## A Mathematical Gift, I: The Interplay Between Topology, Functions, Geometry, and AlgebraThis 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. |

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### 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 |

### Other editions - View all

A Mathematical Gift: The Interplay Between Topology, Functions ..., Volume 3 Kenji Ueno No preview available - 2003 |

### Common terms and phrases

appear assume becomes called Chapter circle closed surface color completely concept consider constant construct continuous coordinate correspondence cover critical points cube curvature curving define definition deformation described dimension dimensional direction discussed disk doughnut edges elliptic equation Euler characteristic example exists expression fact Figure flow four four-dimensional Gauss map Gaussian curvature geometry given gluing hyperbolic idea imagine indices infinite infinity intersection lecture length look mathematics means mechanics moves natural notion observe obtain one-dimensional origin parabolic physics pieces placing plane point of view positive possible problem proof prove quantum mechanics question radius real numbers region represents resulting shape shown in Figure shows sides slightly square string tangent plane theorem theory three-dimensional figures three-dimensional space three-dimensional sphere torus triangle two-dimensional understand unit sphere vector field vertices winding number