Preface xvii Suggestions to instructors xxi About the companion website xxv 1 Overview of Engineering Analysis 1 Chapter Learning Objectives 1 1.1 Introduction 1 1.2 Engineering Analysis and Engineering Practices 2 1.2.1 Creation 2 1.2.2 Problem Solving 2 1.2.3 Decision Making 3 1.3 "Toolbox" for Engineering Analysis 5 1.4 The Four Stages in Engineering Analysis 8 1.5 Examples of the Application of Engineering Analysis in Design 10 1.6 The "Safety Factor" in Engineering Analysis of Structures 17 1.7 Problems 19 2 Mathematical Modeling 21 Chapter Learning Objectives 21 2.1 Introduction 21 2.2 MathematicalModeling Terminology 26 2.2.1 The Numbers 26 2.2.1.1 Real Numbers 26 2.2.1.2 Imaginary Numbers 26 2.2.1.3 Absolute Values 26 2.2.1.4 Constants 26 2.2.1.5 Parameters 26 2.2.2 Variables

26 2.2.3 Functions 27 2.2.3.1 Form 1. Functions with Discrete Values 27 2.2.3.2 Form 2. Continuous Functions 27 2.2.3.3 Form 3.

Piecewise Continuous Functions 28 2.2.4 Curve Fitting Technique in Engineering Analysis 30 2.2.4.1 Curve Fitting Using Polynomial Functions 30 2.2.5 Derivative 31 2.2.5.1 The Physical Meaning of Derivatives 32 2.2.5.2 Mathematical Expression of Derivatives 33 2.2.5.3 Orders of Derivatives 35 2.2.5.4 Higher-order Derivatives in Engineering Analyses 35 2.2.5.5 The Partial Derivatives 36 2.2.6 Integration 36 2.2.6.1 The Concept of Integration 36 2.2.6.2 Mathematical Expression of Integrals 37 2.3 Applications of Integrals 38 2.3.1 Plane Area by Integration 38 2.3.1.1 Plane Area Bounded by Two Curves 41 2.3.2 Volumes of Solids of Revolution 42 2.3.3 Centroids of Plane Areas 47 2.3.3.1 Centroid of a Solid of Plane Geometry with Straight Edges 49 2.3.3.2 Centroid of a Solid with Plane Geometry Defined by Multiple Functions 50 2.3.4 Average Value of Continuous

Functions 52 2.4 Special Functions for MathematicalModeling 54 2.4.1 Special Functions in Solutions in MathematicalModeling 55 2.4.1.1 The Error Function and Complementary Error Function 55 2.4.1.2 The Gamma Function 56 2.4.1.3 Bessel Functions 56 2.4.2 Special Functions for Particular Physical Phenomena 58 2.4.2.1 Step Functions 58 2.4.2.2 Impulsive Functions 60 2.5 Differential Equations 62 2.5.1 The Laws of Physics for Derivation of Differential Equations 62 2.6 Problems 65 3 Vectors and Vector Calculus 73 Chapter Learning Objectives 73 3.1 Vector and Scalar Quantities 73 3.2 Vectors in Rectangular and Cylindrical Coordinate Systems 75 3.2.1 Position Vectors 75 3.3 Vectors in 2D Planes and 3D Spaces 78 3.4 Vector Algebra 79 3.4.1 Addition of Vectors 79 3.4.2 Subtraction of Vectors 79 3.4.3 Addition and Subtraction of Vectors Using

Unit Vectors in Rectangular Coordinate Systems 80 3.4.4 Multiplication of Vectors 81 3.4.4.1 Scalar Multiplier 81 3.4.4.2 Dot Product 82 3.4.4.3 Cross Product 84 3.4.4.4 Cross Product of Vectors for Plane Areas 86 3.4.4.5 Triple product 86 3.4.4.6 Additional Laws of Vector Algebra 87 3.4.4.7 Use of Triple Product of Vectors for Solid Volume 87 3.5 Vector Calculus 88 3.5.1 Vector Functions 88 3.5.2 Derivatives of Vector Functions 89 3.5.3 Gradient, Divergence,

