An Introduction to the Geometrical Analysis of Vector Fields

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 SpaceThe Exponential TheoremThe Composition of Flows of Vector FieldsHadamard's Theorem for FlowsThe CBHD Operation on Finite Dimensional Lie AlgebrasThe Connectivity TheoremThe Carnot-Carathéodory DistanceThe Weak Maximum PrincipleCorollaries of the Weak Maximum PrincipleThe Maximum Propagation PrincipleThe Maximum Propagation along the DriftThe Differential of the Flow wrt its ParametersThe Exponential Theorem for ODEsThe Exponential Theorem for Lie GroupsThe Local Third Theorem of LieConstruction of Carnot GroupsExponentiation of Vector Field Algebras into Lie GroupsOn the Convergence of the CBHD SeriesAppendices:Some Prerequisites of Linear AlgebraDependence Theory for ODEsA Brief Review of Lie Group TheoryFurther ReadingsList of AbbreviationsBibliographyIndex
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 exercisesThe multi-disciplinary nature of the book, covering topics from Analysis (ODE/PDE theory), Geometry (Lie groups, vector fields), Algebra/Linear Algebra (noncommutative structures)