Advanced Engineering
Mathematics, Second Edition
Dennis G. Zill, Loyola Marymount
University, Michael R. Cullen, Late, Loyola
Marymount University
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
Selected Answers for Odd-Numbered Problems