And Curl 91 3.5.3.1 Gradient 91 3.5.3.2 Divergence 91 3.5.3.3 Curl 91 3.6 Applications of Vector Calculus in Engineering Analysis 92 3.6.1 In Heat Transfer 93 3.6.2 In Fluid Mechanics 93 3.6.3 In Electromagnetism with Maxwell's Equations 94 3.7 Application of Vector Calculus in Rigid Body Dynamics 95 3.7.1 Rigid Body in RectilinearMotion 95 3.7.2 Plane CurvilinearMotion in Rectangular Coordinates 97 3.7.3 Application of Vector Calculus in the Kinematics of Projectiles 100 3.7.4 Plane CurvilinearMotion in Cylindrical Coordinates 103 3.7.5 Plane CurvilinearMotion with Normal and Tangential Components 109 3.8 Problems 114 4 Linear Algebra and Matrices 119 Chapter Learning Objectives 119 4.1 Introduction to Linear Algebra and Matrices 119 4.2 Determinants and Matrices 121 4.2.1 Evaluation of Determinants 121 4.2.2 Matrices in Engineering

Analysis 123 4.3 Different Forms of Matrices 123 4.3.1 Rectangular Matrices 123 4.3.2 Square Matrices 124 4.3.3 Row Matrices 124 4.3.4 Column Matrices 124 4.3.5 Upper Triangular Matrices 124 4.3.6 Lower Triangular Matrices 125 4.3.7 Diagonal Matrices 125 4.3.8 Unit Matrices 125 4.4 Transposition of Matrices 125 4.5 Matrix Algebra 126 4.5.1 Addition and Subtraction of Matrices 126 4.5.2 Multiplication of a Matrix by a Scalar Quantity 127 4.5.3 Multiplication of Two Matrices 127 4.5.4 Matrix Representation of Simultaneous Linear Equations 128 4.5.5 Additional Rules for Multiplication of Matrices 129 4.6 Matrix Inversion,

[A]−1 129 4.7 Solution of Simultaneous Linear Equations 131 4.7.1 The Need for Solving Large Numbers of Simultaneous Linear Equations 131 4.7.2 Solution of Large Numbers of Simultaneous Linear Equations Using the Inverse Matrix Technique 133 4.7.3 Solution of Simultaneous Equations Using the Gaussian Elimination Method 135 4.8 Eigenvalues and Eigenfunctions 141 4.8.1 Eigenvalues and Eigenvectors of Matrices 142 4.8.2 Mathematical Expressions of Eigenvalues and Eigenvectors of Square Matrices 142 4.8.3 Application of Eigenvalues and Eigenfunctions in Engineering Analysis 146 4.9 Problems 148 5 Overview of Fourier Series 151 Chapter Learning Objectives 151 5.1 Introduction 151 5.2 Representing Periodic Functions by Fourier Series 152 5.3 Mathematical Expression of Fourier Series 154 5.4 Convergence of Fourier Series 161 5.5 Convergence of Fourier Series at

Discontinuities 164 5.6 Problems 169 6 Introduction to the Laplace Transform and Applications 171 Chapter Learning Objectives 171 6.1 Introduction 171 6.2 Mathematical Operator of Laplace Transform 172 6.3 Properties of the Laplace Transform 174 6.3.1 Linear Operator Property 174 6.3.2 Shifting Property 175 6.3.3 Change of Scale Property 175 6.4 Inverse Laplace Transform 176 6.4.1 Using the Laplace Transform Tables in Reverse 176 6.4.2 The Partial Fraction Method 176 6.4.3 The Convolution Theorem 178 6.5 Laplace Transform of Derivatives 180 6.5.1 Laplace Transform of Ordinary Derivatives 180 6.5.2 Laplace Transform of Partial Derivatives 181 6.6 Solution of Ordinary Differential Equations Using Laplace Transforms 184 6.6.1 Laplace Transform for Solving Nonhomogeneous Differential Equations 184 6.6.2 Differential Equation for the Bending of Beams

186 6.7 Solution of Partial Differential Equations Using Laplace Transforms 192 6.8 Problems 195 7 Application of First-order Differential Equations in Engineering Analysis 199 Chapter Learning Objectives 199 7.1 Introduction 199 7.2 Solution Methods for First-order Ordinary Differential Equations 200 7.2.1 Solution Methods for Separable Differential Equations 200 7.2.2 Solution of Linear, Homogeneous Equations 201 7.2.3 Solution of Linear, Nonhomogeneous Equations