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El. knyga: Fundamentals of Advanced Mathematics V3

(Conservatoire National des Arts et Metiers, France)
  • Formatas: EPUB+DRM
  • Išleidimo metai: 11-Oct-2019
  • Leidėjas: ISTE Press Ltd - Elsevier Inc
  • Kalba: eng
  • ISBN-13: 9780081023860
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Fundamentals of Advanced Mathematics V3Didesnis paveikslėlis
  • Formatas: EPUB+DRM
  • Išleidimo metai: 11-Oct-2019
  • Leidėjas: ISTE Press Ltd - Elsevier Inc
  • Kalba: eng
  • ISBN-13: 9780081023860
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Fundamentals of Advanced Mathematics, Volume Three begins with the study of differential and analytic infinite-dimensional manifolds, then progresses into fibered bundles, in particular, tangent and cotangent bundles. In addition, subjects covered include the tensor calculus on manifolds, differential and integral calculus on manifolds (general Stokes formula, integral curves and manifolds), an analysis on Lie groups, the Haar measure, the convolution of functions and distributions, and the harmonic analysis over a Lie group. Finally, the theory of connections is (linear connections, principal connections, and Cartan connections) covered, as is the calculus of variations in Lagrangian and Hamiltonian formulations.

This volume is the prerequisite to the analytic and geometric study of nonlinear systems.

  • Includes sections on differential and analytic manifolds, vector bundles, tensors, Lie derivatives, applications to algebraic topology, and more
  • Presents an ideal prerequisite resource on the analytic and geometric study of nonlinear systems
  • Provides theory as well as practical information

Recenzijos

"The present volume is the third one of a series which presents the fundamental elements of advanced mathematics that is at the basis of a number of contemporary scientific methods. More precisely, it deals with differential and integral calculus in their local and global components. The book is designed not only for mathematicians, but also for everyone who uses mathematics and needs to understand the control of nonlinear systems (in particular physicists and engineers). The ambitious goal is achieved also thanks to an excellent organization of the topics and the use of a very clear and understandable language. Interesting short historical notes introduce the different topics and help to frame the evolution of concept. The exposition is illustrated with some figures that help a lot in understanding the not easy topics. Very useful attachments are provided: a careful list of notation and term indeces, a reach bibliography, a list of cited authors with biographical notes." --ZBMath

"The book under review is the third volume in a series that lays a solid foundation for advanced mathematics, serving as a fundamental resource for various contemporary scientific methodologies. This particular volume explores the intricate realms of both differential and integral calculus, providing a comprehensive examination of their local and global components. While primarily intended for mathematicians, the book transcends disciplinary boundaries and aims to be read by individuals from diverse fields who utilize mathematics in their work, including physicists and engineers." --MathSciNet

