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by Dennis G. Zill and Michael R. Cullen

ISBN13: 978-0763710651

ISBN10: 0763710652

Edition: 2ND 00

Copyright: 2000

Publisher: Jones & Bartlett Publishers

Published: 2000

International: No

ISBN10: 0763710652

Edition: 2ND 00

Copyright: 2000

Publisher: Jones & Bartlett Publishers

Published: 2000

International: No

Advanced Engineering Mathematics is a compendium of many mathematical topics, all of which are loosely related by the expedient of either being needed or useful in courses and subsequent careers in science and engineering. Consequently, this book represents the most accurate list of what constitutes "engineering mathematics." For flexibility in topic selection, the text is divided into five major sections that illustrate the backbone of science/engineering related mathematics. The first eight chapters of this book constitute a complete short course in ordinary differential equations.

- New to this Edition! The five major sections of the text open with an essay by an acknowledged expert in the field of engineering, this helps to provide students with real-life context to the course material.
- Real-world applications, current examples, and a many illustrations help students visualize important concepts and apply the material to everyday life.
- A complete Solutions Manual is available for the instructor, and a Student Solutions Manual that provides the answers to every third problem.
- Text includes boxed definitions and boxed theorems for easy reference.
- Zill devotes an entire section to Fast Fourier Transforms (FFT), and provides problems using the FFT. Mathematical Models for differential equations are also given special attention.
- Zill has provided Remark sections throughout the text that alert students to certain discussions that require special attention.

Author Bio

**Zill, Dennis G. : Loyola Marymount University**

Dennis G. Zill, Ph.D. in applied mathematics from Iowa State University. Currently professor of mathematics and former chair of the math dept at Loyola Marymount University in Los Angeles.

