O'Reilly logo

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

Advanced Engineering Mathematics, 6th Edition

Book Description

Modern and comprehensive, the new sixth edition of Zill’s Advanced Engineering Mathematics is a full compendium of topics that are most often covered in engineering mathematics courses, and is extremely flexible to meet the unique needs of courses ranging from ordinary differential equations to vector calculus. A key strength of this best-selling text is Zill’s emphasis on differential equation as mathematical models, discussing the constructs and pitfalls of each.

Table of Contents

  1. Cover Page
  2. Title Page
  3. Copyright Page
  4. Contents
  5. Preface
  6. PART 1 Ordinary Differential Equations
    1. 1 Introduction to Differential Equations
      1. 1.1 Definitions and Terminology
      2. 1.2 Initial-Value Problems
      3. 1.3 Differential Equations as Mathematical Models
      4. Chapter 1 in Review
    2. 2 First-Order Differential Equations
      1. 2.1 Solution Curves Without a Solution
        1. 2.1.1 Direction Fields
        2. 2.1.2 Autonomous First-Order DEs
      2. 2.2 Separable Equations
      3. 2.3 Linear Equations
      4. 2.4 Exact Equations
      5. 2.5 Solutions by Substitutions
      6. 2.6 A Numerical Method
      7. 2.7 Linear Models
      8. 2.8 Nonlinear Models
      9. 2.9 Modeling with Systems of First-Order DEs
      10. Chapter 2 in Review
    3. 3 Higher-Order Differential Equations
      1. 3.1 Theory of Linear Equations
        1. 3.1.1 Initial-Value and Boundary-Value Problems
        2. 3.1.2 Homogeneous Equations
        3. 3.1.3 Nonhomogeneous Equations
      2. 3.2 Reduction of Order
      3. 3.3 Homogeneous Linear Equations with Constant Coefficients
      4. 3.4 Undetermined Coefficients
      5. 3.5 Variation of Parameters
      6. 3.6 Cauchy–Euler Equations
      7. 3.7 Nonlinear Equations
      8. 3.8 Linear Models: Initial-Value Problems
        1. 3.8.1 Spring/Mass Systems: Free Undamped Motion
        2. 3.8.2 Spring/Mass Systems: Free Damped Motion
        3. 3.8.3 Spring/Mass Systems: Driven Motion
        4. 3.8.4 Series Circuit Analogue
      9. 3.9 Linear Models: Boundary-Value Problems
      10. 3.10 Green’s Functions
        1. 3.10.1 Initial-Value Problems
        2. 3.10.2 Boundary-Value Problems
      11. 3.11 Nonlinear Models
      12. 3.12 Solving Systems of Linear Equations
      13. Chapter 3 in Review
    4. 4 The Laplace Transform
      1. 4.1 Definition of the Laplace Transform
      2. 4.2 The Inverse Transform and Transforms of Derivatives
        1. 4.2.1 Inverse Transforms
        2. 4.2.2 Transforms of Derivatives
      3. 4.3 Translation Theorems
        1. 4.3.1 Translation on the s-axis
        2. 4.3.2 Translation on the t-axis
      4. 4.4 Additional Operational Properties
        1. 4.4.1 Derivatives of Transforms
        2. 4.4.2 Transforms of Integrals
        3. 4.4.3 Transform of a Periodic Function
      5. 4.5 The Dirac Delta Function
      6. 4.6 Systems of Linear Differential Equations
      7. Chapter 4 in Review
    5. 5 Series Solutions of Linear Differential Equations
      1. 5.1 Solutions about Ordinary Points
        1. 5.1.1 Review of Power Series
        2. 5.1.2 Power Series Solutions
      2. 5.2 Solutions about Singular Points
      3. 5.3 Special Functions
        1. 5.3.1 Bessel Functions
        2. 5.3.2 Legendre Functions
      4. Chapter 5 in Review
    6. 6 Numerical Solutions of Ordinary Differential Equations
      1. 6.1 Euler Methods and Error Analysis
      2. 6.2 Runge–Kutta Methods
      3. 6.3 Multistep Methods
      4. 6.4 Higher-Order Equations and Systems
      5. 6.5 Second-Order Boundary-Value Problems
      6. Chapter 6 in Review
  7. PART 2 Vectors, Matrices, and Vector Calculus
    1. 7 Vectors
      1. 7.1 Vectors in 2-Space
      2. 7.2 Vectors in 3-Space
      3. 7.3 Dot Product
      4. 7.4 Cross Product
      5. 7.5 Lines and Planes in 3-Space
      6. 7.6 Vector Spaces
      7. 7.7 Gram–Schmidt Orthogonalization Process
      8. Chapter 7 in Review
    2. 8 Matrices
      1. 8.1 Matrix Algebra
      2. 8.2 Systems of Linear Algebraic Equations
      3. 8.3 Rank of a Matrix
      4. 8.4 Determinants
      5. 8.5 Properties of Determinants
      6. 8.6 Inverse of a Matrix
        1. 8.6.1 Finding the Inverse
        2. 8.6.2 Using the Inverse to Solve Systems
      7. 8.7 Cramer’s Rule
      8. 8.8 The Eigenvalue Problem
      9. 8.9 Powers of Matrices
      10. 8.10 Orthogonal Matrices
      11. 8.11 Approximation of Eigenvalues
      12. 8.12 Diagonalization
      13. 8.13 LU-Factorization
      14. 8.14 Cryptography
