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A Mathematical Gift, III: The interplay between topology, functions, geometry, and algebra
Koji Shiga, Yokohama, Japan, and Toshikazu Sunada, Tohoku University, Sendai, Japan

Mathematical World
2005; 129 pp; softcover
Volume: 23
ISBN-10: 0-8218-3284-0
ISBN-13: 978-0-8218-3284-4
List Price: US$32
Member Price: US$25.60
Order Code: MAWRLD/23
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This item is also sold as part of the following set: MAWRLD-GSET

This 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.


Advanced high-school students and undergraduates in mathematics.

Table of Contents

The story of the birth of manifolds
  • The prelude to the birth of manifolds
  • The birth of manifolds
The story of area and volume from everyday notions to mathematical concepts
  • Transition from the notion of "size" to the concept of "area"
  • Scissors-congruent polygons
  • Scissors-congruent polyhedra
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