Preface |
Preliminaries / 1: |
Self-Adjoint Operators / 1-1: |
Fourier Coefficients |
Exercises |
Curvilinear Coordinates / 1-2: |
Scaling Factors |
Volume Integrals |
The Gradient |
The Laplacian |
Spherical Coordinates |
Other Curvilinear Systems |
Applications |
An Alternate Approach (Optional) |
Approximate Identities and the Dirac-δ Function / 1-3: |
Approximate Identities |
The Dirac-δ Function in Physics |
Some Calculus for the Dirac-δ Function |
The Dirac-δ Function in Curvilinear Coordinates |
The Issue of Convergence / 1-4: |
Series of Real Numbers |
Convergence versus Absolute Convergence |
Series of Functions |
Power Series |
Taylor Series |
Some Important Integration Formulas / 1-5: |
Other Facts We Will Use Later |
Another Important Integral |
Vector Calculus / 2: |
Vector Integration / 2-1: |
Path Integrals |
Line Integrals |
Surfaces |
Parameterized Surfaces |
Integrals of Scalar Functions Over Surfaces |
Surface Integrals of Vector Functions |
Divergence and Curl / 2-2: |
Cartesian Coordinate Case |
Cylindrical Coordinate Case |
Spherical Coordinate Case |
The Curl |
The Curl in Cartesian Coordinates |
The Curl in Cylindrical Coordinates |
The Curl in Spherical Coordinates |
Green's Theorem, the Divergence Theorem, and Stokes' Theorem / 2-3: |
The Divergence (Gauss') Theorem |
Stokes' Theorem |
An Application of Stokes' Theorem |
An Application of the Divergence Theorem |
Conservative Fields |
Green's Functions / 3: |
Introduction |
Construction of Green's Function Using the Dirac-δ Function / 3-1: |
Construction of Green's Function Using Variation of Parameters / 3-2: |
Construction of Green's Function from Eigenfunctions / 3-3: |
More General Boundary Conditions / 3-4: |
The Fredholm Alternative (or, What If 0 Is an Eigenvalue?) / 3-5: |
Green's Function for the Laplacian in Higher Dimensions / 3-6: |
Fourier Series / 4: |
Basic Definitions / 4-1: |
Methods of Convergence of Fourier Series / 4-2: |
Fourier Series on Arbitrary Intervals |
The Exponential Form of Fourier Series / 4-3: |
Fourier Sine and Cosine Series / 4-4: |
Double Fourier Series / 4-5: |
Exercise |
Three Important Equations / 5: |
Laplace's Equation / 5-1: |
Derivation of the Heat Equation in One Dimension / 5-2: |
Derivation of the Wave Equation in One Dimension / 5-3: |
An Explicit Solution of the Wave Equation / 5-4: |
Converting Second-Order PDEs to Standard Form / 5-5: |
Sturm-Liouville Theory / 6: |
The Self-Adjoint Property of a Sturm-Liouville Equation / 6-1: |
Completeness of Eigenfunctions for Sturm-Liouville Equations / 6-2: |
Uniform Convergence of Fourier Series / 6-3: |
Separation of Variables in Cartesian Coordinates / 7: |
Solving Laplace's Equation on a Rectangle / 7-1: |
Laplace's Equation on a Cube / 7-2: |
Solving the Wave Equation in One Dimension by Separation of Variables / 7-3: |
Solving the Wave Equation in Two Dimensions in Cartesian Coordinates by Separation of Variables / 7-4: |
Solving the Heat Equation in One Dimension Using Separation of Variables / 7-5: |
The Initial Condition Is the Dirac-δ Function |
Steady State of the Heat Equation / 7-6: |
Checking the Validity of the Solution / 7-7: |
Solving Partial Differential Equations in Cylindrical Coordinates Using Separation of Variables / 8: |
An Example Where Bessel Functions Arise |
The Solution to Bessel's Equation in Cylindrical Coordinates / 8-1: |
Solving Laplace's Equation in Cylindrical Coordinates Using Separation of Variables / 8-2: |
The Wave Equation on a Disk (Drum Head Problem) / 8-3: |
The Heat Equation on a Disk / 8-4: |
Solving Partial Differential Equations in Spherical Coordinates Using Separation of Variables / 9: |
An Example Where Legendre Equations Arise / 9-1: |
The Solution to Bessel's Equation in Spherical Coordinates / 9-2: |
Legendre's Equation and Its Solutions / 9-3: |
Associated Legendre Functions / 9-4: |
Laplace's Equation in Spherical Coordinates / 9-5: |
The Fourier Transform / 10: |
The Fourier Transform as a Decomposition / 10-1: |
The Fourier Transform from the Fourier Series / 10-2: |
Some Properties of the Fourier Transform / 10-3: |
Solving Partial Differential Equations Using the Fourier Transform / 10-4: |
The Spectrum of the Negative Laplacian in One Dimension / 10-5: |
The Fourier Transform in Three Dimensions / 10-6: |
The Laplace Transform / 11: |
Properties of the Laplace Transform / 11-1: |
Solving Differential Equations Using the Laplace Transform / 11-2: |
Solving the Heat Equation Using the Laplace Transform / 11-3: |
The Wave Equation and the Laplace Transform / 11-4: |
Solving PDEs with Green's Functions / 12: |
Solving the Heat Equation Using Green's Function / 12-1: |
Green's Function for the Nonhomogeneous Heat Equation |
The Method of Images / 12-2: |
Method of Images for a Semi-infinite Interval |
Method of Images for a Bounded Interval |
Green's Function for the Wave Equation / 12-3: |
Green's Function and Poisson's Equation / 12-4: |
Appendix: Computing the Laplacian with the Chain Rule |
References |
Index |
Preface |
Preliminaries / 1: |
Self-Adjoint Operators / 1-1: |