Introduction to Methods of Applied Mathematics

Written by: Sean Mauch (Caltech)

The author might have called this huge mathematics textbook an introduction but with over 3,000 pages it certainly goes beyond a mere Introduction to Methods of Applied Mathematics. With more than a little humor, the author explains that most STEM related textbooks that can actually be comprehended by students have titles that include things like “Introduction” or “Elementary” in the titles. Texts with “Intermediate” or “Advanced” in the title tend to be incomprehensible and of poor production quality.

The author is Sean Mauch of the Center for Advanced Computing Research at the California Institute of Technology. He has generously allowed his textbook to be used by virtually anyone, anywhere in any fashion. I think I’ve linked to the original source where Mauch maintains the original PDF file. He began writing the textbook while still an undergraduate student at Caltech. He claims it is “neither complete nor polished” and yet it appears to have been adopted by a number of professors and universities as an approved textbook or at least a recommendation for supplemental reading.

Each of the chapters close with a series of exercises, hints and solutions. Some chapters also contain quizzes with solutions.

The text of Introduction to Methods of Applied Mathematics displays a copyright date of January 24, 2004.

Table of Contents for Introduction to Methods of Applied Mathematics

  1. Algebra – Sets and Functions
    1. Sets
    2. Single Valued Functions
    3. Inverses and Multi-Valued Functions
    4. Transforming Equiations
  2. Algebra – Vectors
    1. Vectors
      1. Scalars and Vectors
      2. The Kronecker Delta and Einstein Summation Convention
      3. The Dot and Cross Product
    2. Sets of Vectors in n Dimensions
  3. Calculus – Differential Calculus
    1. Limits of Functions
    2. Continuous Functions
    3. The Derivative
    4. Implicit Differentiation
    5. Maxima and Minima
    6. Mean Value Theorems
      1. ApplicationL Using Taylor’s Theorem to Approximate Functions
      2. Application: Finite Difference Schemes
    7. L’Hospital’s Rule
  4. Calculus – Integral Calculus
    1. The Indefinite Integral
    2. The Definite Integral
      1. Definition
      2. Properties
    3. The Fundamental Theorem of Integral Calculus
    4. Techniques of Integration
      1. Partial Fractions
    5. Improper Integrals
  5. Calculus – Vector Calculus
    1. Vector Functions
    2. Gradient, Divergence and Curl
  6. Functions of a Complex Variable – Complex Numbers
    1. Complex Numbers
    2. The Complex Plane
    3. Polar Form
    4. Arithmetic and Vectors
    5. Integer Exponents
    6. Rational Exponents
  7. Functions of a Complex Variable
    1. Curves and Regions
    2. The Point at Infinity and the Stereographic Projections
    3. A Gentle Introduction to Branch Points
    4. Cartesian and Modulus-Argument Form
    5. Graphing Functions of a Complex Variable
    6. Trigonometric Functions
    7. Inverse Trigonometric Functions
    8. Riemann Surfaces
    9. Branch Points
  8. Functions of a Complex Variable – Analytic Functions
    1. Complex Derivatives
    2. Cauchy-Riemann Equations
    3. Harmonic Functions
    4. Singularities
      1. Categorization of Singularities
      2. Isolated and Non-Isolated Singularities
    5. Application: Potential Flow
  9. Functions of a Complex Variable – Analytic Continuation
    1. Analytic Continuation
    2. Analytic Continuation of Sums
    3. Analytic Functions Defined in Terms of Real Variables
      1. Polar Coordinates
      2. Analytic Functions Defined in Terms of Their Real or Imaginary Parts
  10. Functions of a Complex Variable – Contour Integration and the Cauchy-Goursat Theorem
    1. Line Integrals
    2. Contour Integrals
      1. Maximum Modulus Integral Bound
    3. The Cauchy-Goursat Theorem
    4. Contour Deformation
    5. Morera’s Theorem
    6. Indefinite Integrals
    7. Fundamental Theorem of Calculus via Primitives
      1. Line Integrals and Primitives
      2. Contour Integrals
    8. Fundamental Theorem of Calculus via Complex Calculus
  11. Functions of a Complex Variable – Cauchy’s Integral Formula
    1. Cauchy’s Integral Formula
    2. The Argument Theorem
    3. Rouche’s Theorem
  12. Functions of a Complex Variable – Series and Convergence
