偏微分方程（第一卷）

內(nèi)容概要

　　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.　　此書為英文版！

書籍目錄

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   References2  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  References3  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  References4  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  References5  Linear Elliptic Equations  Introduction  ……6  Linear Evolution EquationsA  Outline of Functional AnalysisB  Manifolds,Vector Bundles,and Lie GroupsIndex

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