PART A Ordinary Differential Equations (ODEs)

CHAPTER 1 First-Order ODEs

1.1 Basic Concepts. Modeling

1.2 Geometric Meaning of y' = ¿(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

17.4 Conformal Mapping by Other Functions

17.5 Riemann Surfaces. Optional

Sustainability and Ethical Considerations

Chapter 17 Review Questions and Problems

Summary of Chapter 17

CHAPTER 18 Complex Analysis and Potential Theory

18.1 Electrostatic Fields

18.2 Use of...