Part I: Ordinary Differential Equations

Chapter 1. Introduction to Differential Equations

1.1. Definitions and Terminology

1.2. Initial-Value Problems

1.3. Differential Equations as Mathematical Models

Chapter 1. Review Exercises

Chapter 2. First-Order Differential Equations

2.1. Solution Curves Without the Solution

2.2. Separable Variables

2.3. Linear Equations

2.4. Exact Equations

2.5. Solutions by Substitutions

2.6. A Numerical Solutions

2.7. Linear Models

2.8. Nonlinear

2.9. Systems: Linear and Nonlinear Models

Chapter 2. Review Exercises

Chapter 3. Higher-Order Differential Equations

3.1. Preliminary Theory: Linear and Nonlinear Models

3.2. Reduction of Order

3.3. Homogenous Linear Equations with Constant Coefficients

3.4. Undetermined

3.5. Variations of Parameters

3.6. Cauchy-Euler Equation

3.7. Nonlinear Equations

3.8. Linear Models: Initial-Value Problems

3.9. Linear Models: Boundary-Value Problems

3.10. Nonlinear Models

3.11. Solving Systems of Linear Models

Chapter 3. Review Exercises

Chapter 4. The Laplace Transform

4.1. Definition of the Laplace Transform

4.2. The Inverse Transform and Transforms of Derivations

4.3. Translation Theorems

4.4. Additional Operational Properties

4.5. Dirac Delta Function

4.6. Solving Systems of Linear Equations

Chapter 4. Review Exercises

Chapter 5. Series Solutions of Linear Equations

5.1. Solutions about Ordinary Points

5.2. Solutions about Singular Points

5.3. Two Special Equations

Chapter 5. Review Exercises

Chapter 6. Numerical Solutions of Ordinary Differential Equations

6.1. Euler Methods and Error Analysis

6.2. Runge-Kutta Methods

6.3. Methods

6.4. Higher-Order Equations and Systems

6.5. Second-Order Boundary-Value Problems

Part II: Vectors, Matrices, and Vector Calculus

Chapter 7. Vectors

7.1. Vectors in 2-Space

7.2. Vectors in 3-Space

7.3. The Dot Product

7.4. The Cross Product

7.5. Lines and Planes in 3-Space

7.6. Vector Spaces

Chapter 7. Review Exercises

Chapter 8. Matrices

8.1. Matrix Algebra

8.2. Systems of Linear Algebraic Equations

8.3. Rank of a Matrix

8.4. Determinants

8.5. Properties of Determinants

8.6. Inverse of a Matrix

8.7. Cramer's Rule

8.8. The Eigenvalue Problem

8.9. Power of Matrices

8.10. Orthogonal Matrices

8.11. Approximation of Eigenvalues

8.12. Diagonalization

8.13. Cryptography

8.14. An Error-Correcting Code

8.15. Method of Least Squares

8.16. Discrete Compartmental Models

Chapter 8. Review Exercises

Chapter 9. Vector Calculus

9.1. Vector Functions

9.2. Motion on a Curve

9.3. Curvature and Components of Acceleration

9.4. Functions of Several Variables

9.5. The Directional Derivative

9.6. Planes and Normal Lines

9.7. Divergence and Curl

9.8. Line Integrals

9.9. Line Integrals Independent of the Path

9.10. Review of Double Integrals

9.11. Double Integrals in Polar Coordinates

9.12. Green's Theorem

9.13. Surface Integrals

9.14. Strokes' Theorem

9.15. Review of Triple Integrals

9.16. Divergence Theorem

9.17. Change of Variables in Multiple Integrals

Chapter 9. Review Exercises

Part III: Systems of Differential Equations

Chapter 10. System of Linear Differential Equations

10.1. Preliminary

10.2. Homogeneous Linear Systems

10.3. Solution by Diagonalization

10.4. Nonhomogenous Linear Systems

10.5. Matrix Exponential

Chapter 10. Review Exercise

Chapter 11. Systems of Nonlinear Differential Equations

11.1. Autonomous Systems, Critical Points, and Periodic Solutions

11.2. Stability of Linear Systems

11.3. Linearization and Local Stability

11.4. Modeling Using Autonomous Systems

11.5. Periodic Solutions, Limit Cycles, and Global Stability

Chapter 11. Review Exercise

Part IV: Fourier Series and Partial Differential Equations

Chapter 12. Orthogonal Functions and Fourier Series

12.1. Orthogonal Functions

12.2. Fourier Series

12.3. Fourier Cosine and Sine Series

12.4. Complex Fourier Series and Frequency Spectrum

12.5. Sturm-Liouville Problem

12.6. Bessel and Legendre Series

Chapter 12. Review Exercises

Chapter 13. Boundary-Value Problems in Rectangular Coordinates

13.1. Separable Partial Differential Equations

13.2. Classical Equations and Boundary-Value Problems

13.3. Heat Equation

