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.
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"With A Mathematical Gift, I, there is no reason why every undergraduate student should not be exposed to some topology ... accessible to even high school students ... beautiful illustrations and straightforward explanations of sophisticated ideas. Real world and concrete scenarios are used ... elegant explanations ... exercises ... are friendly and non-threatening ... the perfect choice for anyone who conducts summer workshops for high school students ... an ideal supplement for graduate students studying topology for the first time ... also excellent for undergraduate independent studies ... I enjoyed reading this book ... fun to look at ... instructive and motivating ... impressed ... wowed by the detail and clarity presented by the authors. Readers of A Mathematical Gift, I will want to read A Mathematical Gift, II." ---- MAA Online
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