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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