13.4. Wave Equation

13.5. Laplace's Equation

13.6. Nonhomogeneous Equations and Boundary Conditions

13.7. Orthogonal Series Expansions

13.8. Fourier Series in Two Variable

Chapter 13. Review Exercises

Chapter 14. Boundary-Value Problems in Other Coordinate Systems

14.1. Problems Involving Laplace's Equation in Polar Coordinates

14.2. Problems in Polar and Cylindrical Coordinates: Bessel Functions

14.3. Problems in Spherical Coordinates: Legendre Polynomials

Chapter 14. Review Exercises

Chapter 15. Integral Transform Method

15.1. Error Function

15.2. Applications of the Laplace Transform

15.3. Fourier Integral

15.4. Fourier Transforms

15.5. Fast Fourier Transform

Chapter 15. Review Exercises

Chapter 16. Numerical Solutions to Partial Differential Equations

16.1. Elliptic Equations

16.2. Parabolic Equations

16.3. Hyperbolic Equations

Chapter 16. Review Exercises

Part V: Complex Analysis

Chapter 17. Functions of a Complex Variable

17.1. Complex Numbers

17.2. Form of Complex Numbers; Power and Roots

17.3. Sets of Points in the Complex Plane

17.4. Functions of a Complex Variable; Analyticity

17.5. Cauchy-Reimann Equations

17.6. Exponential and Logarithmic Functions

17.7. Trigonometric and Hyperbolic Functions

17.8. Inverse Trigonometric and Hyperbolic Functions

Chapter 17. Review Exercise

Chapter 18. Integration in the Complex Plane

18.1. Contour Integrals

18.2. Cauchy-Goursat Theorem

18.3. Independence of Path

18.4. Cauchy's Integral Formula

Chapter 18. Review Exercises

Chapter 19. Series and Residues

19.1. Sequences and Series

19.2. Taylor Series

19.3. Laurent Series

19.4. Zeros and Poles

19.5. Residues and Residue Theorem

19.6. Evaluation of Real Integrals

Chapter 19. Review Exercises

Chapter 20. Conformal Mappings and Applications

20.1. Complex Functions as Mappings

20.2. Conformal Mapping and the Dirichlet Problem

20.3. Linear Fractional Transformations

20.4. Schwarz-Christoffel Transformations

20.5. Poisson Integral Formulas

20.6. Applications

Chapter 20. Review Exercise

Appendix I Some Derivative and Integral Formulas

Appendix II Gamma Function; Exercises

Appendix III Table of Laplace Transforms

Appendix IV Conformal Mappings

Appendix V Some BASIC Programs for Numerical Methods

Selected Answers for Odd-Numbered Problems

Dennis G. Zill and Michael R. Cullen

ISBN13: 978-0763710651ISBN10: 0763710652

Edition: 2ND 00

Copyright: 2000

Publisher: Jones & Bartlett Publishers

Published: 2000

International: No

Advanced Engineering Mathematics is a compendium of many mathematical topics, all of which are loosely related by the expedient of either being needed or useful in courses and subsequent careers in science and engineering. Consequently, this book represents the most accurate list of what constitutes "engineering mathematics." For flexibility in topic selection, the text is divided into five major sections that illustrate the backbone of science/engineering related mathematics. The first eight chapters of this book constitute a complete short course in ordinary differential equations.

- New to this Edition! The five major sections of the text open with an essay by an acknowledged expert in the field of engineering, this helps to provide students with real-life context to the course material.
- Real-world applications, current examples, and a many illustrations help students visualize important concepts and apply the material to everyday life.
- A complete Solutions Manual is available for the instructor, and a Student Solutions Manual that provides the answers to every third problem.
- Text includes boxed definitions and boxed theorems for easy reference.
- Zill devotes an entire section to Fast Fourier Transforms (FFT), and provides problems using the FFT. Mathematical Models for differential equations are also given special attention.
- Zill has provided Remark sections throughout the text that alert students to certain discussions that require special attention.

Author Bio

**Zill, Dennis G. : Loyola Marymount University**

Dennis G. Zill, Ph.D. in applied mathematics from Iowa State University. Currently professor of mathematics and former chair of the math dept at Loyola Marymount University in Los Angeles.

