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TABLE OF CONTENTS
PART A Ordinary Differential Equations (ODEs)
CHAPTER 1 First-Order ODEs
1.1 Basic Concepts. Modeling
1.2 Geometric Meaning of y'' = f( x , y ). Direction Fields, Euler''s Method
1.3 Separable ODEs. Modeling
1.4 Exact ODEs. Integrating Factors
1.5 Linear ODEs. Bernoulli Equation. Population Dynamics
1.6 Orthogonal Trajectories. Optional
1.7 Existence and Uniqueness of Solutions for Initial Value Problems Sustainability and Ethical Considerations
Chapter 1 Review Questions and Problems Summary of
Chapter 1
CHAPTER 2 Second-Order Linear ODEs
2.1 Homogeneous Linear ODEs of Second Order
2.2 Homogeneous Linear ODEs with Constant Coefficients
2.3 Differential Operators. Optional
2.4 Modeling of Free Oscillations of a Mass-Spring System
2.5 Euler-Cauchy Equations
2.6 Existence and Uniqueness of Solutions. Wronskian
2.7 Nonhomogeneous ODEs
2.8 Modeling: Forced Oscillations. Resonance
2.9 Modeling: Electric Circuits
2.10 Solution by Variation of Parameters Sustainability and Ethical Considerations
Chapter 2 Review Questions and Problems Summary of
Chapter 2
CHAPTER 3 Higher Order Linear ODEs
3.1 Homogeneous Linear ODEs
3.2 Homogeneous Linear ODEs with Constant Coefficients
3.3 Nonhomogeneous Linear ODEs Sustainability and Ethical Considerations
Chapter 3 Review Questions and Problems Summary of
Chapter 3
CHAPTER 4 Systems of ODEs. Phase Plane. Qualitative Methods
4.0 For Reference: Basics of Matrices and Vectors
4.1 Systems of ODEs as Models in Engineering Applications
4.2 Basic Theory of Systems of ODEs. Wronskian
4.3 Constant-Coefficient Systems. Phase Plane Method
4.4 Criteria for Critical Points. Stability
4.5 Qualitative Methods for Nonlinear Systems
4.6 Nonhomogeneous Linear Systems of ODEs Sustainability and Ethical Considerations
Chapter 4 Review Questions and Problems Summary of
Chapter 4
CHAPTER 5 Series Solutions of ODEs. Special Functions
5.1 Power Series Method
5.2 Legendre''s Equation. Legendre Polynomials (x)
5.3 Extended Power Series Method: Frobenius Method
5.4 Bessel''s Equation. Bessel Functions (x)
5.5 Bessel Functions of the ( x ). General Solution Sustainability and Ethical Considerations
Chapter 5 Review Questions and Problems Summary of
Chapter 5
CHAPTER 6 Laplace Transforms
6.1 Laplace Transform. Linearity. First Shifting Theorem ( s -Shifting)
6.2 Transforms of Derivatives and Integrals. ODEs
6.3 Unit Step Function (Heaviside Function). Second Shifting Theorem ( t -Shifting)
6.4 Short Impulses. Dirac''s Delta Function. Partial Fractions
6.5 Convolution. Integral Equations
6.6 Differentiation and Integration of Transforms. ODEs with Variable Coefficients
6.7 Systems of ODEs
6.8 Laplace Transform: General Formulas
6.9 Table of Laplace Transforms Sustainability and Ethical Considerations
Chapter 6 Review Questions and Problems Summary of
Chapter 6
PART B Linear Algebra. Vector Calculus
CHAPTER 7 Linear Algebra: Matrices, Vectors, Determinants. Linear Systems
7.1 Matrices, Vectors: Addition and Scalar Multiplication
7.2 Matrix Multiplication
7.3 Linear Systems of Equations. Gauss Elimination
7.4 Linear Independence. Rank of a Matrix. Vector Space
7.5 Solutions of Linear Systems: Existence, Uniqueness
7.6 For Reference: Second- and Third-Order Determinants
7.7 Determinants. Cramer''s Rule
7.8 Inverse of a Matrix. Gauss-Jordan Elimination
7.9 Vector Spaces, Inner Product Spaces. Linear Transformations. Optional Sustainability and Ethical Considerations
Chapter 7 Review Questions and Problems Summary of
Chapter 7
CHAPTER 8 Linear Algebra: Matrix Eigenvalue Problems
8.1 The Matrix Eigenvalue Problem. Determining Eigenvalues and Eigenvectors
8.2 Some Applications of Eigenvalue Problems
8.3 Symmetric, Skew-Symmetric, and Orthogonal Matrices
8.4 Eigenbases. Diagonalization. Quadratic Forms
8.5 Complex Matrices and Forms. Optional Sustainability and Ethical Considerations
Chapter 8 Review Questions and Problems Summary of
Chapter 8
CHAPTER 9 Vector Differential Calculus. Grad, Div, Curl
9.1 Vectors in 2-Space and 3-Space
