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
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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