Includes bibliographical references and index.

PART A Ordinary Differential Equations (ODEs)



CHAPTER 1 First-Order ODEs

1.1 Basic Concepts. Modeling

1.2 Geometric Meaning of y’ = ƒ(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 Conformal Mapping. Modeling

18.3 Heat Problems

18.4 Fluid Flow

18.5 Poisson’s Integral Formula for Potentials

18.6 General Properties of Harmonic Functions. Uniqueness Theorem for the Dirichlet Problem

Sustainability and Ethical Considerations

Chapter 18 Review Questions and Problems

Summary of Chapter 18



PART E Numeric Analysis

Software



CHAPTER 19 Numerics in General

19.1 Introduction

19.2 Solution of Equations by Iteration

19.3 Interpolation

19.4 Spline Interpolation

19.5 Numeric Integration and Differentiation

Sustainability and Ethical Considerations

Chapter 19 Review Questions and Problems

Summary of Chapter 19



CHAPTER 20 Numeric Linear Algebra

20.1 Linear Systems: Gauss Elimination

20.2 Linear Systems: LU-Factorization, Matrix Inversion

20.3 Linear Systems: Solution by Iteration

20.4 Linear Systems: Ill-Conditioning, Norms

20.5 Least Squares Method

20.6 Matrix Eigenvalue Problems: Introduction

20.7 Inclusion of Matrix Eigenvalues

20.8 Power Method for Eigenvalues

20.9 Tridiagonalization and QR-Factorization

Sustainability and Ethical Considerations

Chapter 20 Review Questions and Problems

Summary of Chapter 20



CHAPTER 21 Numerics for ODEs and PDEs

21.1 Methods for First-Order ODEs

21.2 Multistep Methods

21.3 Methods for Systems and Higher Order ODEs

21.4 Methods for Elliptic PDEs

21.5 Neumann and Mixed Problems. Irregular Boundary

21.6 Methods for Parabolic PDEs

21.7 Method for Hyperbolic PDEs

Sustainability and Ethical Considerations

Chapter 21 Review Questions and Problems

Summary of Chapter 21



PART F Optimization, Graphs



CHAPTER 22 Unconstrained Optimization. Linear Programming

22.1 Basic Concepts. Unconstrained Optimization: Method of Steepest Descent

22.2 Linear Programming

22.3 Simplex Method

22.4 Simplex Method: Difficulties

Sustainability and Ethical Considerations

Chapter 22 Review Questions and Problems

Summary of Chapter 22



CHAPTER 23 Graphs. Combinatorial Optimization

23.1 Graphs and Digraphs

23.2 Shortest Path Problems. Complexity

23.3 Bellman’s Principle. Dijkstra’s Algorithm

23.4 Shortest Spanning Trees: Greedy Algorithm

23.5 Shortest Spanning Trees: Prim’s Algorithm

23.6 Flows in Networks

23.7 Maximum Flow: Ford–Fulkerson Algorithm

23.8 Bipartite Graphs. Assignment Problems

Sustainability and Ethical Considerations

Chapter 23 Review Questions and Problems

Summary of Chapter 23



PART G Probability, Statistics (available online)

Software



CHAPTER 24 Data Analysis. Probability Theory

24.1 Data Representation. Average. Spread

24.2 Experiments, Outcomes, Events

24.3 Probability

24.4 Permutations and Combinations

24.5 Random Variables. Probability Distributions

24.6 Mean and Variance of a Distribution

24.7 Binomial, Poisson, and Hypergeometric Distributions

24.8 Normal Distribution

24.9 Distributions of Several Random Variables

Sustainability and Ethical Considerations

Chapter 24 Review Questions and Problems

Summary of Chapter 24



CHAPTER 25 Mathematical Statistics

25.1 Introduction. Random Sampling

25.2 Point Estimation of Parameters

25.3 Confidence Intervals

25.4 Testing Hypotheses. Decisions

25.5 Quality Control

25.6 Acceptance Sampling

25.7 Goodness of Fit. χ2-Test

25.8 Nonparametric Tests

25.9 Regression. Fitting Straight Lines. Correlation

Sustainability and Ethical Considerations

Chapter 25 Review Questions and Problems

Summary of Chapter 25



PROJECTs



APPENDIX 1 References



APPENDIX 2 Answers to Odd-Numbered Problems (available online)



APPENDIX 3 Auxiliary Material

A3.1 Formulas for Special Functions

A3.2 Partial Derivatives

A3.3 Sequences and Series

A3.4 Grad, Div, Curl, in Curvilinear Coordinates



APPENDIX 4 Additional Proofs



APPENDIX 5 Tables



APPENDIX 6 Emerging Topics in Applied Mathematics



INDEX

Advanced Engineering Mathematics, 11th Edition, is known for its comprehensive coverage, careful and correct mathematics, outstanding exercises, and self-contained subject matter parts for maximum flexibility. It opens with ordinary differential equations and ends with the topic of mathematical statistics. The analysis chapters address: Fourier analysis and partial differential equations, complex analysis, and numeric analysis. The book is written by a pioneer in the field of applied mathematics. This comprehensive volume is designed to equip students and professionals with the mathematical tools necessary to tackle complex engineering challenges and drive innovation.



This edition of the text maintains those aspects of the previous editions that have led to the book being so successful. In addition to introducing a new appendix on emerging topics in applied mathematics, each chapter now features a dedicated section on how mathematical modeling and engineering can address environmental and societal challenges, promoting sustainability and ethical practices. This edition includes a revision of the problem sets, making them even more effective, useful, and up-to-date by adding the problems on open-source mathematical software.