    1. Series of Constants
      1. Definitions
      2. Special Series
      3. Convergence Tests
    2. Uniform Convergence
      1. Tests for Uniform Convergence
      2. Uniform Convergence and Continuous Functions
    3. Uniformly Convergent Power Series
    4. Integration and Differentiation of Power Series
    5. Taylor Series
      1. Newton’s Binomial Formula
    6. Laurent Series
  13. Functions of a Complex Variable – The Residue Theorem
    1. The Residue Theorem
    2. Cauchy Principle Value for Real Integrals
      1. The Cauchy Principal Value
    3. Cauchy Principal Value for Contour Integrals
    4. Integrals on the Real Axis
    5. Fourier Integrals
    6. Fourier Cosine and Sine Integrals
    7. Contour Integration and Branch Cuts
    8. Exploiting Symmetry
      1. Wedge Contours
      2. Box Contours
    9. Definite Integrals Involving Sine and Cosine
    10. Infinite Sums
  14. Ordinary Differential Equations – First Order Differential Equations
    1. Notation
    2. Example Problems
      1. Growth and Decay
    3. One Parameter Families of Functions
    4. Integrable Forms
      1. Separable Equations
      2. Exact Equations
      3. Homogeneous Coefficient Equations
    5. The First Order, Linear Differential Equation
      1. Homogeneous Equations
      2. Inhomogeneous Equations
      3. Variation of Parameters
    6. Initial Conditions
      1. Piecewise Continuous Coefficients and Inhomogeneities
    7. Well-Posed Problems
    8. Equations in the Complex Plane
      1. Ordinary Points
      2. Regular Singular Points
      3. Irregular Singular Points
      4. The Point at Infinity
  15. Ordinary Differential Equations – First Order Linear Systems of Differential Equations
    1. Introduction
    2. Using Eigenvalues and Eigenvectors to find Homogeneous Solutions
    3. Matrices and Jordan Canonical Form
    4. Using the Matrix Exponential
  16. Ordinary Differential Equations – Theory of Linear Ordinary Differential Equations
    1. Exact Equations
    2. Nature of Solutions
    3. Transformation to a First Order System
    4. The Wronskian
      1. Derivative of a Determinant
      2. The Wronskian of a Set of Functions
      3. The Wronskian of the Solutions to a Differential Equation
    5. Well-Posed Problems
    6. The Fundamental Set of Solutions
    7. Adjoint Equations
  17. Ordinary Differential Equations – Techniques for Linear Differential Equations
    1. Constant Coefficient Equations
      1. Second Order Equations
      2. Real-Valued Solutions
      3. Higher Order Equations
    2. Euler Equations
      1. Real-Valued Solutions
    3. Exact Equations
    4. Equations Without Explicit Dependence on y
    5. Reduction of Order
    6. * Reduction of Order and the Adjoint Equation
  18. Ordinary Differential Equations – Techniques for Nonlinear Differential Equations
    1. Bernoulli Equations
    2. Riccati Equations
    3. Exchanging the Dependent and Independent Variables
    4. Autonomous Equations
    5. * Equidimensional-in-x Equations
    6. * Equidimensional-in-y Equations
    7. * Scale-Invariant Equations
  19. Ordinary Differential Equations – Transformations and Canonical Forms
    1. The Constant Coefficient Equation
    2. Normal Form
      1. Second Order Equations
      2. Higher Order Differential Equations
    3. Transformations of the Independent Value
      1. Transformation to the form u” + a(x) u = 0
      2. Transformation to a Constant Coefficient Equation
    4. Integral Equations
      1. Initial Value Problems
      2. Boundary Value Problems
  20. Ordinary Differential Equations – The Dirac Delta Function
    1. Derivative of the Heaviside Function
    2. The Delta Function as a Limit
    3. Higher Dimensions
    4. Non-Rectangular Coordinate Systems
  21. Ordinary Differential Equations – Inhomogeneous Differential Equations
    1. Particular Solutions
    2. Method of Undetermined Coefficients
    3. Variation of Parameters
      1. Second Order Differential Equations
      2. Higher Order Differential Equations
    4. Piecewise Continuous Coefficients and Inhomogeneities
    5. Inhomogeneous Boundary Conditions
      1. Eliminating Inhomogeneous Boundary Conditions
      2. Separating Inhomogeneous Equations and Inhomogeneous Boundary Conditions
