分析 第2卷 英文PDF格式文档图书下载

Chapter Ⅵ Integral calculus in one variable 4

1 Jump continuous functions 4

Staircase and jump continuous functions 4

A characterization of jump continuous functions 6

The Banach space of jump continuous functions 7

2 Continuous extensions 10

The extension of uniformly continuous functions 10

Bounded linear operators 12

The continuous extension of bounded linear operators 15

3 The Cauchy-Riemann Integral 17

The integral of staircase functions 17

The integral of jump continuous functions 19

Riemann sums 20

4 Properties of integrals 25

Integration of sequences of functions 25

The oriented integral 26

Positivity and monotony of integrals 27

Componentwise integration 30

The first fundamental theorem of calculus 30

The indefinite integral 32

The mean value theorem for integrals 33

5 The technique of integration 38

Variable substitution 38

Integration by parts 40

The integrals of rational functions 43

6 Sums and integrals 50

The Bernoulli numbers 50

Recursion formulas 52

The Bernoulli polynomials 53

The Euler-Maclaurin sum formula 54

Power sums 56

Asymptotic equivalence 57

The Riemann ζ function 59

The trapezoid rule 64

7 Fourier series 67

The L2 scalar product 67

Approximating in the quadratic mean 69

Orthonormal systems 71

Integrating periodic functions 72

Fourier coefficients 73

Classical Fourier series 74

Bessel's inequality 77

Complete orthonormal systems 79

Piecewise continuously differentiable functions 82

Uniform convergence 83

8 Improper integrals 90

Admissible functions 90

Improper integrals 90

The integral comparison test for series 93

Absolutely convergent integrals 94

The majorant criterion 95

9 The gamma function 98

Euler's integral representation 98

The gamma function on C＼（-N） 99

Gauss's representation formula 100

The reflection formula 104

The logarithmic convexity of the gamma function 105

Stirling's formula 108

The Euler beta integral 110

Chapter Ⅶ Multivariable differential calculus 118

1 Continuous linear maps 118

The completeness of L（E,F） 118

Finite-dimensional Banach spaces 119

Matrix representations 122

The exponential map 125

Linear difierential equations 128

Gronwall's lemma 129

The variation of constants formula 131

Determinants and eigenvalues 133

Fundamental matrices 136

Second order linear differential equations 140

2 Differentiability 149

The definition 149

The derivative 150

Directional derivatives 152

Partial derivatives 153

The Jacobi matrix 155

A differentiability criterion 156

The Riesz representation theorem 158

The gradient 159

Complex differentiability 162

3 Multivariable differentiation rules 166

Linearity 166

The chain rule 166

The product rule 169

The mean value theorem 169

The differentiability of limits of sequences of functions 171

Necessary condition for local extrema 171

4 Multilinear maps 173

Continuous multilinear maps 173

The canonical isomorphism 175

Symmetric multilinear maps 176

The derivative of multilinear maps 177

5 Higher derivatives 180

Definitions 180

Higher order partial derivatives 183

The chain rule 185

Taylor's formula 185

Functions of m variables 186

Sufficient criterion for local extrema 188

6 Nemytskii operators and the calculus of variations 195

Nemytskii operators 195

The continuity of Nemytskii operators 195

The differentiability of Nemytskii operators 197

The differentiability of parameter-dependent integrals 200

Variational problems 202

The Euler-Lagrange equation 204

Classical mechanics 207

7 Inverse maps 212

The derivative of the inverse of linear maps 212

The inverse function theorem 214

Diffeomorphisms 217

The solvability of nonlinear systems of equations 218

8 Implicit functions 221

Differentiable maps on product spaces 221

The implicit function theorem 223

Regular values 226

Ordinary differential equations 226

Separation of variables 229

Lipschitz continuity and uniqueness 233

The Picard-Lindel？f theorem 235

9 Manifolds 242

Submanifolds of Rn 242

Graphs 243

The regular value theorem 243

The immersion theorem 244

Embeddings 247

Local charts and parametrizations 252

Change of charts 255

10 Tangents and normals 260

The tangential in Rn 260

The tangential space 261

Characterization of the tangential space 265

Differentiable maps 266

The differential and the gradient 269

Normals 271

Constrained extrema 272

Applications of Lagrange multipliers 273

Chapter Ⅷ Line integrals 281

1 Curves and their lengths 281

The total variation 281

Rectifiable paths 282

Differentiable curves 284

Rectifiable curves 286

2 Curves in Rn 292

Unit tangent vectors 292

Parametrization by arc length 293

Oriented bases 294

The Frenet n-frame 295

Curvature of plane curves 298

Identifying lines and circles 300

Instantaneous circles along curves 300

The vector product 302

The curvature and torsion of space curves 303

3 Pfaff forms 308

Vector fields and Pfaff forms 308

The canonical basis 310

Exact forms and gradient fields 312

The Poincaré lemma 314

Dual operators 316

Transformation rules 317

Modules 321

4 Line integrals 326

The definition 326

Elementary properties 328

The fundamental theorem of line integrals 330

Simply connected sets 332

The homotopy invariance of line integrals 333

5 Holomorphic functions 339

Complex line integrals 339

Holomorphism 342

The Cauchy integral theorem 343

The orientation of circles 344

The Cauchy integral formula 345

Analytic functions 346

Liouville's theorem 348

The Fresnel integral 349

The maximum principle 350

Harmonic functions 351

Goursat's theorem 353

The Weierstrass convergence theorem 356

6 Meromorphic functions 360

The Laurent expansion 360

Removable singularities 364

Isolated singularities 365

Simple poles 368

The winding number 370

The continuity of the winding number 374

The generalized Cauchy integral theorem 376

The residue theorem 378

Fourier integrals 379

References 387

Index 389

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