      15. 8.15 An Error-Correcting Code
      16. 8.16 Method of Least Squares
      17. 8.17 Discrete Compartmental Models
      18. Chapter 8 in Review
    3. 9 Vector Calculus
      1. 9.1 Vector Functions
      2. 9.2 Motion on a Curve
      3. 9.3 Curvature and Components of Acceleration
      4. 9.4 Partial Derivatives
      5. 9.5 Directional Derivative
      6. 9.6 Tangent Planes and Normal Lines
      7. 9.7 Curl and Divergence
      8. 9.8 Line Integrals
      9. 9.9 Independence of the Path
      10. 9.10 Double Integrals
      11. 9.11 Double Integrals in Polar Coordinates
      12. 9.12 Green’s Theorem
      13. 9.13 Surface Integrals
      14. 9.14 Stokes’ Theorem
      15. 9.15 Triple Integrals
      16. 9.16 Divergence Theorem
      17. 9.17 Change of Variables in Multiple Integrals
      18. Chapter 9 in Review
  8. PART 3 Systems of Differential Equations
    1. 10 Systems of Linear Differential Equations
      1. 10.1 Theory of Linear Systems
      2. 10.2 Homogeneous Linear Systems
        1. 10.2.1 Distinct Real Eigenvalues
        2. 10.2.2 Repeated Eigenvalues
        3. 10.2.3 Complex Eigenvalues
      3. 10.3 Solution by Diagonalization
      4. 10.4 Nonhomogeneous Linear Systems
        1. 10.4.1 Undetermined Coefficients
        2. 10.4.2 Variation of Parameters
        3. 10.4.3 Diagonalization
      5. 10.5 Matrix Exponential
      6. Chapter 10 in Review
    2. 11 Systems of Nonlinear Differential Equations
      1. 11.1 Autonomous Systems
      2. 11.2 Stability of Linear Systems
      3. 11.3 Linearization and Local Stability
      4. 11.4 Autonomous Systems as Mathematical Models
      5. 11.5 Periodic Solutions, Limit Cycles, and Global Stability
      6. Chapter 11 in Review
  9. PART 4 Partial Differential Equations
    1. 12 Orthogonal Functions and Fourier Series
      1. 12.1 Orthogonal Functions
      2. 12.2 Fourier Series
      3. 12.3 Fourier Cosine and Sine Series
      4. 12.4 Complex Fourier Series
      5. 12.5 Sturm–Liouville Problem
      6. 12.6 Bessel and Legendre Series
        1. 12.6.1 Fourier–Bessel Series
        2. 12.6.2 Fourier–Legendre Series
      7. Chapter 12 in Review
    2. 13 Boundary-Value Problems in Rectangular Coordinates
      1. 13.1 Separable Partial Differential Equations
      2. 13.2 Classical PDEs and Boundary-Value Problems
      3. 13.3 Heat Equation
      4. 13.4 Wave Equation
      5. 13.5 Laplace’s Equation
      6. 13.6 Nonhomogeneous Boundary-Value Problems
      7. 13.7 Orthogonal Series Expansions
      8. 13.8 Fourier Series in Two Variables
      9. Chapter 13 in Review
    3. 14 Boundary-Value Problems in Other Coordinate Systems
      1. 14.1 Polar Coordinates
      2. 14.2 Cylindrical Coordinates
      3. 14.3 Spherical Coordinates
      4. Chapter 14 in Review
    4. 15 Integral Transform Method
      1. 15.1 Error Function
      2. 15.2 Applications of the Laplace Transform
      3. 15.3 Fourier Integral
      4. 15.4 Fourier Transforms
      5. 15.5 Fast Fourier Transform
      6. Chapter 15 in Review
    5. 16 Numerical Solutions of Partial Differential Equations
      1. 16.1 Laplace’s Equation
      2. 16.2 Heat Equation
      3. 16.3 Wave Equation
      4. Chapter 16 in Review
  10. PART 5 Complex Analysis
    1. 17 Functions of a Complex Variable
      1. 17.1 Complex Numbers
      2. 17.2 Powers and Roots
      3. 17.3 Sets in the Complex Plane
      4. 17.4 Functions of a Complex Variable
      5. 17.5 Cauchy–Riemann Equations
      6. 17.6 Exponential and Logarithmic Functions
      7. 17.7 Trigonometric and Hyperbolic Functions
      8. 17.8 Inverse Trigonometric and Hyperbolic Functions
      9. Chapter 17 in Review
    2. 18 Integration in the Complex Plane
      1. 18.1 Contour Integrals
      2. 18.2 Cauchy–Goursat Theorem
      3. 18.3 Independence of the Path
      4. 18.4 Cauchy’s Integral Formulas
      5. Chapter 18 in Review
    3. 19 Series and Residues
      1. 19.1 Sequences and Series
      2. 19.2 Taylor Series
      3. 19.3 Laurent Series
      4. 19.4 Zeros and Poles
      5. 19.5 Residues and Residue Theorem
      6. 19.6 Evaluation of Real Integrals
      7. Chapter 19 in Review
    4. 20 Conformal Mappings
      1. 20.1 Complex Functions as Mappings
      2. 20.2 Conformal Mappings
      3. 20.3 Linear Fractional Transformations
      4. 20.4 Schwarz–Christoffel Transformations
      5. 20.5 Poisson Integral Formulas
      6. 20.6 Applications
      7. Chapter 20 in Review
  11. Appendices
    1. I Derivative and Integral Formulas
    2. II Gamma Function
    3. III Table of Laplace Transforms
    4. IV Conformal Mappings
  12. Answers to Selected Odd-Numbered Problems
  13. Index