Mathematical Methods in Engineering and Physics

Mathematical Methods in Engineering and Physics:
Contents

by Gary N. Felder and Kenny M. Felder

Below is a list of the 14 chapters and 13 appendices in the book. You can download the preface, three complete chapters (Chapters 4, 11, and 14), scattered complete sections from various chapters, and all of the Motivating and Discovery Exercises in the book. (Click here for an explanation of what "Exercises" are and how to use them.)

Click the [+] next to any chapter to see the sections and/or download content. (All links are in blue.)


Preface (a brief overview of the philosophy and structure of the book)
       •  Available online and in the printed book

  1. Introduction to Ordinary Differential Equations [+]
    1. Motivating Exercise: The Simple Harmonic Oscillator
             •  Entire section available in the printed book and online
             We also have two alternate Motivating Exercises built around a fluid flow scenario:
             •  a relatively short and straightforward fluid flow exercise that is typical of our Motivating Exercises, and
             •  a longer and more involved fluid flow exercise that steps through the derivation of a differential equation.
    2. Overview of Differential Equations
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    3. Arbitrary Constants
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    4. Slope Fields and Equilibrium
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    5. Separation of Variables
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    6. Guess and Check, and Linear Superposition
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    7. Coupled Equations
             •  Entire section available online only
             •  Discovery Exercise available online only
    8. Differential Equations on a Computer
             •  Entire section available online only
             •  Discovery Exercise available online only
    9. Additional Problems
             •  Entire section available online only
  2. Taylor Series and Series Convergence [+]
    1. Motivating Exercise: Vibrations in a Crystal
             •  Entire section available in the printed book and online
    2. Linear Approximations
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    3. Maclaurin Series
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    4. Taylor Series
             •  Entire section available in the printed book
    5. Finding One Series From Another
             •  Entire section available in the printed book
    6. Sequences and Series
             •  Entire section available in the printed book
             •  Discovery Exercise on geometric series available in the printed book and online
             •  Discovery Exercise on infinite series available in the printed book and online
    7. Tests for Series Convergence
             •  Entire section available in the printed book
    8. Asymptotic Expansions
             •  Entire section available online only
    9. Additional Problems
             •  Entire section available online only
  3. Complex Numbers [+]
    1. Motivating Exercise: The Underdamped Harmonic Oscillator
             •  Entire section available in the printed book and online
         Alternate Motivating Exercise: An RLC Circuit
             •  The same math with a different physical setup, this exercise is available online only.
    2. Complex Numbers
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    3. The Complex Plane
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    4. Euler's Formula I—The Complex Exponential Function
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    5. Euler's Formula II—Modeling Oscillations
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    6. Special Application: Electric Circuits
             •  Entire section available online only
    7. Additional Problems
             •  Entire section available online only
  4. Partial Derivatives [+]
    Click here to download a copy of the entire chapter
    1. Motivating Exercise: The Wave Equation
             •  Entire section available in the printed book and online
    2. Partial Derivatives
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    3. The Chain Rule
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    4. Implicit Differentiation
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    5. Directional Derivatives
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    6. The Gradient
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    7. Tangent Plane Approximations and Power Series
             •  Entire section available online only
             •  Discovery Exercise available online only
    8. Optimization and the Gradient
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    9. Lagrange Multipliers
             •  Entire section available in the printed book
             •  Discovery Exercise available online only
             •  An alternative explanation of why Lagrange multipliers work -- available online only
    10. Special Application: Thermodynamics
             •  Entire section available online only
    11. Additional Problems
             •  Entire section available online only
  5. Integrals in Two or More Dimensions [+]
    1. Motivating Exercise: Newton's Problem, (or) The Gravitational Field of a Sphere
             •  Entire section available in the printed book and online
    2. Setting Up Integrals
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    3. Cartesian Double Integrals over a Rectangular Region
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    4. Cartesian Double Integrals over a Non-Rectangular Region
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    5. Triple Integrals in Cartesian Coordinates
