimination 5.10.2 Pivoting 5.10.3 Tridiagonal Systems 5.11 LU Factorization 5.12 Gauss–Seidel Iteration 5.13 Matrix Inversion 5.14 Least Squares Regression Problems Chapter 6 Nonlinear Algebra: Root Finding 6.1 Introduction 6.2 Applications 6.2.1 Simple Interest 6.2.2 Thermodynamic Equations of State 6.2.3 Heat Transfer: Thermal Radiation 6.2.4 Design of an Electric Circuit 6.3 Root Finding Methods 6.4 Graphical Method 6.5 Bisection Method 6.6 False Position Method 6.7 Newton–Raphson Method 6.8 Secant Method 6.9 Roots of Simultaneous Nonlinear Equations Problems Chapter 7 Introduction to Ordinary Differential Equations 7.1 Classification of Ordinary Differential Equations 7.1.1 Autonomous versus Nonautonomous Systems 7.1.2 Initial Value and Boundary Value Problems 7.2 First-Order Ordinary Differential Equations 7.2.1 First-Order Phase Portraits 7.2.2 Nonautonomous Systems 7.2.3 First-Order Linear Equations 7.2.4 Lumped Thermal Models 7.2.5 RC Electrical Circuit 7.2.6 First-Order Nonlinear Equations 7.2.7 Population Dynamics 7.3 Second-Order Initial Value Problems 7.3.1 Second-Order Phase Portraits 7.3.2 Second-Order Linear Equations 7.3.3 Mechanical Vibrations 7.3.4 Mechanical and Electrical Circuits 7.3.5 Second-Order Nonlinear Equations 7.3.6 The Pendulum 7.3.7 Predator–Prey Models 7.4 Second-Order Boundary Value Problems 7.5 Higher-Order Systems Problems Chapter 8 Laplace Transforms 8.1 Definition of the Laplace Transform 8.2 Laplace Transform Pairs 8.3 Properties of the Laplace Transform 8.4 The Inverse Laplace Transformation 8.4.1 Partial-Fraction Expansion Method 8.4.2 Partial-Fraction Expansion for Distinct Poles 8.4.3 Partial-Fraction Expansion for Multiple Poles 8.5 Solutions of Linear Ordinary Differential Equation 8.5.1 General Strategy 8.5.2 First-Order Ordinary Differential Equations 8.5.3 Second-Order Ordinary Differential Equations 8.6 The Transfer Function 8.6.1 The Impulse Response 8.6.2 First-Order Ordinary Differential Equations Problems Chapter 9 Numerical Solutions of Ordinary Differential Equations 9.1 Introduction to Numerical Solutions 9.2 Runge–Kutta Methods 9.2.1 Euler’s Method 9.2.2 Heun’s Method 9.2.3 Higher-Order Runge–Kutta Methods 9.2.4 Numerical Comparison of Runge–Kutta Schemes 9.3 Coupled Systems of First-Order Differential Equations 9.4 Second-Order Initial Value Problems 9.5 Implicit Schemes 9.6 Second-Order Boundary Value Problems: The Shooting Method Problems Chapter 10 First-Order Ordinary Differential Equations 10.1 Stability of the Fixed Points 10.1.1 RC Electrical Circuit 10.1.2 Population Model 10.2 Characteristics of Linear Systems 10.3 Solution Using Integrating Factors 10.4 First-Order Nonlinear Systems and Bifurcations 10.4.1 Saddle-Node Bifurcation 10.4.2 Transcritical Bifurcation 10.4.3 Example of a Transcritical Bifurcation: Laser Threshold 10.4.4 Supercritical Pitchfork Bifurcation 10.4.5 Subcritical Pitchfork Bifurcation Problems Chapter 11 Second-Order Ordinary Differential Equations 11.1 Linear Systems 11.2 Classification of Linear Systems 11.3 Classical Spring-Mass-Damper 11.4 Stability Analysis of the Fixed Points 11.5 Pendulum 11.5.1 Fixed Points: No Forcing, No Damping 11.5.2 Fixed Points: General Case 11.6 Competition Models 11.6.1 Coexistence 11.6.2 Extinction 11.7 Limit Cycles 11.7.1 van der Pol Oscillator 11.7.2 Poincare–Bendixson Theorem 11.8 Bifurcations 11.8.1 Saddle-Node Bifurcation 11.8.2 Transcritical Bifurcation 11.8.3 Supercritical Pitchfork Bifurcation 11.8.4 Subcritical Pitchfork Bifurcation 11.8.5 Hopf Bifurcations 11.8.6 Supercritical Hopf Bifurcation 11.8.7 Subcritical Hopf Bifurcation 11.9 Coupled Oscillators Problems: LINEAR SYSTEMS Problems: NONLINEAR SYSTEMS Index Undergraduate Engineering Students Need Good Mathematics Skills, And This Textbook Supports This With A Strong Emphasis On Visualization And The Methods And Tools Needed Across The Whole Of Engineering. The Visual Approach Is Emphasised, And Excessive Proofs And Derivations Are Avoided. The Visual Images Explain And Teach The Mathematical Methods. The Book's Website Www.crcpress.com/9780367432768 Provides Dynamic And Interactive Codes In Mathematica To Accompany The Examples For The Reader To Explore On Their Own With Mathematica Or The Free Computational Document Format Player, And Access For Instructors To A Solutions Manual. This textbook for undergraduates across the range of engineering has a strong emphasis on visualization and the methods and tools needed across the whole of engineering. The book's website provides dynamic and interactive codes in Mathematica to accompany the examples for the reader to explore on their own. In 1860, fifteen-year-old third-grader Simon Green attempts to herd one thousand turkeys from Missouri to Denver, Colorado, in hopes of selling them at a profit.