**Mathematical World**

Volume: 23;
2005;
129 pp;
Softcover
Secondary 53

**Print ISBN: 978-0-8218-3284-4
Product Code: MAWRLD/23**

List Price: $34.00

Individual Member Price: $27.20

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#### Supplemental Materials

# A Mathematical Gift, III: The interplay between topology, functions, geometry, and algebra

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*Koji Shiga; Toshikazu Sunada*

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.

#### Readership

Advanced high-school students and undergraduates in mathematics.