Preface xi
Errata for Volume 1 and Volume 2 xv
List of Notations xix
Chapter 1 Differential Calculus
1.1 Introduction
1(1)
1.2 Frechet differential calculus
2(25)
1.2.1 General conventions
2(3)
1.2.2 Frechet differential
5(4)
1.2.3 Mappings of class CP
9(3)
1.2.4 Taylor's formulas
12(4)
1.2.5 Analytic functions
16(3)
1.2.6 The implicit function theorem and its consequences
19(8)
1.3 Other approaches to differential calculus
27(8)
1.3.1 Lagrange variations and Gateaux differentials
27(2)
1.3.2 Calculus of variations: elementary concepts
29(3)
1.3.3 "Convenient" differentials
32(3)
1.4 Smooth partitions of unity
35(2)
1.4.1 Cinfinity-paracompactness of Banach spaces
35(1)
1.4.2 cinfinity-paracompactness
36(1)
1.5 Ordinary differential equations
37(12)
1.5.1 Existence and uniqueness theorems
37(6)
1.5.2 Linear differential equations
43(2)
1.5.3 Parameter dependence of solutions
45(4)
Chapter 2 Differential and Analytic Manifolds 49(44)
2.1 Introduction
49(1)
2.2 Manifolds: tangent space of a manifold at a point
50(15)
2.2.1 Notion of a manifold
50(6)
2.2.2 Morphisms of manifolds
56(2)
2.2.3 Tangent mappings
58(1)
2.2.4 Tangent vectors
58(7)
2.3 Tangent linear mappings; submanifolds
65(16)
2.3.1 Tangent linear mapping; rank
65(1)
2.3.2 Differential
66(1)
2.3.3 Submanifolds
67(1)
2.3.4 Immersions and embeddings
68(3)
2.3.5 Submersions, subimmersions and etale mappings
71(3)
2.3.6 Submanifolds of Kn
74(1)
2.3.7 Products of manifolds
75(1)
2.3.8 Transversal morphisms and manifolds
76(2)
2.3.9 Fiber product of manifolds
78(1)
2.3.10 Covectors and cotangent spaces
79(1)
2.3.11 Cotangent linear mapping
80(1)
2.4 Lie groups and their actions
81(12)
2.4.1 Lie groups
81(7)
2.4.2 Manifolds of orbits and homogeneous manifolds
88(5)
Chapter 3 Fiber Bundles 93(38)
3.1 Introduction
93(1)
3.2 Tangent bundle and cotangent bundle
94(4)
3.2.1 Tangent bundle
94(2)
3.2.2 Cotangent bundle
96(2)
3.2.3 Tangent bundle and cotangent bundle functors
98(1)
3.3 Fibrations
98(10)
3.3.1 Notion of a fibration
99(2)
3.3.2 Fiber product and preimage of fibrations
101(2)
3.3.3 Coverings
103(4)
3.3.4 Sections
107(1)
3.4 Vector bundles
108(13)
3.4.1 Vector bundles
108(4)
3.4.2 Dual of a vector bundle
112(1)
3.4.3 Subbundles and quotient bundles
113(1)
3.4.4 Whitney sum and tensor product
114(1)
3.4.5 The category of vector bundles
115(5)
3.4.6 Preimage of a fiber bundle
120(1)
3.5 Principal bundles
121(10)
3.5.1 Notion of a principal bundle
121(2)
3.5.2 Vertical tangent vectors
123(1)
3.5.3 Morphisms of principal bundles
124(1)
3.5.4 Principal bundles defined by cocycles
124(1)
3.5.5 Fiber bundle associated with a principal bundle
125(1)
3.5.6 Extension, restriction, quotientization of the structural group
126(2)
3.5.7 Examples of trivial principal bundles
128(3)
Chapter 4 Tensor Calculus on Manifolds 131(42)
4.1 Introduction
131(1)
4.2 Tensor calculus
132(13)
4.2.1 Tensors
132(3)
4.2.2 Symmetric tensors and antisymmetric tensors
135(3)
4.2.3 Exterior algebra
138(1)
4.2.4 Duality in the exterior algebra
139(2)
4.2.5 Interior products
141(2)
4.2.6 Tensors on Banach spaces
143(2)
4.3 Tensor fields
145(3)
4.3.1 Vector fields
145(1)
4.3.2 Covector field
146(1)
4.3.3 Tensor fields and scalar fields
146(2)
4.4 Differential forms
148(22)
4.4.1 Differential forms of degree p
148(1)
4.4.2 Preimage of a differential p-form
149(2)
4.4.3 Differential forms taking values in a fiber bundle. List of formulas
151(3)
4.4.4 Orientation
154(3)
4.4.5 Integral of a differential form of maximal degree
157(6)
4.4.6 Differential forms of odd type
163(3)
4.4.7 Integration of a differential form over a chain
166(4)
4.5 Pseudo-Riemannian manifolds
