| Partial differential equations is a many-faceted subject.Created to describe the mechanical behavior of objects such as vibrating strings and blowing winds,it has developed into a body of material that interacts with many branches of math-ematics,such as differential geometry,complex analysis,and harmonic analysis,as well as a ubiquitous factor in the description and elucidation of problems in mathematical physics. 此书为英文版！ |
| Contents of Volumes Ⅱ and Ⅲ Introduction 1 Basic Theory of ODE and Vector Fields Introduction 1 The derivative 2 Fundamental local existence theorem for ODE 3 Inverse function and implicit function theorems 4 Constant-coefficient linear systems； exponentiation of matrices 5 Variable-coefficient linear systems of ODE: Duhamel‘s principle 6 Dependence of solutions on initial data and on other parameters 7 Flows and vector fields 8 Lie brackets 9 Commuting flows； Frobenius‘s theorem 10 Hamiltonian systems 11 Geodesics 12 Variational problems and the stationary action principle 13 Differential forms 14 The symplectic form and canonical transformations 15 First-order， scalar， nonlinear PDE 16 Completely integrable Hamiltonian systems 17 Examples of integrable systems:central force problems 18 Relativistic motion 19 Topological applications of differential forms 20 Critical points and inedxof a vector field A ZZNonsmooth vector fields References 2 The Laplace Equation and Wae Equation Introduction 1 Vibrating strings and membranes 2 The divergence of a vector field 3 The covariant derivative and divergence of tensor fields 4 The Laplace operator on a Riemannian manifold 5 The wave equation on a product manifold and energy conservation 6 Uniqueness nad finite propagation speed 7 Lorentz manifolds and stress-energy tensors 8 More general hyperbolic equations;energy estimates 9 The symbol of a differential operatorand a general Green-Stokes formula 10 The Hodge Laplacian on k-forms 11 Maxwells equations References 3 Fourier Analysis,Distribution,and Constant-Coefficient Linear PDE Introduction 1 Fouier series 2 Harmonic functions and holomorphic functions in the plane 3 The Fourier transform 4 Distributions and tempered distributions 5 The classical evolution equations 6 Radial distributions,polar coordinates,and Bessel functions 7 The method of images and Poisson s summation formula 8 Homogeneous distributions and principal value distributions 9 Elliptic operators 10 Local solvability of constant-coefficient PDE 11 The discrete Fourier transform 12 The fast Fourier transform A The mighty Gaussian and the sublime gamma function References 4 Sobolev Spaces Introduction 1 Sobolev spaceson Rn 2 The complex interpolation method 3 Sobolev spaces on compact manifolds 4 Sobolev spaces on bounded dmains 5 The Sobolev spaces H5/0 6 The Schwartz kernel theorem References 5 Linear Elliptic Equations Introduction …… 6 Linear Evolution Equations A Outline of Functional Analysis B Manifolds,Vector Bundles,and Lie Groups Index |
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