Contents of Volumes II and III
Preface
Basic Theory of ODE and Vector Fields
l 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 index of a vector field
A Nonsmooth vector fields
References
The Laplace Equation and Wave Equation
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 and finite propagation speed
7 Lorentz manifolds and stress-energy tensors
8 More general hyperbolic equations; energy estimates
……
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