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分析 第3卷 英文PDF|Epub|txt|kindle电子书版本网盘下载
- （德）阿莫恩（HERBERTAMANN），JOACHIMESCHER著 著
- 出版社： 北京：世界图书北京出版公司
- ISBN：9787510047985
- 出版时间：2012
- 标注页数：468页
- 文件大小：71MB
- 文件页数：481页
- 主题词：分析（数学）－英文
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图书目录
Chapter Ⅸ Elements of measure theory3
1 Measurable spaces3
σ-algebras3
The Borel σ-algebra5
The second countability axiom6
Generating the Borel σ-algebra with intervals8
Bases of topological spaces9
The product topology10
Product Borelσ-algebras12
Measurability of sections13
2 Measures17
Set functions17
Measure spaces18
Properties of measures18
Null sets20
3 Outer measures24
The construction of outer measures24
The Lebesgue outer measure25
The Lebesgue-Stieltjes outer measure28
Hausdorff outer measures29
4 Measurable sets32
Motivation32
Theσ-algebra of μ-measurable sets33
Lebesgue measure and Hausdorff measure35
Metric measures36
5 The Lebesgue measure40
The Lebesgue measure space40
The Lebesgue measure is regular41
A characterization of Lebesgue measurability44
Images of Lebesgue measurable sets44
The Lebesgue measure is translation invariant47
A characterization of Lebesgue measure48
The Lebesgue measure is invariant under rigid motions50
The substitution rule for linear maps51
Sets without Lebesgue measure53
Chapter Ⅹ Integration theory62
1 Measurable functions62
Simple functions and measurable functions62
A measurability criterion64
Measurable ？-valued functions67
The lattice of measurable ？-valued functions68
Pointwise limits of measurable functions73
Radon measures74
2 Integrable functions80
The integral of a simple function80
The L1-seminorm82
The Bochner-Lebesgue integral84
The completeness of L187
Elementary properties of integrals88
Convergence in L191
3 Convergence theorems97
Integration of nonnegative ？-valued functions97
The monotone convergence theorem100
Fatou's lemma101
Integration of ？-valued functions103
Lebesgue's dominated convergence theorem104
Parametrized integrals107
4 Lebesgue spaces110
Essentially bounded functions110
The H？lder and Minkowski inequalities111
Lebesgue spaces are complete114
Lp-spaces116
Continuous functions with compact support118
Embeddings119
Continuous linear functionals on Lp121
5 The n-dimensional Bochner-Lebesgue integral128
Lebesgue measure spaces128
The Lebesgue integral of absolutely integrable functions129
A characterization of Riemann integrable functions132
6 Fubini's theorem137
Maps defined almost everywhere137
Cavalieri's principle138
Applications of Cavalieri's principle141
Tonelli's theorem144
Fubini's theorem for scalar functions145
Fubini's theorem for vector-valued functions148
Minkowski's inequality for integrals152
A characterization of Lp(Rm+n,E)157
A trace theorem158
7 The convolution162
Defining the convolution162
The translation group165
Elementary properties of the convolution168
Approximations to the identity170
Test functions172
Smooth partitions of unity173
Convolutions of E-valued functions177
Distributions177
Linear differential operators181
Weak derivatives184
8 The substitution rule191
Pulling back the Lebesgue measure191
The substitution rule:general case195
Plane polar coordinates197
Polar coordinates in higher dimensions198
Integration of rotationally symmetric functions202
The substitution rule for vector-valued functions203
9 The Fourier transform206
Definition and elementary properties206
The space of rapidly decreasing functions208
The convolution algebra S211
Calculations with the Fourier transform212
The Fourier integral theorem215
Convolutions and the Fourier transform218
Fourier multiplication operators220
Plancherel's theorem223
Symmetric operators225
The Heisenberg uncertainty relation227
Chapter Ⅺ Manifolds and differential forms235
1 Submanifolds235
Definitions and elementary properties235
Submersions241
Submanifolds with boundary246
Local charts250
Tangents and normals251
The regular value theorem252
One-dimensional manifolds256
Partitions of unity256
2 Multilinear algebra260
Exterior products260
Pull backs267
The volume element268
The Riesz isomorphism271
The Hodge star operator273
Indefinite inner products277
Tensors281
3 The local theory of differential forms285
Definitions and basis representations285
Pull backs289
The exterior derivative292
The Poincaré lemma295
Tensors299
4 Vector fields and differential forms304
Vector fields304
Local basis representation306
Differential forms308
Local representations311
Coordinate transformations316
The exterior derivative319
Closed and exact forms321
Contractions322
Orientability324
Tensor fields330
5 Riemannian metrics333
The volume element333
Riemannian manifolds337
The Hodge star348
The codifferential350
6 Vector analysis358
The Riesz isomorphism358
The gradient361
The divergence363
The Laplace-Beltrami operator367
The curl372
The Lie derivative375
The Hodge-Laplace operator379
The vector product and the curl382
Chapter Ⅻ Integration on manifolds391
1 Volume measure391
The Lebesgue σ-algebra of M391
The definition of the volume measure392
Properties397
Integrability398
Calculation of several volumes401
2 Integration of differential forms407
Integrals of m-forms407
Restrictions to submanifolds409
The transformation theorem414
Fubini's theorem415
Calculations of several integrals418
Flows of vector fields421
The transport theorem425
3 Stokes's theorem430
Stokes's theorem for smooth manifolds430
Manifolds with singularities432
Stokes's theorem with singularities436
Planar domains439
Higher-dimensional problems441
Homotopy invariance and applications443
Gauss's law446
Green's formula448
The classical Stokes's theorem450
The star operator and the coderivative451
References457