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
www.crcpress.com/9780367432768.
Author(s): Brian Vick
Publisher: CRC Press
Year: 2020
Language: English
Pages: xiv+232
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
