1 Basic Theory of ODE and Vector Fields 1
1 The derivative 3
2 Fundamental local existence theorem for ODE 9
3 Inverse function and implicit function theorems 12
4 Constant-coefficient linear systems;exponentiation of matrices 16
5 Variable-coefficient linear systems of ODE：Duhamel's principle 26
6 Dependence of solutions on initial data and on other parameters 31
7 Flows and vector fields 35
8 Lie brackets 40
9 Commuting flows;Frobenius's theorem 43
10 Hamiltonian systems 47
11 Geodesics 51
12 Variational problems and the stationary action principle 59
13 Differential forms 70
14 The symplectic form and canonical transformations 83
15 First-order,scalar,nonlinear PDE 89
16 Completely integrable hamiltonian systems 96
17 Examples of integrable systems;central force problems 101
18 Relativistic motion 105
19 Topological applications of differential forms 110
20 Critical points and index of a vector field 118
A Nonsmooth vector fields 122
References 125
2 The Laplace Equation and Wave Equation 127
1 Vibrating strings and membranes 129
2 The divergence of a vector field 140
3 The covariant derivative and divergence of tensor fields 145
4 The Laplace operator on a Riemannian manifold 153
5 The wave equation on a product manifold and energy conservation 156
6 Uniqueness and finitepropagation speed 162
7 Lorentz manifolds and stress-energy tensors 166
8 More general hyperbolic equations;energy estimates 172
9 The symbol of a differential operator and a general Green-Stokes formula 176
10 The Hodge Laplacian on k-forms 180
11 Maxwell's equations 184
References 194
3 Fourier Analysis,Distributions,and Constant-Coefficient LinearPDE 197
1 Fourier series 198
2 Harmonic functions and holomorphic functions in the plane 209
3 The Fourier transform 222
4 Distributions and tempered distributions 230
5 The classical evolution equations 244
6 Radial distributions,polar coordinates,and Bessel functions 263
7 The method ofimages and Poisson's summation formula 273
8 Homogeneous distributions and principal value distributions 278
9 Elliptic operators 286
10 Local solvability of constant-coefficient PDE 289
11 The discrete Fourier transform 292
12 The fast Fourier transform 301
A The mighty Gaussian and the sublime gamma function 306
References 312
4 Sobolev Spaces 315
1 Sobolev spaces on Rn 315
2 The complex interpolation method 321
3 Sobolev spaces on compact manifolds 328
4 Sobolev spaces on bounded domains 331
5 The Sobolev spaces Hs 0（Ω） 338
6 The Schwartz kernel theorem 345
7 Sobolev spaces on rough domains 349
References 351
5 Linear Elliptic Equations 353
1 Existence and regularity of solutions to the Dirichlet problem 354
2 The weak and strong maximum principles 364
3 The Dirichlet problem on the ball in Rn 373
4 The Riemann mapping theorem（smooth boundary） 379
5 The Dirichlet problem on a domain with a rough boundary 383
6 The Riemann mapping theorem（rough boundary） 398
7 The Neumann boundary problem 402
8 The Hodge decomposition and harmonic forms 410
9 Natural boundary problems for the Hodge Laplacian 421
10 Isothermal coordinates and conformal structures on surfaces 438
11 General elliptic boundary problems 441
12 Operator properties of regular boundary problems 462
A Spaces of generalized functions on manifolds with boundary 471
B The Mayer-Vietoris sequence in de Rham cohomology 475
References 478
6 Linear Evolution Equations 481
1 The heat equation and the wave equation on bounded domains 482
2 The heat equation and wave equation on unbounded domains 490
3 Maxwell's equations 496
4 The Cauchy-Kowalewsky theorem 499
5 Hyperbolic systems 504
6 Geometrical optics 510
7 The formation of caustics 518
8 Boundary layer phenomena for the heat semigroup 535
A Some Banach spaces of harmonic functions 541
B The stationary phase method 543
References 545
A Outline of Functional Analysis 549
1 Banach spaces 549
2 Hilbert spaces 556
3 Fréchet spaces;locally convex spaces 561
4 Duality 564
5 Linear operators 571
6 Compact operators 579
7 Fredholm operators 593
8 Unbounded operators 596
9 Semigroups 603
References 615
B Manifolds,Vector Bundles,and Lie Groups 617
1 Metric spaces and topological spaces 617
2 Manifolds 622
3 Vector bundles 624
4 Sard's theorem 626
5 Lie groups 627
6 The Campbell-Hausdorff formula 630
7 Representations of Lie groups and Lie algebras 632
8 Representations of compact Lie groups 636
9 Representations of SU（2）and related groups 641
References 647
Index 649