Partial Differential Equations

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-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 Geodesies.- 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.- 2 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.- 9 The symbol of a differential operator and a general Green-Stokes formula.- 10 The Hodge Laplacian on k-forms.- 11 Maxwell’s equations.- References.- 3 Fourier Analysis, Distributions, 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 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.- The mighty Gaussian and the sublime gamma function.- References.- 4 Sobolev Spaces.- 1 Sobolev spaces on ?n.- 2 The complex interpolation method.- 3 Sobolev spaces on compact manifolds.- 4 Sobolev spaces on bounded domains.- 5 The Sobolev spaces Hs0(?).- 6 The Schwartz kernel theorem.- 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 ball in ?n.- 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 of regular boundary problems.- Spaces of generalized functions on manifolds with boundary.- The Mayer-Vietoris sequence 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 The Cauchy-Kowalewsky theorem.- 5 Hyperbolic systems.- 6 Geometrical optics.- 7 The formation of caustics.- Some Banach spaces of harmonic functions.- The stationary phase method.- References.- A Outline of Functional Analysis.- 1 Banach spaces.- 2 Hilbert spaces.- 3 Fréchet 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-Hausdorff formula.- 7 Representations of Lie groups and Lie algebras.- 8 Representations of compact Lie groups.- 9 Representations of SU(2) and related groups.- References.