             •  Entire section available in the printed book
    6. Double Integrals in Polar Coordinates
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    7. Cylindrical and Spherical Coordinates
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    8. Line Integrals
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    9. Parametrically Expressed Surfaces
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    10. Surface Integrals
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    11. Special Application: Gravitational Forces
             •  Entire section available online only
    12. Additional Problems
             •  Entire section available online only
  6. Linear Algebra I [+]
    1. The Motivating Example On Which We Base the Whole Chapter: The Three-Spring Problem
             •  Entire section available in the printed book and online
    2. Matrices: The Easy Stuff
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    3. Matrix Times Column
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    4. Basis Vectors
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    5. Matrix Times Matrix
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    6. The Identity and Inverse Matrices
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    7. Linear Dependence and the Determinant
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    8. Eigenvectors and Eigenvalues
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    9. Putting It Together: Revisiting the Three-Spring Problem
             •  Entire section available in the printed book and online
    10. Additional Problems
             •  Entire section available online only
  7. Linear Algebra II [+]
    1. Geometric Transformations
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    2. Tensors
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    3. Vector Spaces and Complex Vectors
             •  Entire section available in the printed book
    4. Row Reduction
             •  Entire section available online only
    5. Linear Programming and the Simplex Method
             •  Entire section available online only
             •  Discovery Exercise available online only
    6. Additional Problems
             •  Entire section available online only
  8. Vector Calculus [+]
    1. Motivating Exercise: Flowing Fluids
             •  Entire section available in the printed book and online
    2. Scalar and Vector Fields
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    3. Potential in One Dimension
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    4. From Potential to Gradient
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    5. From Gradient to Potential: The Gradient Theorem
             •  Entire section available in the printed book
    6. Divergence, Curl, and Laplacian
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    7. Divergence and Curl II: The Math Behind the Pictures
             •  Entire section available in the printed book
    8. Vectors in Curvilinear Coordinates
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    9. The Divergence Theorem
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    10. Stokes' Theorem
             •  Entire section available in the printed book
    11. Conservative Vector Fields
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    12. Additional Problems
             •  Entire section available online only
  9. Fourier Series and Transforms [+]
    1. Motivating Exercise: Discovering Extrasolar Planets
             •  Entire section available in the printed book and online
    2. Introduction to Fourier Series
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
             •  Click here for a set of Mathematica assignments showing the convergence of Fourier series
    3. Deriving the Formula for a Fourier Series
             •  Entire section available in the printed book
    4. Different Periods and Finite Domains
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    5. Fourier Series with Complex Exponentials
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    6. Fourier Transforms
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    7. Discrete Fourier Transforms
             •  Entire section available online only
    8. Multivariate Fourier Series
             •  Entire section available online only
             •  Discovery Exercise available online only
    9. Additional Problems
             •  Entire section available online only
  10. Methods of Solving Ordinary Differential Equations [+]
    1. Motivating Exercise: A Damped, Driven Oscillator
             •  Entire section available in the printed book and online
    2. Guess and Check
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    3. Phase Portraits
             •  Entire section available online only
    4. Linear First Order Differential Equations
             •  Entire section available online only
             •  Discovery Exercise available online only
    5. Exact Differential Equations
             •  Entire section available online only
             •  Discovery Exercise available online only
    6. Linearly Independent Solutions and the Wronskian
             •  Entire section available online only
             •  Discovery Exercise available online only
    7. Variable Substitution
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    8. Three Special Cases of Variable Substitution
             •  Entire section available in the printed book
    9. Reduction of Order and Variation of Parameters
             •  Entire section available online only
             •  Discovery Exercise available online only
    10. Heaviside, Dirac, and Laplace
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    11. Using Laplace Transforms to Solve Differential Equations
