文档介绍：Topics in Mathematical Physics
Prof.
Spring semester 2002
Contents
Chapter 1. Differential equations of Mathematical Physics
Differential equations of elliptic type
Diffusion equations
Wave equations
Systems
Nonlinear equations
Hamilton-Jacobi theory
Relativistic field theory
Classification
Initial and boundary value problems
Inverse problems
Chapter 2. Elementary methods
Change of variables
Bilinear integrals
Conservation laws
Method of plane waves
Fourier transform
Theory of distributions
Chapter 3. Fundamental solutions
Basic definition and properties
Fundamental solutions for elliptic operators
More examples
Hyperbolic polynomials and source functions
Wave propagators
Inhomogeneous hyperbolic operators
Riesz groups
Chapter 4. The Cauchy problem
Definitions
Cauchy problem for distributions
Hyperbolic Cauchy problem
Solution of the Cauchy problem for wave equations
Domain of dependence
2
Chapter 5. Helmholtz equation and scattering
Time-harmonic waves
Source functions
Radiation conditions
Scattering on obstacle
Interference and diffraction
Chapter 6. Geometry of waves
Wave fronts
Hamilton-Jacobi theory
Geometry of rays
An integrable case
Legendre transformation and geometric duality
Ferm´at principle
The major Huygens principle
Geometrical optics
Caustics
Geometrical conservation law
Chapter 7. The method of Fourier integrals
Elements of symplectic geometry
Generating functions
Fourier integrals
Lagrange distributions
Hyperbolic Cauchy problem revisited
Chapter 8. ic waves
Vector analysis
Maxwell equations
Harmonic analysis of solutions
Cauchy problem
Local conservation laws
3
Chapter 1
Differential equations of
Mathematical Physics
Differential equations of elliptic type
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