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An Introduction to the Geometrical Analysis of Vector Fields (eBook)

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  • 121,790 Words
  • 452 Pages

This book provides the reader with a gentle path through the multifaceted theory of vector fields, starting from the definitions and the basic properties of vector fields and flows, and ending with some of their countless applications, in the framework of what is nowadays called Geometrical Analysis. Once the background material is established, the applications mainly deal with the following meaningful settings:


Contents:
  • Flows of Vector Fields in Space
  • The Exponential Theorem
  • The Composition of Flows of Vector Fields
  • Hadamard's Theorem for Flows
  • The CBHD Operation on Finite Dimensional Lie Algebras
  • The Connectivity Theorem
  • The Carnot-Carath√©odory Distance
  • The Weak Maximum Principle
  • Corollaries of the Weak Maximum Principle
  • The Maximum Propagation Principle
  • The Maximum Propagation along the Drift
  • The Differential of the Flow wrt its Parameters
  • The Exponential Theorem for ODEs
  • The Exponential Theorem for Lie Groups
  • The Local Third Theorem of Lie
  • Construction of Carnot Groups
  • Exponentiation of Vector Field Algebras into Lie Groups
  • On the Convergence of the CBHD Series
  • Appendices:
    • Some Prerequisites of Linear Algebra
    • Dependence Theory for ODEs
    • A Brief Review of Lie Group Theory
  • Further Readings
  • List of Abbreviations
  • Bibliography
  • Index

Readership: Graduate students and researchers in geometrical analysis.
Key Features:
  • Its original point of view: Ordinary Differential Equation Theory is used as a basis to develop, in a UNITARY WAY, all the topics of the book: from Maximum Principles (maximum propagation, etc.), to Geometrical Analysis (flows, differentials, etc.), from Lie Group Theory (construction of Lie groups, etc.), to Control Theory (connectivity, composition of flows, etc.)
  • Its teachability at many levels (graduate and undergraduate, PhD, research book), due to its essential SELF-CONTAINEDNESS and the presence of several exercises
  • The multi-disciplinary nature of the book, covering topics from Analysis (ODE/PDE theory), Geometry (Lie groups, vector fields), Algebra/Linear Algebra (noncommutative structures)

This book provides the reader with a gentle path through the multifaceted theory of vector fields, starting from the definitions and the basic properties of vector fields and flows, and ending with some of their countless applications, in the framework of what is nowadays called Geometrical Analysis. Once the background material is established, the applications mainly deal with the following meaningful settings:


Contents:
  • Flows of Vector Fields in Space
  • The Exponential Theorem
  • The Composition of Flows of Vector Fields
  • Hadamard's Theorem for Flows
  • The CBHD Operation on Finite Dimensional Lie Algebras
  • The Connectivity Theorem
  • The Carnot-Carath√©odory Distance
  • The Weak Maximum Principle
  • Corollaries of the Weak Maximum Principle
  • The Maximum Propagation Principle
  • The Maximum Propagation along the Drift
  • The Differential of the Flow wrt its Parameters
  • The Exponential Theorem for ODEs
  • The Exponential Theorem for Lie Groups
  • The Local Third Theorem of Lie
  • Construction of Carnot Groups
  • Exponentiation of Vector Field Algebras into Lie Groups
  • On the Convergence of the CBHD Series
  • Appendices:
    • Some Prerequisites of Linear Algebra
    • Dependence Theory for ODEs
    • A Brief Review of Lie Group Theory
  • Further Readings
  • List of Abbreviations
  • Bibliography
  • Index

Readership: Graduate students and researchers in geometrical analysis.
Key Features:
  • Its original point of view: Ordinary Differential Equation Theory is used as a basis to develop, in a UNITARY WAY, all the topics of the book: from Maximum Principles (maximum propagation, etc.), to Geometrical Analysis (flows, differentials, etc.), from Lie Group Theory (construction of Lie groups, etc.), to Control Theory (connectivity, composition of flows, etc.)
  • Its teachability at many levels (graduate and undergraduate, PhD, research book), due to its essential SELF-CONTAINEDNESS and the presence of several exercises
  • The multi-disciplinary nature of the book, covering topics from Analysis (ODE/PDE theory), Geometry (Lie groups, vector fields), Algebra/Linear Algebra (noncommutative structures)


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by stefano biagi, andrea bonfiglioli

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