Contents

• • 1. Graphs
    • 1.1 Introduction
    • 1.2 Example
      • 1.2.1 Using reasoning
      • 1.2.2 Using brute force
    • 1.3 Example
      • 1.3.1 Solution
  • 2. Optimization
    • 2.1 Introduction
    • 2.2 Example
      • 2.2.1 Definition
      • 2.2.2 Reduction
      • 2.2.3 Further reduction
      • 2.2.4 Finding the length
      • 2.2.5 Finding the optimum angle
      • 2.2.6 Finding the optimum length
    • 2.3 General Approach
      • 2.3.1 Formulation
      • 2.3.2 Interior minima
      • 2.3.3 Boundary minima
  • 3. Approximations
    • 3.1 Introduction
    • 3.2 Example
      • 3.2.1 Identification
      • 3.2.2 Results
      • 3.2.3 Other way
    • 3.3 Example
      • 3.3.1 Identification
      • 3.3.2 Finish
  • 4. Limits
    • 4.1 Introduction
    • 4.2 Example
      • 4.2.1 Observations
      • 4.2.2 L'Hopital
      • 4.2.3 Better
    • 4.3 Example
      • 4.3.1 Grinding it out
      • 4.3.2 Using insight
  • 5. Combined Changes in Variables
    • 5.1 Introduction
    • 5.2 Example
      • 5.2.1 Identification
      • 5.2.2 Results
    • 5.3 Example
      • 5.3.1 Solution
  • 6. Curvilinear Motion
    • 6.1 Introduction
    • 6.2 Example
      • 6.2.1 Position
      • 6.2.2 Velocity
      • 6.2.3 Acceleration
  • 7. Line Integrals
    • 7.1 Introduction
    • 7.2 Example
      • 7.2.1 Identification
      • 7.2.2 Solution
  • 8. Surface and Volume Integrals
    • 8.1 Introduction
    • 8.2 Example
      • 8.2.1 Region
      • 8.2.2 Approach
      • 8.2.3 Results
    • 8.3 Example
      • 8.3.1 Approach
      • 8.3.2 Results
  • 9. Numerical Integration
    • 9.1 Introduction
    • 9.2 Example
      • 9.2.1 Solution
  • 10. Geometry using vectors
    • 10.1 Introduction
    • 10.2 Example
      • 10.2.1 Identification
      • 10.2.2 Solution
    • 10.3 Example
      • 10.3.1 Identification
      • 10.3.2 Solution
  • 11. Vector Analysis
    • 11.1 Coordinate Changes
      • 11.1.1 General
      • 11.1.2 Orthogonal coordinates
  • 12. Gaussian Elimination
    • 12.1 Elimination Procedure
    • 12.2 Partial Pivoting
    • 12.3 Back substitution
    • 12.4 LU Theorem
    • 12.5 Row Canonical Form
    • 12.6 Null Spaces and Solution Spaces
  • 13. Inverse Matrices
    • 13.1 Finding inverses using GE
    • 13.2 Finding inverses using minors
    • 13.3 Finding inverses using transposing
  • 14. Eigenvalues and Eigenvectors
    • 14.1 Finding Eigenvalues
    • 14.2 Eigenvectors of nonsymmetric matrices
    • 14.3 Eigenvectors of symmetric matrices
  • 15. Change of Basis
    • 15.1 General Procedure
    • 15.2 Diagonalization of nonsymmetric matrices
    • 15.3 Diagonalization of symmetric matrices
  • 16. Laplace Transformation
    • 16.1 Partial Fractions
    • 16.2 Completing the square
  • 17. More on Systems
    • 17.1 Solution of systems using diagonalization
    • 17.2 Solving Partial Differential Equations
    • 17.3 More details on the extension
  • 18. Introduction
    • 18.1 Basic Concepts
      • 18.1.1 The prevalence of partial differential equations
      • 18.1.2 Definitions
      • 18.1.3 Typical boundary conditions
    • 18.2 The Standard Examples
      • 18.2.1 The Laplace equation
        • 18.2.1.1 Solution stanexl-a
        • 18.2.1.2 Solution stanexl-b
        • 18.2.1.3 Solution stanexl-b1
        • 18.2.1.4 Solution stanexl-b2
        • 18.2.1.5 Solution stanexl-b3
        • 18.2.1.6 Solution stanexl-b5
        • 18.2.1.7 Solution stanexl-c
        • 18.2.1.8 Solution stanexl-d
        • 18.2.1.9 Solution stanexl-e
      • 18.2.2 The heat equation
        • 18.2.2.1 Solution stanexh-a
        • 18.2.2.2 Solution stanexh-b
      • 18.2.3 The wave equation