170(3)
4.5.1 Metric
170(1)
4.5.2 Pseudo-Riemannian volume element
171(2)
Chapter 5 Differential and Integral Calculus on Manifolds 173(72)
5.1 Introduction
173(1)
5.2 Currents and differential operators
174(9)
5.2.1 Currents and distributions
174(7)
5.2.2 Differential operators and point distributions
181(2)
5.3 Manifolds of mappings
183(4)
5.3.1 The Banach framework
183(3)
5.3.2 The "convenient" framework
186(1)
5.4 Lie derivatives
187(8)
5.4.1 Lie algebras
187(3)
5.4.2 Lie derivative of a function
190(2)
5.4.3 Lie brackets
192(1)
5.4.4 Lie derivative of vector, covector and tensor fields
193(1)
5.4.5 Lie derivative of a p-form
194(1)
5.5 Exterior differential
195(5)
5.5.1 E. Cartan's theorem
195(3)
5.5.2 Application to vector calculus
198(2)
5.6 Stokes' formula and applications
200(24)
5.6.1 Stokes' formula on a chain
200(3)
5.6.2 Ostrogradsky and Green formulas
203(3)
5.6.3 Hodge duality and codifferentials
206(7)
5.6.4 Gauss' theorem and Poisson's formula
213(2)
5.6.5 Homology, cohomology and duality
215(9)
5.7 Integral curves and manifolds
224(21)
5.7.1 First-order differential equations
224(4)
5.7.2 Second-order differential equations
228(1)
5.7.3 Sprays
229(2)
5.7.4 Straightening of vector fields and frames
231(2)
5.7.5 Integral manifolds, foliations
233(12)
Chapter 6 Analysis on Lie Groups 245(70)
6.1 Introduction
245(1)
6.2 Convolution
246(10)
6.2.1 Convolution of distributions
246(4)
6.2.2 Haar measure and convolution of functions
250(6)
6.3 Classification of Lie algebras
256(17)
6.3.1 Additional notions from algebra
256(3)
6.3.2 Classical Lie algebras
259(1)
6.3.3 General notions about Lie algebras
260(3)
6.3.4 Nilpotent Lie algebras
263(2)
6.3.5 Solvable Lie algebras
265(2)
6.3.6 Simple and semi-simple Lie algebras
267(4)
6.3.7 Reductive Lie algebras
271(1)
6.3.8 Real compact Lie algebras
272(1)
6.4 Relation between Lie groups and Lie algebras
273(11)
6.4.1 Lie algebra of a Lie group
273(5)
6.4.2 Passing from a Lie algebra to a Lie group
278(3)
6.4.3 Dictionary
281(3)
6.5 Harmonic analysis
284(31)
6.5.1 Introduction
284(2)
6.5.2 Harmonic analysis on Rn
286(10)
6.5.3 Fourier series and Fourier transforms on the torus
296(6)
6.5.4 Fourier transform on a locally compact commutative group
302(8)
6.5.5 Overview of non-commutative harmonic analysis
310(5)
Chapter 7 Connections 315(54)
7.1 Introduction
315(2)
7.2 Linear connections
317(16)
7.2.1 Curvilinear coordinates
317(6)
7.2.2 Linear connection on a vector bundle
323(2)
7.2.3 Linear connection on a manifold
325(2)
7.2.4 Parallel transport and geodesics
327(3)
7.2.5 Covariant exterior differential
330(1)
7.2.6 Curvature and torsion of a linear connection
331(2)
7.3 Method of moving frames
333(25)
7.3.1 Moving frame and gauge potential
334(3)
7.3.2 Curvature, torsion and covariant exterior differential of a G-connection
337(3)
7.3.3 Quasi-parallelogram method
340(4)
7.3.4 Fundamental equalities
344(1)
7.3.5 Connection form on the bundle of G-frames
345(2)
7.3.6 Principal connections and parallel transport
347(3)
7.3.7 Covariant exterior differentiation on a principal bundle
350(1)
7.3.8 Characterization of a G-connection
351(1)
7.3.9 Curvature and torsion forms of a principal connection
352(3)
7.3.10 Cartan connections
355(3)
7.4 Riemannian geometry
358(11)
7.4.1 Levi-Civita connection
358(2)
7.4.2 Geodesics
360(1)
7.4.3 Flat pseudo-Riemannian manifolds
361(2)
7.4.4 Ricci tensor and Einstein tensor
363(6)
References 369(10)
Cited Authors 379(8)
Index 387
Henri Bourlčs is Full Professor and Chair at the Conservatoire National des Arts et Métiers, Paris, France.
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