9.2 Inner Product (Dot Product)
9.3 Vector Product (Cross Product)
9.4 Vector and Scalar Functions and Their Fields. Vector Calculus: Derivatives
9.5 Curves. Arc Length. Curvature. Torsion
9.6 Calculus Review: Functions of Several Variables. Optional
9.7 Gradient of a Scalar Field. Directional Derivative
9.8 Divergence of a Vector Field
9.9 Curl of a Vector Field Sustainability and Ethical Considerations
Chapter 9 Review Questions and Problems Summary of
Chapter 9
CHAPTER 10 Vector Integral Calculus. Integral Theorems
10.1 Line Integrals
10.2 Path Independence of Line Integrals
10.3 Calculus Review: Double Integrals. Optional
10.4 Green''s Theorem in the Plane
10.5 Surfaces for Surface Integrals
10.6 Surface Integrals
10.7 Triple Integrals. Divergence Theorem of Gauss
10.8 Further Applications of the Divergence Theorem
10.9 Stokes''s Theorem Sustainability and Ethical Considerations
Chapter 10 Review Questions and Problems Summary of
Chapter 10
PART C Fourier Analysis. Partial Differential Equations (PDEs)
CHAPTER 11 Fourier Analysis
11.1 Fourier Series
11.2 Arbitrary Period. Even and Odd Functions. Half-Range Expansions
11.3 Forced Oscillations
11.4 Approximation by Trigonometric Polynomials
11.5 Sturm-Liouville Problems. Orthogonal Functions
11.6 Orthogonal Series. Generalized Fourier Series
11.7 Fourier Integral
11.8 Fourier Cosine and Sine Transforms
11.9 Fourier Transform. Discrete and Fast Fourier Transforms
11.10 Tables of Transforms Sustainability and Ethical Considerations
Chapter 11 Review Questions and Problems Summary of
Chapter 11
CHAPTER 12 Partial Differential Equations (PDEs)
12.1 Basic Concepts of PDEs
12.2 Modeling: Vibrating String, Wave Equation
12.3 Solution by Separating Variables. Use of Fourier Series
12.4 D''Alembert''s Solution of the Wave Equation. Characteristics
12.5 Modeling: Heat Flow from a Body in Space. Heat Equation
12.6 Heat Equation: Solution by Fourier Series. Steady Two-Dimensional Heat Problems. Dirichlet Problem
12.7 Heat Equation: Modeling Very Long Bars. Solution by Fourier Integrals and Transforms
12.8 Modeling: Membrane, Two-Dimensional Wave Equation
12.9 Rectangular Membrane. Double Fourier Series
12.10 Laplacian in Polar Coordinates. Circular Membrane. Fourier-Bessel Series
12.11 Laplace''s Equation in Cylindrical and Spherical Coordinates. Potential
12.12 Solution of PDEs by Laplace Transforms Sustainability and Ethical Considerations
Chapter 12 Review Questions and Problems Summary of
Chapter 12
PART D Complex Analysis
CHAPTER 13 Complex Numbers and Functions. Complex Differentiation
13.1 Complex Numbers and Their Geometric Representation
13.2 Polar Form of Complex Numbers. Powers and Roots
13.3 Derivative. Analytic Function
13.4 Cauchy-Riemann Equations. Laplace''s Equation
13.5 Exponential Function
13.6 Trigonometric and Hyperbolic Functions. Euler''s Formula
13.7 Logarithm. General Power. Principal Value Sustainability and Ethical Considerations
Chapter 13 Review Questions and Problems Summary of
Chapter 13
CHAPTER 14 Complex Integration
14.1 Line Integral in the Complex Plane
14.2 Cauchy''s Integral Theorem
14.3 Cauchy''s Integral Formula
14.4 Derivatives of Analytic Functions Sustainability and Ethical Considerations
Chapter 14 Review Questions and Problems Summary of
Chapter 14
CHAPTER 15 Power Series, Taylor Series
15.1 Sequences, Series, Convergence Tests
15.2 Power Series
15.3 Functions Given by Power Series
15.4 Taylor and Maclaurin Series
15.5 Uniform Convergence. Optional Sustainability and Ethical Considerations
Chapter 15 Review Questions and Problems Summary of
Chapter 15
CHAPTER 16 Laurent Series. Residue Integration
16.1 Laurent Series
16.2 Singularities and Zeros. Infinity
16.3 Residue Integration Method
16.4 Residue Integration of Real Integrals Sustainability and Ethical Considerations
Chapter 16 Review Questions and Problems Summary of
Chapter 16
CHAPTER 17 Conformal Mapping
17.1 Geometry of Analytic Functions: Conformal Mapping
17.2 Linear Fractional Transformations (Möbius Transformations)
17.3 Special Linear Fractional Transformations 17