        • 18.2.3.1 Solution stanexw-a
        • 18.2.3.2 Solution stanexw-b
        • 18.2.3.3 Solution stanexw-c
        • 18.2.3.4 Solution stanexw-e
        • 18.2.3.5 Solution stanexw-f
    • 18.3 Properly Posedness
      • 18.3.1 The conditions for properly posedness
        • 18.3.1.1 Solution ppc-a
        • 18.3.1.2 Solution ppc-b
        • 18.3.1.3 Solution ppc-c
      • 18.3.2 An improperly posed parabolic problem
      • 18.3.3 An improperly posed elliptic problem
        • 18.3.3.1 Solution ppe-a
        • 18.3.3.2 Solution ppe-b
        • 18.3.3.3 Solution ppe-c
        • 18.3.3.4 Solution ppe-d
        • 18.3.3.5 Solution ppe-e
        • 18.3.3.6 Solution ppe-f
      • 18.3.4 Improperly posed hyperbolic problems
        • 18.3.4.1 Solution pph-a
        • 18.3.4.2 Solution pph-b
    • 18.4 Energy methods
      • 18.4.1 The Poisson equation
        • 18.4.1.1 Solution emp-a
        • 18.4.1.2 Solution emp-b
      • 18.4.2 The heat equation
      • 18.4.3 The wave equation
    • 18.5 Variational methods [None]
    • 18.6 Classification
      • 18.6.1 Introduction
      • 18.6.2 Scalar second order equations
        • 18.6.2.1 Solution clasnd-a
    • 18.7 Changes of Coordinates
      • 18.7.1 Introduction
      • 18.7.2 The formulae for coordinate transformations
      • 18.7.3 Rotation of coordinates
        • 18.7.3.1 Solution rotcoor-a
      • 18.7.4 Explanation of the classification
        • 18.7.4.1 Solution expclass-a
    • 18.8 Two-Dimensional Coordinate Transforms
      • 18.8.1 Characteristic Coordinates
      • 18.8.2 Parabolic equations in two dimensions
      • 18.8.3 Elliptic equations in two dimensions
        • 18.8.3.1 Solution 2dcanel-a
  • 19. Green’s Functions
    • 19.1 Introduction
      • 19.1.1 The one-dimensional Poisson equation
        • 19.1.1.1 Solution gf1d-a
        • 19.1.1.2 Solution gf1d-b
      • 19.1.2 More on delta and Green’s functions
    • 19.2 The Poisson equation in infinite space
      • 19.2.1 Overview
      • 19.2.2 Loose derivation
        • 19.2.2.1 Solution pninfl-a
      • 19.2.3 Rigorous derivation
    • 19.3 The Poisson or Laplace equation in a finite region
      • 19.3.1 Overview
      • 19.3.2 Intro to the solution procedure
      • 19.3.3 Derivation of the integral solution
        • 19.3.3.1 Solution pnfd-a
      • 19.3.4 Boundary integral (panel) methods
      • 19.3.5 Poisson’s integral formulae
      • 19.3.6 Derivation
        • 19.3.6.1 Solution pnifd-a
        • 19.3.6.2 Solution pnifd-b
      • 19.3.7 The integral formula for the Neumann problem
      • 19.3.8 Smoothness of the solution
  • 20. First Order Equations
    • 20.1 Classification and characteristics
    • 20.2 Numerical solution
    • 20.3 Analytical solution
    • 20.4 Using the boundary or initial condition
    • 20.5 The inviscid Burgers’ equation
      • 20.5.1 Wave steepening
      • 20.5.2 Shocks
      • 20.5.3 Conservation laws
      • 20.5.4 Shock relation
      • 20.5.5 The entropy condition
    • 20.6 First order equations in more dimensions
    • 20.7 Systems of First Order Equations (None)
  • 21. D'Alembert Solution of the Wave equation
    • 21.1 Introduction
    • 21.2 Extension to finite regions
      • 21.2.1 The physical problem
      • 21.2.2 The mathematical problem
      • 21.2.3 Dealing with the boundary conditions
      • 21.2.4 The final solution
  • 22. Separation of Variables
    • 22.1 A simple example
      • 22.1.1 The physical problem
      • 22.1.2 The mathematical problem
      • 22.1.3 Outline of the procedure
      • 22.1.4 Step 1: Find the eigenfunctions
      • 22.1.5 Should we solve the other equation?
