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open 2 no From the introduction to the monograph: Students of scientific disciplines usually meet vector fields during their undergraduate courses in connection with physics, when studying conservative forces and potentials. Then the vector fields reappear in the context of the differential geometry, in the manifold theory, and again in the theory of Lie groups and in the advanced theory of ordinary differential equations. The aim of this book is to provide the reader with a gentle path through the multifaceted theory of vector fields, starting from the definition and the basic properties of vector fields and flows, and ending with some of their countless applications. The building blocks of rich background material comprise the following topics: ∙ commutators and Lie derivatives; the semi-group property and the equation of variation; global vector fields, C^0,C^k,C^ω-dependence; ∙ relatedness, invariance and commutability; Hadamard-type formulas; ∙ composition of flows: Taylor approximation and exact formulas; ∙ the algebraic Campbell-Baker-Hausdorff-Dynkin Theorem; the relevant series and convergence; the Campbell-Baker-Hausdorff-Dynkin operation and its local associativity; Poincaré-type ordinary differential equations; ∙ iterated commutators and the Hörmander bracket-generating condition; connectivity; sub-unit curves and the control distance. Once the background material is established in the first part of the monograph, the application mainly deals, according to the choice of the authors, with the following three settings: ∙ ordinary differential equations theory; ∙ weak and strong maximum principles, propagation principles; ∙ Lie groups, with an emphasis on the construction of Lie groups. This book also provides an introduction to the basic theory of geometrical analysis with a new foundational presentation based on ordinary differential equations techniques, in a unitary and self-contained way. open S Biagi; A Bonfiglioli S Biagi; A Bonfiglioli