Foundation Engineering Mathematics

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