A Mathematical Gift, I: The Interplay Between Topology, Functions, Geometry, and Algebra

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American Mathematical Soc., 2003 - Mathematics - 136 pages
This book will bring the beauty and fun of learning mathematics to the classroom. It offers serious mathematics in a lively, reader-friendly style. Included are exercises and many figures that illustrate the main concepts. The first chapter presents the geometry and topology of surfaces. Among other topics, the authors discuss the Poincare-Hopf theorem on critical points of vector fields on surfaces and the Gauss-Bonnet theorem on the relation between curvature and topology (the Euler characteristics). The second chapter addresses various aspects of the concept of dimension, including the Peano curve and the Poincare approach. Also addressed are the structure of three-dimensional manifolds, in particular, provided that the three-dimensional 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). It is suitable for classroom use for high school mathematics teachers and undergraduate courses in science and liberal arts.
 

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Contents

Invitation to Topology Viewing Figures Globally
1
Introduction
3
The Euler Characteristic
5
Vortices Created by Winds and the Euler Characteristic
25
Curvature of a Surface and the Euler Characteristic
43
The Story of Dimension
75
Introduction
77
Learning to Appreciate Dimension
79
What is Dimension?
89
ThreeDimensional Figures
107
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