             •  Entire section available in the printed book
    12. Green's Functions
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    13. Additional Problems
             •  Entire section available online only
  11. Partial Differential Equations [+]
    Click here to download a copy of the entire chapter
    1. Motivating Exercise: The Heat Equation
             •  Entire section available in the printed book and online
    2. Overview of Partial Differential Equations
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    3. Normal Modes
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
             •  Click here for a set of Mathematica animations illustrating combinations of normal modes
    4. Separation of Variables—The Basic Method
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    5. Separation of Variables—More than Two Variables
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    6. Separation of Variables—Polar Coordinates and Bessel Functions
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    7. Separation of Variables—Spherical Coordinates and Legendre Polynomials
             •  Entire section available in the printed book
    8. Inhomogeneous Boundary Conditions
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    9. The Method of Eigenfunction Expansion
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    10. The Method of Fourier Transforms
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    11. The Method of Laplace Transforms
             •  Entire section available in the printed book
    12. Additional Problems
             •  Entire section available online only
  12. Special Functions and ODE Series Solutions [+]
    1. Motivating Exercise: The Circular Drum
             •  Entire section available in the printed book and online
    2. Some Handy Summation Tricks
             •  Entire section available in the printed book
    3. A Few Special Functions
             •  Entire section available in the printed book
    4. Solving Differential Equations with Power Series
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    5. Legendre Polynomials
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    6. The Method of Frobenius
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    7. Bessel Functions
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    8. Sturm-Liouville Theory and Series Expansions
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    9. Proof of the Orthogonality of Sturm-Liouville Eigenfunctions
             •  Entire section available online only
    10. Special Application: The Quantum Harmonic Oscillator and Ladder Operators
             •  Entire section available online only
             •  Discovery Exercise available online only
    11. Additional Problems
             •  Entire section available online only
  13. Calculus with Complex Numbers [+]
    1. Motivating Exercise: Laplace's Equation
             •  Entire section available in the printed book and online
             •  We also have an alternate Motivating Exercises built around a fluid flow scenario.
    2. Functions of Complex Numbers
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    3. Derivatives, Analytic Functions, and Laplace's Equation
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    4. Contour Integration
             •  Entire section available in the printed book
    5. Some Uses of Contour Integration
             •  Entire section available in the printed book
    6. Integrating Along Branch Cuts and Through Poles
             •  Entire section available online only
    7. Complex Power Series
             •  Entire section available in the printed book
    8. Mapping Curves and Regions
             •  Entire section available in the printed book
             •  Discovery Exercise available in the printed book and online
    9. Conformal Mapping and Laplace's Equation
             •  Entire section available in the printed book
    10. Special Application: Fluid Flow
             •  Entire section available online only
    11. Additional Problems
             •  Entire section available online only
  14. Calculus of Variations [+]
    Click here to download a copy of the entire chapter
    1. Motivating Exercise: Rescuing the Swimmer
             •  Available as part of the online chapter (click above) or for individual download.
    2. Variational Problems and the Euler-Lagrange Equation
             •  Entire section available as part of the online chapter (click above).
             •  Discovery Exercise available as part of the online chapter (click above) or for individual download.
    3. Why the Euler-Lagrange Equation Works
             •  Available in the online chapter (click above).
    4. Special Application: Lagrangian Mechanics
             •  Available in the online chapter (click above).
    5. Additional Problems
             •  Available in the online chapter (click above).

Appendix A: Different Types of Differential Equations
Appendix B: Taylor Series
Appendix C: Tests for Series Convergence
Appendix D: Curvilinear Coordinates
Appendix E: Matrices
Appendix F: Vector Calculus
Appendix G: Fourier Series and Transforms
Appendix H: Laplace Transforms
Appendix I: Which PDE Technique do I Choose?
Appendix J: Some Common Differential Equations and their Solutions
Appendix K: Special Functions
Appendix L: Answers to "Check Yourself" in Exercises
Appendix M: Answers to Odd-Numbered Problems



Click here for the home page for the book, http://www.felderbooks.com/mathmethods.
Professors can request a digital evaluation copy or contact a Wiley rep to request a print evaluation copy.
If you have questions, we would love to hear from you. Click here to email the authors at GFelder@Smith.edu.