      • 22.1.6 Step 2: Solve the problem
    • 22.2 Comparison with D'Alembert
    • 22.3 Understanding the Procedure
      • 22.3.1 An ordinary differential equation as a model
      • 22.3.2 Vectors versus functions
      • 22.3.3 The inner product
      • 22.3.4 Matrices versus operators
      • 22.3.5 Some limitations
    • 22.4 Handling Periodic Boundary Conditions
      • 22.4.1 The physical problem
      • 22.4.2 The mathematical problem
      • 22.4.3 Outline of the procedure
      • 22.4.4 Step 1: Find the eigenfunctions
      • 22.4.5 Step 2: Solve the problem
      • 22.4.6 Summary of the solution
    • 22.5 Finding the Green's function
    • 22.6 Inhomogeneous boundary conditions
      • 22.6.1 The physical problem
      • 22.6.2 The mathematical problem
      • 22.6.3 Outline of the procedure
      • 22.6.4 Step 0: Fix the boundary conditions
      • 22.6.5 Step 1: Find the eigenfunctions
      • 22.6.6 Step 2: Solve the problem
      • 22.6.7 Summary of the solution
    • 22.7 Finding the Green's functions
    • 22.8 An alternate procedure
      • 22.8.1 The physical problem
      • 22.8.2 The mathematical problem
      • 22.8.3 Step 0: Fix the boundary conditions
      • 22.8.4 Step 1: Find the eigenfunctions
      • 22.8.5 Step 2: Solve the problem
      • 22.8.6 Summary of the solution
    • 22.9 A Summary of Separation of Variables
      • 22.9.1 The form of the solution
      • 22.9.2 Limitations of the method
      • 22.9.3 The procedure
      • 22.9.4 More general eigenvalue problems
    • 22.10 More general eigenfunctions
      • 22.10.1 The physical problem
      • 22.10.2 The mathematical problem
      • 22.10.3 Step 0: Fix the boundary conditions
      • 22.10.4 Step 1: Find the eigenfunctions
      • 22.10.5 Step 2: Solve the problem
      • 22.10.6 Summary of the solution
      • 22.10.7 An alternative procedure
    • 22.11 A Problem in Three Independent Variables
      • 22.11.1 The physical problem
      • 22.11.2 The mathematical problem
      • 22.11.3 Step 1: Find the eigenfunctions
      • 22.11.4 Step 2: Solve the problem
      • 22.11.5 Summary of the solution
  • 23. Fourier Transforms [None]
  • 24. Laplace Transforms
    • 24.1 Overview of the Procedure
      • 24.1.1 Typical procedure
      • 24.1.2 About the coordinate to be transformed
    • 24.2 A parabolic example
      • 24.2.1 The physical problem
      • 24.2.2 The mathematical problem
      • 24.2.3 Transform the problem
      • 24.2.4 Solve the transformed problem
      • 24.2.5 Transform back
    • 24.3 A hyperbolic example
      • 24.3.1 The physical problem
      • 24.3.2 The mathematical problem
      • 24.3.3 Transform the problem
      • 24.3.4 Solve the transformed problem
      • 24.3.5 Transform back
      • 24.3.6 An alternate procedure
  • A. Addenda
    • A.1 Distributions
  • B. Derivations
    • B.1 Orthogonal coordinate derivatives
    • B.2 Harmonic functions are analytic
    • B.3 Some properties of harmonic functions
    • B.4 Coordinate transformation derivation
    • B.5 2D coordinate transformation derivation
    • B.6 2D elliptical transformation
  • C. Notes
    • C.1 Why this book?
    • C.2 History and wish list
  
  
• Web Pages
• References