Contents of Volumes II and III Preface 1 Basic Theory of ODE and Vector Fields 1 The derivative 2 Fundamental local existence theorem for ODE 3 Inverse function and implicit function theorems 4 Constant-coefficientlinear systems; exponentiation of matrices 5 Variable-coefficientlinear systems of ODE: Duhamels principle 6 Dependence of solutions on initial data and on other parameters 7 Flows and vector fields 8 Lie brackets 9 Commuting flows; Frobeniuss theorem 10 Hamiltoniansystems 11 Geodesics 12 Variational problems and the stationary action principle 13 Differential forms N 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 index of a vector field A Nonsmooth vector fields References 2 The Laplace Equation and Wave Equation 1 Vibrating strings and membranes 2 The divergence of a vector field 3The 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 and finite propagation speed 7 Lorentz manifolds and stress-energy tensors 8 More general hyperbolic equations; energy estimates 9 The symbol of a differential operator and a general Green-Stokes formula 10 The Hodge Laplacian on k-forms 11 Maxwells equations References 3 FourierAnalysisDistributions and Constant-Coefficient Linear PDE 1 Fourier 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 ofimages and Poissons summation formula 8 Homogeneous distributions and principal value distributions 9 Elliptic operators 10 Local solvability ofconstant-coefficientPDE 11 The discrete Fourier transform
12 The fast Fourier transform A The mighty Gaussian and the sublime gamma function References 4 SobolevSpaces 1 Sobolev spaces on Rn 2 The complex interpolation method 3 Sobolev spaces on compact manifolds 4 Sobolev spaces on bounded domains 5 The Sobolev spaces H50（Ω） 6 The Schwartzkerneltheorem 7 Sobolev spaces on rough domains References 5 Linear Elliptic Equations 1 Existence and regularity of solutions to the Dirichlet problem 2 The weak and strong maximum principles 3 The Dirichlet problem on the ba 4 The Riemann mapping theorem （smooth boundary） 5 The Dirichlet problem on a domain with a rough boundary 6 The Riemann mapping theorem （rough boundary） 7 The Neumann boundary problem 8 The Hodge decomposition and harmonic forms 9 Natural boundary problems for the Hodge Laplacian 10 Isothermal coordinates and conformal structures on surfaces 11 General elliptic boundary problems 12 Operator properties ofregular boundary problems A Spaces of generalized functions on manifolds with boundary B The Mayer-Vietoris sequ6nce in deRham cohomology References 6 Linear Evolution Equations 1 The heat equation and the wave equation on bounded domains 2 The heat equation and wave equation on unbounded domains 3 Maxwell's equations 4 TheCauchy-Kowalewsky theorem 5 Hyperbolic systems 6 Geometrical optics 7 The formation of caustics 8 Boundary layer phenomena for the heat semigroup A Some Banach spaces of harmonic functions B The stationary phase method References A Outline of Functional Analysis 1 Banach spaces 2 Hilbert spaces 3 Fr6chet spaces; locally convex spaces 4 Duality 5 Linear operators 6 Compact operators 7 Fredholm operators 8 Unbounded operators 9 Semigroups
References B Manifolds, Vector Bundles, and Lie Groups 1 Metric spaces and topological spaces 2 Manifolds 3 Vector bundles 4 Sard's theorem 5 Lie groups 6 The Campbell-Hausdorffformula 7 Representations of Lie groups and Lie algebras 8 Representations of compact Lie groups 9 Representations of SU(2) and related groups References Index
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