Advanced Engineering Mathematics, International Adaptation

Autorentext

ERWIN KREYSZIG, Professor of Mathematics, Ohio State University, Columbus, Ohio.

Klappentext

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.

Inhalt

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' = (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. Do…