## A Mathematical Gift: The Interplay Between Topology, Functions, Geometry, and Algebra, Volume 3This book brings the beauty and fun of mathematics to the classroom. It offers serious mathematics in a lively, reader-friendly style. Included are exercises and many figures illustrating the main concepts. The first chapter talks about the theory of manifolds. It includes discussion of smoothness, differentiability, and analyticity, the idea of local coordinates and coordinate transformation, and a detailed explanation of the Whitney imbedding theorem (both in weak and in strong form). The second chapter discusses the notion of the area of a figure on the plane and the volume of a solid body in space. It includes the proof of the Bolyai-Gerwien theorem about scissors-congruent polynomials and Dehn's solution of the Third Hilbert Problem. This is the third volume originating from a series of lectures given at Kyoto University (Japan). It is suitable for classroom use for high school mathematics teachers and for undergraduate mathematics courses in the sciences and liberal arts. The first and second volumes are available as Volume 19 and Volume 20 in the AMS series, Mathematical World. |

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### Contents

The Birth of Manifolds | 35 |

The Story of Area and Volume from Everyday | 61 |

ScissorsCongruent Polygons | 89 |

ScissorsCongruent Polyhedra | 109 |

### Other editions - View all

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

### Common terms and phrases

20th century AABC ABCD additive function algebra analytic functions area function Axiom of Choice Banach-Tarski Paradox base called complicated figures concept of area congruent consider construct continuous functions coordinate plane coordinate systems coordinate transformations corresponding curve definition Dehn,s Theorem denote differentiable functions dihedral angles dimension discuss edges embedded equation example exists formula geometry given graph Grassmann Manifold height Hence idea infinite intersection introduced invariant irrational number Klein Bottle lecture Lemma local coordinates m(Ki mapping math mathematical concept mathematicians method metric space n-dimensional manifold natural numbers normal plane notion number line number of squares obtained open sets parcels of land partial derivatives pieces polygon polyhedra polyhedron prism problem projective plane projective space proof property of smoothness prove question rational number readers real numbers rectangle region scissors-congruent sequence of points set theory side square decomposition Suppose surface topological space variables volume Whitney Whitney Embedding Theorem words