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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