Foundation Engineering Mathematics

Mathematics plays a central role in modern culture, and a basic understanding of the nature of mathematics is required for scientific literacy. This new textbook will prepare readers to continue to develop analytical and numerical skills through the study of a variety of mathematical techniques. The statistical element of this textbook enhances readers’ ability to organise and interpret data. Most of the topics covered in this textbook are widely used in various areas of engineering, including industrial engineering, to analyse complex systems, optimise processes and make informed decisions to improve efficiency, productivity and reliability in various industrial settings. From the complexities of double integration and ordinary differential equations to the complexities of linear systems of differential equations, Fourier series and Laplace transform, Foundation Engineering Mathematics unfolds with careful attention to detail, offering readers a structured approach to mastering these fundamental topics. Each chapter of this book is carefully presented to provide a balance between theoretical foundations and practical applications, ensuring that readers not only grasp the underlying principles but also appreciate their relevance in real-world engineering scenarios. Each chapter is accompanied by practical examples, illustrative diagrams and engineering applications to reinforce understanding and demonstrate the relevance of mathematical concepts in engineering practice. Whether you’re a student embarking on your journey into the world of mathematics or an experienced engineer seeking to deepen your understanding of mathematical concepts, this book serves as an invaluable resource, guiding you through the complexities of mathematical theory and its engineering applications. A solutions manual and a set of PowerPoint slides are available for qualified textbook adoptions. Cover Half Title Series Title Copyright Contents Preface About the Authors Chapter 1 Double Integration 1.1 Introduction 1.2 Double Integral over Rectangular Regions 1.2.1 Exercise 1.3 Double Integrals over Non-Rectangular Regions 1.4 Reversing the Order of Integration 1.4.1 Exercises 1.5 Double Integration over Polar Rectangular Regions 1.6 Double Integration over General Polar Regions 1.6.1 Exercises 1.7 Engineering Applications 1.7.1 Density Distributions of a Two-Dimensional Thin Layer 1.7.2 Centre of Mass of a Two-Dimensional Thin Layer (Lamina) 1.7.3 Moment of Inertia of a Two-Dimensional Thin Layer (Lamina) 1.7.4 Exercises Chapter 2 Ordinary Differential Equations 2.1 Introduction 2.2 Vibrating Spring 2.3 ODEs – The Basics 2.4 Separable ODEs 2.4.1 Exercises 2.5 Linear Equations 2.6 First-Order Linear Equations 2.6.1 Exercises 2.7 Second-Order Linear ODEs with Constant Coefficients 2.7.1 Homogeneous Equations with Constant Coefficients 2.7.2 Exercises 2.8 Homogeneous Equations, the Initial Value Problems 2.8.1 Exercise 2.9 Non-Homogeneous Equations 2.9.1 The Method of Undetermined Coefficients 2.9.2 Exercises 2.10 Numerical Solutions of Differential Equations 2.10.1 Euler Method 2.10.2 Implementation of the Euler Method Using Excel 2.10.3 Exercises 2.11 Engineering Applications 2.11.1 Falling Objects 2.11.2 Mixture of Solutions 2.11.3 Vibration of a Spring-Mass System 2.11.4 Fluid Flow Streamlines 2.11.5 Heat Transfer – Fourier’s Law of Heat Conduction 2.11.6 Electrical Circuit 2.11.7 Exercises Chapter 3 Laplace Transform 3.1 Introduction 3.2 Definition of the Laplace Transform 3.3 Linear Property of the Laplace Transform 3.4 Inverse Laplace Transform 3.5 The First Shifting Theorem 3.5.1 Exercises 3.6 Inverse Laplace Transform Using Completing the Square 3.6.1 Exercises 3.7 Derivatives and the Laplace Transform 3.7.1 Using Laplace Transform to Solve ODEs 3.7.2 Exercises 3.8 The Unit Step Function 3.8.1 Products Involving Unit Step Functions 3.8.2 Laplace Transform of the Unit Step Function 3.9 The Second Shifting Theorem 3.9.1 Exercises 3.10 Dirac Delta Function 3.10.1 Exercises 3.11 Convolution 3.11.1 Visual Explanation 3.11.2 Some Useful Property of Convolution 3.11.3 Laplace Transform of f * g 3.11.4 Exercises 3.12 Application in Control Engineering 3.13 Table of Laplace Transforms Chapter 4 Linear Systems of Differential Equations 4.1 Introduction 4.2 Eigenvalues 4.3 