Table of Contents

Part I: Ordinary Differential Equations

Chapter 1. Introduction to Differential Equations

1.1. Definitions and Terminology

1.2. Initial-Value Problems

1.3. Differential Equations as Mathematical Models

Chapter 1. Review Exercises

Chapter 2. First-Order Differential Equations

2.1. Solution Curves Without the Solution

2.2. Separable Variables

2.3. Linear Equations

2.4. Exact Equations

2.5. Solutions by Substitutions

2.6. A Numerical Solutions

2.7. Linear Models

2.8. Nonlinear

2.9. Systems: Linear and Nonlinear Models

Chapter 2. Review Exercises

Chapter 3. Higher-Order Differential Equations

3.1. Preliminary Theory: Linear and Nonlinear Models

3.2. Reduction of Order

3.3. Homogenous Linear Equations with Constant Coefficients

3.4. Undetermined

3.5. Variations of Parameters

3.6. Cauchy-Euler Equation

3.7. Nonlinear Equations

3.8. Linear Models: Initial-Value Problems

3.9. Linear Models: Boundary-Value Problems

3.10. Nonlinear Models

3.11. Solving Systems of Linear Models

Chapter 3. Review Exercises

Chapter 4. The Laplace Transform

4.1. Definition of the Laplace Transform

4.2. The Inverse Transform and Transforms of Derivations

4.3. Translation Theorems

4.4. Additional Operational Properties

4.5. Dirac Delta Function

4.6. Solving Systems of Linear Equations

Chapter 4. Review Exercises

Chapter 5. Series Solutions of Linear Equations

5.1. Solutions about Ordinary Points

5.2. Solutions about Singular Points

5.3. Two Special Equations

Chapter 5. Review Exercises

Chapter 6. Numerical Solutions of Ordinary Differential Equations

6.1. Euler Methods and Error Analysis

6.2. Runge-Kutta Methods

6.3. Methods

6.4. Higher-Order Equations and Systems

6.5. Second-Order Boundary-Value Problems

Part II: Vectors, Matrices, and Vector Calculus

Chapter 7. Vectors

7.1. Vectors in 2-Space

7.2. Vectors in 3-Space

7.3. The Dot Product

7.4. The Cross Product

7.5. Lines and Planes in 3-Space

7.6. Vector Spaces

Chapter 7. Review Exercises

Chapter 8. Matrices

8.1. Matrix Algebra

8.2. Systems of Linear Algebraic Equations

8.3. Rank of a Matrix

8.4. Determinants

8.5. Properties of Determinants

8.6. Inverse of a Matrix

8.7. Cramer's Rule

8.8. The Eigenvalue Problem

8.9. Power of Matrices

8.10. Orthogonal Matrices

8.11. Approximation of Eigenvalues

8.12. Diagonalization

8.13. Cryptography

8.14. An Error-Correcting Code

8.15. Method of Least Squares

8.16. Discrete Compartmental Models

Chapter 8. Review Exercises

Chapter 9. Vector Calculus

9.1. Vector Functions

9.2. Motion on a Curve

9.3. Curvature and Components of Acceleration

9.4. Functions of Several Variables

9.5. The Directional Derivative

9.6. Planes and Normal Lines

9.7. Divergence and Curl

9.8. Line Integrals

9.9. Line Integrals Independent of the Path

9.10. Review of Double Integrals

9.11. Double Integrals in Polar Coordinates

9.12. Green's Theorem

9.13. Surface Integrals

9.14. Strokes' Theorem

9.15. Review of Triple Integrals

9.16. Divergence Theorem

9.17. Change of Variables in Multiple Integrals

Chapter 9. Review Exercises

Part III: Systems of Differential Equations

Chapter 10. System of Linear Differential Equations

10.1. Preliminary

10.2. Homogeneous Linear Systems

10.3. Solution by Diagonalization

10.4. Nonhomogenous Linear Systems

10.5. Matrix Exponential

Chapter 10. Review Exercise

Chapter 11. Systems of Nonlinear Differential Equations

11.1. Autonomous Systems, Critical Points, and Periodic Solutions

11.2. Stability of Linear Systems

11.3. Linearization and Local Stability

11.4. Modeling Using Autonomous Systems

11.5. Periodic Solutions, Limit Cycles, and Global Stability

Chapter 11. Review Exercise

Part IV: Fourier Series and Partial Differential Equations

Chapter 12. Orthogonal Functions and Fourier Series

12.1. Orthogonal Functions

12.2. Fourier Series

12.3. Fourier Cosine and Sine Series

12.4. Complex Fourier Series and Frequency Spectrum

12.5. Sturm-Liouville Problem

12.6. Bessel and Legendre Series

Chapter 12. Review Exercises

Chapter 13. Boundary-Value Problems in Rectangular Coordinates

13.1. Separable Partial Differential Equations

13.2. Classical Equations and Boundary-Value Problems

13.3. Heat Equation

13.4. Wave Equation

13.5. Laplace's Equation

13.6. Nonhomogeneous Equations and Boundary Conditions

13.7. Orthogonal Series Expansions

13.8. Fourier Series in Two Variable

Chapter 13. Review Exercises

Chapter 14. Boundary-Value Problems in Other Coordinate Systems

14.1. Problems Involving Laplace's Equation in Polar Coordinates

14.2. Problems in Polar and Cylindrical Coordinates: Bessel Functions

14.3. Problems in Spherical Coordinates: Legendre Polynomials

Chapter 14. Review Exercises

Chapter 15. Integral Transform Method

15.1. Error Function

15.2. Applications of the Laplace Transform

15.3. Fourier Integral

15.4. Fourier Transforms

15.5. Fast Fourier Transform

Chapter 15. Review Exercises

Chapter 16. Numerical Solutions to Partial Differential Equations

16.1. Elliptic Equations

16.2. Parabolic Equations

16.3. Hyperbolic Equations

Chapter 16. Review Exercises

Part V: Complex Analysis

Chapter 17. Functions of a Complex Variable

17.1. Complex Numbers

17.2. Form of Complex Numbers; Power and Roots

17.3. Sets of Points in the Complex Plane

17.4. Functions of a Complex Variable; Analyticity

17.5. Cauchy-Reimann Equations

17.6. Exponential and Logarithmic Functions

17.7. Trigonometric and Hyperbolic Functions

17.8. Inverse Trigonometric and Hyperbolic Functions

Chapter 17. Review Exercise

Chapter 18. Integration in the Complex Plane

18.1. Contour Integrals

18.2. Cauchy-Goursat Theorem

18.3. Independence of Path

18.4. Cauchy's Integral Formula

Chapter 18. Review Exercises

Chapter 19. Series and Residues

19.1. Sequences and Series

19.2. Taylor Series

19.3. Laurent Series

19.4. Zeros and Poles

19.5. Residues and Residue Theorem

19.6. Evaluation of Real Integrals

Chapter 19. Review Exercises

Chapter 20. Conformal Mappings and Applications

20.1. Complex Functions as Mappings

20.2. Conformal Mapping and the Dirichlet Problem

20.3. Linear Fractional Transformations

20.4. Schwarz-Christoffel Transformations

20.5. Poisson Integral Formulas

20.6. Applications

Chapter 20. Review Exercise

Appendix I Some Derivative and Integral Formulas

Appendix II Gamma Function; Exercises

Appendix III Table of Laplace Transforms

Appendix IV Conformal Mappings

Appendix V Some BASIC Programs for Numerical Methods

Selected Answers for Odd-Numbered Problems

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