Title
A Mathematical Gift, 1: The Interplay Between Topology, Functions, Geometry, and Algebra (Mathematical World) (v. 1),New
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This book will bring the beauty and fun of mathematics to the classroom. It offers serious mathematics in a lively, readerfriendly style. Included are exercises and many figures illustrating the main concepts.The first chapter presents the geometry and topology of surfaces. Among other topics, the authors discuss the PoincarHopf theorem on critical points of vector fields on surfaces and the GaussBonnet theorem on the relation between curvature and topology (the Euler characteristic). The second chapter addresses various aspects of the concept of dimension, including the Peano curve and the Poincar approach. Also addressed is the structure of threedimensional manifolds. In particular, it is proved that the threedimensional sphere is the union of two doughnuts.This is the first of three volumes originating from a series of lectures given by the authors at Kyoto University (Japan).
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