Eigenvectors 4.3.1 Exercises 4.4 Repeated Eigenvalues 4.5 Complex Eigenvalues and Associated Eigenvectors 4.5.1 Exercises 4.6 Diagonalisation of a Matrix 4.6.1 Exercises 4.7 Homogeneous Linear Systems 4.7.1 Solving 2 × 2 Systems (Distinct Real Eigenvalues) 4.7.2 Solving 2 × 2 Systems (Algebraic Multiplicity 2, Geometric Multiplicity 1) 4.7.3 Solving 2 × 2 Second-Order Systems 4.7.4 Solving 3 × 3 Systems (3 Distinct Real Eigenvalues) 4.7.5 Solving 3 × 3 Systems (Algebraic Multiplicity 2 and Geometric Multiplicity 2) 4.7.6 Solving 3 × 3 Systems (Algebraic Multiplicity 2 and Geometric Multiplicity 1) 4.7.7 Solving 3 × 3 Systems (Algebraic Multiplicity 3 and Geometric Multiplicity 2) 4.7.8 Solving 3 × 3 Systems (Algebraic Multiplicity 3 and Geometric Multiplicity 1) 4.7.9 Exercises 4.8 Systems with Complex Eigenvalues 4.8.1 Exercises 4.9 Engineering Applications 4.9.1 A Two-Tank Mixing Problem 4.9.2 A Three-Tank Mixing Problem 4.9.3 Modelling 2 Degrees-of-Freedom Spring-Mass-Damper System Chapter 5 Fourier Series 5.1 Introduction 5.2 Periodic Functions 5.3 Adding Periodic Signals with Unequal Frequencies 5.4 Adding Signals When Frequencies Are Integer Multiple of Smallest Frequency 5.5 The Beat Phenomenon 5.6 Frequency Domain (Frequency Spectrum) 5.6.1 Exercises 5.7 Fourier Series of the Square Wave 5.8 Fourier Series of Periodic Functions 5.9 Frequency Spectrum of the Fourier Series 5.9.1 Exercises 5.10 Engineering Applications 5.10.1 Vibration Analysis 5.10.2 Voltage Output of a Rectifier 5.10.3 Exercises Chapter 6 Statistics 6.1 Introduction 6.2 Population 6.3 Sample 6.4 Random Variable 6.5 Qualitative Variables 6.6 Quantitative Variables 6.7 Descriptive Statistics 6.7.1 Summarising Data 6.7.2 Frequency Distribution (Discrete Variable) 6.8 The Mean, Standard Deviation and Median 6.8.1 Exercises 6.9 Continuous Quantitative Variables and Probability Distribution 6.10 Normal Distribution 6.10.1 Standard Normal Distribution 6.10.2 Exercises 6.10.3 Standardising the Random Variable X 6.10.4 Exercises 6.10.5 Calculating the X-Value by Knowing the Probability 6.10.6 Exercises 6.11 Estimation and Confidence Intervals 6.11.1 Point Estimation 6.11.2 Distribution of the Sample Mean 6.11.3 Exercises 6.11.4 Interval Estimates 6.11.5 Exercises 6.11.6 The t-Distribution 6.11.7 Exercises 6.12 Hypothesis Testing 6.12.1 Exercises 6.13 Correlation and Regression 6.13.1 Introduction 6.13.2 Correlation 6.13.3 Regression 6.13.4 Exercises Index Mathematics plays a central role in modern culture, and a basic understanding of the nature of mathematics is required for scientific literacy. This new textbook will prepare readers to continue to develop analytical and numerical skills through the study of a variety of mathematical techniques. The statistical element of this textbook enhances the readers’ ability to organize and interpret data. Most of the topics covered in this textbook are widely used in various areas of engineering, including industrial engineering, to analyze complex systems, optimize processes and make informed decisions to improve efficiency, productivity and reliability in various industrial settings. From the complexities of double integration and ordinary differential equations to the complexities of linear systems of differential equations, Fourier series and Laplace transform, Foundation Engineering Mathematics unfolds with careful attention to detail, offering readers a structured approach to mastering these fundamental topics. Each chapter book is carefully presented to provide a balance between theoretical foundations and practical applications, ensuring that readers not only grasp the underlying principles but also appreciate their relevance in real-world engineering scenarios. Each chapter is accompanied by practical examples, illustrative diagrams and engineering applications to reinforce understanding and demonstrate the relevance of mathematical concepts in engineering practice. Whether you're a student embarking on your journey into the world of mathematics or a experienced engineer seeking to deepen your understanding of mathematical concepts, this book serves as an invaluable resource, guiding you through the complexities of mathematical theory and its engineering applications. A solutions manual and a set of PowerPoint slides are available for qualified textbook adoptions.