      3. Existence of Solutions of Problems with Inhomogeneous Boundary Conditions
    6. Green Functions for First Order Equations
    7. Green Functions for Second Order Equations
      1. Green Functions for Sturm-Liouville Problems
      2. Initial Value Problems
      3. Problems with Unmixed Boundary Conditions
      4. Problems with Mixed Boundary Conditions
    8. Green Functions for Higher Order Problems
    9. Fredholm Alternative Theorem
  22. Ordinary Differential Equations – Difference Equations
    1. Introduction
    2. Exact Equations
    3. Homogeneous First Order
    4. Inhomogeneous First Order
    5. Homogeneous Constant Coefficient Equations
    6. Reduction of Order
  23. Ordinary Differential Equations – Series Solutions fo Differential Equations
    1. Ordinary Points
      1. Taylor Series Expansion for a Second Order Differential Equation
    2. Regular Singular Points of Second Order Equations
      1. Indicial Equation
      2. The Case: Double Root
      3. The Case: Roots Differ by an Integer
    3. Irregular Singular Points
    4. The Point at Infinity
  24. Ordinary Differential Equations – Asymptotic Expansions
    1. Asymptotic Relations
    2. Leading Order Behavior of Differential Equations
    3. Integration by Parts
    4. Asymptotic Series
    5. Asymptotic Expansions of Differential Equations
      1. The Parabolic Cylinder Equation
  25. Ordinary Differential Equations – Hilbert Spaces
    1. Linear Spaces
    2. Inner Products
    3. Norms
    4. Linear Independence
    5. Orthogonality
    6. Gramm-Schmidt Orthogonalization
    7. Orthonormal Function Expansion
    8. Sets Of Functions
    9. Least Squares Fit to a Function and Completeness
    10. Closure Relation
    11. Linear Operators
  26. Ordinary Differential Equations – Self-Adjoint Linear Operators
    1. Adjoint Operators
    2. Self-Adjoint Operators
  27. Ordinary Differential Equations – Self-Adjoint Boundary Value Problems
    1. Summary of Adjoint Operators
    2. Formally Self-Adjoint Operators
    3. Self-Adjoint Problems
    4. Self-Adjoint Eigenvalue Problems
    5. Inhomogeneous Equations
  28. Ordinary Differential Equations – Fourier Series
    1. An Eigenvalue Problem
    2. Fourier Series
    3. Least Squares Fit
    4. Fourier Series for Functions Defined on Arbitrary Ranges
    5. Fourier Cosine Series
    6. Fourier Sine Series
    7. Complex Fourier Series and Parseval’s Theorem
    8. Behavior of Fourier Coefficients
    9. Gibb’s Phenomenon
    10. Integrating and Differentiating Fourier Series
  29. Ordinary Differential Equations – Regular Sturm-Liouville Problems
    1. Derivation of the Sturm-Liouville Form
    2. Properties of Regular Sturm-Liouville Problems
    3. Solving Differential Equations with Eigenfunction Expansions
  30. Ordinary Differential Equations – Integrals and Convergence
    1. Uniform Convergence of Integrals
    2. The Riemann-Lebesgue Lemma
    3. Cauchy Principal Value
      1. Integrals on an Infinite Domain
      2. Singular Functions
  31. Ordinary Differential Equations – The Laplace Transform
    1. The Laplace Transform
    2. The Inverse Laplace Transform
      1. f(s) with Poles
      2. f(s) with Branch Points
      3. Asymptotic Behavior of f(s)
    3. Properties of the Laplace Transform
    4. Constant Coefficient Differential Equations
    5. Systems of Constant Coefficient Differential Equations
  32. Ordinary Differential Equations – The Fourier Transform
    1. Derivation from a Fourier Series
    2. The Fourier Transform
      1. A Word of Caution
    3. Evaluating Fourier Integrals
      1. Integrals that Converge
      2. Cauchy Principal Value and Integrals that are Not Absolutely Convergent
      3. Analytic Continuation
    4. Properties of the Fourier Transform
      1. Closure Relation
      2. Fourier Transform of a Derivative
      3. Fourier Convolution Theorem
      4. Parseval’s Theorem
      5. Shift Property
      6. Fourier Transform of x f(x)
    5. Solving Differential Equations with the Fourier Transform
    6. The Fourier Cosine and Sine Transform
      1. The Fourier Cosine Transform
      2. The Fourier Sine Transform
    7. Properties of the Fourier Cosine and Sine Transform
      1. Transforms of Derivatives
      2. Convolution Theorems
      3. Cosine and Sine Transform in Terms of the Fourier Transform
