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Ⅰ Differentiation and Integration on Manifolds 1

1 The Weierstraв approximation theorem 2

2 Parameter-invariant integrals and differential forms 12

3 The exterior derivative of differential forms 23

4 The Stokes integral theorem for manifolds 30

5 The integral theorems of Gauв and Stokes 39

6 Curvilinear integrals 56

7 The lemma of Poincaré 67

8 Co-derivatives and the Laplace-Beltrami operator 72

9 Some historical notices to chapter Ⅰ 89

Ⅱ Foundations of Functional Analysis 91

1 Daniell's integral with examples 91

2 Extension of Daniell's integral to Lebesgue's integral 96

3 Measurable sets 109

4 Measurable functions 121

5 Riemann's and Lebesgue's integral on rectangles 134

6 Banach and Hilbert spaces 140

7 The Lebesgue spaces Lp（X） 151

8 Bounded linear functionals on Lp（X） and weak convergence 161

9 Some historical notices to chapter Ⅱ 172

Ⅲ Brouwer's Degree of Mapping with Geometric Applications 175

1 The winding number 175

2 The degree of mapping in In 184

3 Geometric existence theorems 193

4 The index of a mapping 195

5 The product theorem 204

6 Theorems of Jordan-Brouwer 210

Ⅳ Generalized Analytic Functions 215

1 The Cauchy-Riemann differential equation 215

2 Holomorphic functions in Cn 219

3 Geometric behavior of holomorphic functions in C 233

4 Isolated singularities and the general residue theorem 242

5 The inhomogeneous Cauchy-Riemann differential equation 255

6 Pseudoholomorphic functions 266

7 Conformal mappings 270

8 Boundary behavior of conformal mappings 285

9 Some historical notices to chapter Ⅳ 295

Ⅴ Potential Theory and Spherical Harmonics 297

1 Poisson's differential equation in Rn 297

2 Poisson's integral formula with applications 310

3 Dirichlet's problem for the Laplace equation in Rn 321

4 Theory of spherical harmonics: Fourier series 334

5 Theory of spherical harmonics in n variables 340

Ⅵ Linear Partial Differential Equations in 1n 355

1 The maximum principle for elliptic differential equations 355

2 Quasilinear elliptic differential equations 365

3 The heat equation 370

4 Characteristic surfaces 384

5 The wave equation in Rn for n＝1,3,2 395

6 The wave equation in Rn forn≥2 403

7 The inhomogeneous wave equation and an initial-boundary-value problem 414

8 Classification, transformation and reduction of partial differential equations 419

9 Some historical notices to the chapters Ⅳ and Ⅵ 428

References 431

Index 433