分析 第2卷 英文PDF|Epub|txt|kindle电子书版本网盘下载

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Chapter Ⅵ Integral calculus in one variable4

1 Jump continuous functions4

Staircase and jump continuous functions4

A characterization of jump continuous functions6

The Banach space of jump continuous functions7

2 Continuous extensions10

The extension of uniformly continuous functions10

Bounded linear operators12

The continuous extension of bounded linear operators15

3 The Cauchy-Riemann Integral17

The integral of staircase functions17

The integral of jump continuous functions19

Riemann sums20

4 Properties of integrals25

Integration of sequences of functions25

The oriented integral26

Positivity and monotony of integrals27

Componentwise integration30

The first fundamental theorem of calculus30

The indefinite integral32

The mean value theorem for integrals33

5 The technique of integration38

Variable substitution38

Integration by parts40

The integrals of rational functions43

6 Sums and integrals50

The Bernoulli numbers50

Recursion formulas52

The Bernoulli polynomials53

The Euler-Maclaurin sum formula54

Power sums56

Asymptotic equivalence57

The Riemann ζ function59

The trapezoid rule64

7 Fourier series67

The L2 scalar product67

Approximating in the quadratic mean69

Orthonormal systems71

Integrating periodic functions72

Fourier coefficients73

Classical Fourier series74

Bessel's inequality77

Complete orthonormal systems79

Piecewise continuously differentiable functions82

Uniform convergence83

8 Improper integrals90

Admissible functions90

Improper integrals90

The integral comparison test for series93

Absolutely convergent integrals94

The majorant criterion95

9 The gamma function98

Euler's integral representation98

The gamma function on C＼（-N）99

Gauss's representation formula100

The reflection formula104

The logarithmic convexity of the gamma function105

Stirling's formula108

The Euler beta integral110

Chapter Ⅶ Multivariable differential calculus118

1 Continuous linear maps118

The completeness of L（E,F）118

Finite-dimensional Banach spaces119

Matrix representations122

The exponential map125

Linear difierential equations128

Gronwall's lemma129

The variation of constants formula131

Determinants and eigenvalues133

Fundamental matrices136

Second order linear differential equations140

2 Differentiability149

The definition149

The derivative150

Directional derivatives152

Partial derivatives153

The Jacobi matrix155

A differentiability criterion156

The Riesz representation theorem158

The gradient159

Complex differentiability162

3 Multivariable differentiation rules166

Linearity166

The chain rule166

The product rule169

The mean value theorem169

The differentiability of limits of sequences of functions171

Necessary condition for local extrema171

4 Multilinear maps173

Continuous multilinear maps173

The canonical isomorphism175

Symmetric multilinear maps176

The derivative of multilinear maps177

5 Higher derivatives180

Definitions180

Higher order partial derivatives183

The chain rule185

Taylor's formula185

Functions of m variables186

Sufficient criterion for local extrema188

6 Nemytskii operators and the calculus of variations195

Nemytskii operators195

The continuity of Nemytskii operators195

The differentiability of Nemytskii operators197

The differentiability of parameter-dependent integrals200

Variational problems202

The Euler-Lagrange equation204

Classical mechanics207

7 Inverse maps212

The derivative of the inverse of linear maps212

The inverse function theorem214

Diffeomorphisms217

The solvability of nonlinear systems of equations218

8 Implicit functions221

Differentiable maps on product spaces221

The implicit function theorem223

Regular values226

Ordinary differential equations226

Separation of variables229

Lipschitz continuity and uniqueness233

The Picard-Lindel？f theorem235

9 Manifolds242

Submanifolds of Rn242

Graphs243

The regular value theorem243

The immersion theorem244

Embeddings247

Local charts and parametrizations252

Change of charts255

10 Tangents and normals260

The tangential in Rn260

The tangential space261

Characterization of the tangential space265

Differentiable maps266

The differential and the gradient269

Normals271

Constrained extrema272

Applications of Lagrange multipliers273

Chapter Ⅷ Line integrals281

1 Curves and their lengths281

The total variation281

Rectifiable paths282

Differentiable curves284

Rectifiable curves286

2 Curves in Rn292

Unit tangent vectors292

Parametrization by arc length293

Oriented bases294

The Frenet n-frame295

Curvature of plane curves298

Identifying lines and circles300

Instantaneous circles along curves300

The vector product302

The curvature and torsion of space curves303

3 Pfaff forms308

Vector fields and Pfaff forms308

The canonical basis310

Exact forms and gradient fields312

The Poincaré lemma314

Dual operators316

Transformation rules317

Modules321

4 Line integrals326

The definition326

Elementary properties328

The fundamental theorem of line integrals330

Simply connected sets332

The homotopy invariance of line integrals333

5 Holomorphic functions339

Complex line integrals339

Holomorphism342

The Cauchy integral theorem343

The orientation of circles344

The Cauchy integral formula345

Analytic functions346

Liouville's theorem348

The Fresnel integral349

The maximum principle350

Harmonic functions351

Goursat's theorem353

The Weierstrass convergence theorem356

6 Meromorphic functions360

The Laurent expansion360

Removable singularities364

Isolated singularities365

Simple poles368

The winding number370

The continuity of the winding number374

The generalized Cauchy integral theorem376

The residue theorem378

Fourier integrals379

References387

Index389