    8. Solving Differential Equations with the Fourier Cosine and Sine Transformations
  33. Ordinary Differential Equations – The Gamma Function
    1. Euler’s Formula
    2. Hankel’s Formula
    3. Gauss’ Formula
    4. Weierstrass’ Formula
    5. Stirling’s Approximation
  34. Ordinary Differential Equations – Bessel Functions
    1. Bessel’s Equation
    2. Frobeneius Series Solution about z = 0
      1. Behavior at Infinity
    3. Bessel Functions of the First Kind
      1. The Bessel Function Satisfies Bessel’s Equation
      2. Series Expansion of the Bessel Function
      3. Bessel Functions of Non-Integer Order
      4. Recursion Formulas
      5. Bessel Functions of Half-Integer Order
    4. Neumann Expansions
    5. Bessel Functions of the Second Kind
    6. Hankel Functions
    7. The Modified Bessel Equation
  35. Partial Differential Equations – Transforming Equations
  36. Partial Differential Equations – Classification of Partial Differential Equations
    1. Classification of Second Order Quasi-Linear Equations
      1. Hyperbolic Equations
      2. Parabolic Equations
      3. Elliptic Equations
    2. Equilibrium Solutions
  37. Partial Differential Equations – Separation of Variables
    1. Eigensolutions of Homogeneous Equations
    2. Homogeneous Equations with Homogeneous Boundary Conditions
    3. Time-Independent Sources and Boundary Conditions
    4. Inhomogeneous Equations with Homogeneous Boundary Conditions
    5. Inhomogeneous Boundary Conditions
    6. The Wave Equations
    7. General Method
  38. Partial Differential Equations – Finite Transforms
  39. Partial Differential Equations – The Diffusion Equation
  40. Partial Differential Equations – Laplace’s Equation
    1. Introduction
    2. Fundamental Solution
      1. Two Dimensional Space
  41. Partial Differential Equations – Waves
  42. Partial Differential Equations – Similarity Methods
  43. Partial Differential Equations – Method of Characteristics
    1. First Order Linear Equations
    2. First Order Quasi-Linear Equations
    3. The Method of Characteristics and the Wave Equation
    4. The Wave Equation for an Infinite Domain
    5. The Wave Equation for a Semi-Infinite Domain
    6. The Wave Equation for a Finite Domain
    7. Envelopes of Curves
  44. Partial Differential Equations – Transform Methods
    1. Fourier Transform for Partial Differential Equations
    2. The Fourier Sine Transform
    3. Fourier Transform
  45. Partial Differential Equations – Green Functions
    1. Inhomogeneous Equations and Homogeneous Boundary Conditions
    2. Homogeneous Equations and Inhomogeneous Boundary Conditions
    3. Eigenfunction Expansions for Elliptic Equations
    4. The Method of Images
  46. Partial Differential Equations – Conformal Mapping
  47. Partial Differential Equations – Non-Cartesian Coordinates
    1. Spherical Coordinates
    2. Laplace’s Equation in a Disk
    3. Laplace’s Equation in an Annulus
  48. Calculus of Variations
  49. Nonlinear Differential Equations
  50. Nonlinear Partial Differential Equations

But, Wait, There’s More

Just in case the first 50 chapters of this so-called Introduction to Methods of Applied Mathematics didn’t cover enough material, there’s a huge collection of appendices with even more great stuff for calculus students.

  1. Greek Letters
  2. Notation
  3. Formulas from Complex Variables
  4. Table of Derivatives
  5. Table of Integrals
  6. Table of Sums
  7. Table of Taylor Series
  8. Continuous Transforms
    1. Properties of Laplace Transforms
    2. Table of Laplace Transforms
    3. Table of Fourier Transforms
    4. Table of Fourier Transforms in n Dimensions
    5. Table of Fourier Cosine Transforms
    6. Table of Fourier Sine Transforms
  9. Table of Wronskians
  10. Sturm-Liouville Eigenvalue Problems
  11. Green Functions for Ordinary Differential Equations
  12. Trigonometric Identities
    1. Circular Functions
    2. Hyperbolic Functions
  13. Bessel Functions
    1. Definite Integrals
  14. Formulas from Linear Algebra
  15. Vector Analysis
  16. Partial Fractions
  17. Finite Math
  18. Physics
  19. Probability
    1. Independent Events
    2. Playing the Odds
  20. Economics
  21. Glossary

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Introduction to Methods of Applied Mathematics

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