This three-volume set addresses the interplay between topology, functions, geometry, and algebra. Bringing the beauty and fun of mathematics to the classroom, the authors offer serious mathematics in a lively, reader-friendly style. Included are exercises and many figures illustrating the main concepts. It is suitable for advanced high-school students, graduate students, and researchers. The three-volume set includes A Mathematical Gift I, II, and III.
Part I: Invitation to topology (Viewing figures globally): Introduction The Euler characteristic Vortices created by winds and the Euler characteristic Curvature of a surface and the Euler characteristic The story of dimension: Introduction Learning to appreciate dimension What is dimension? Three-dimensional figures Physics and dimension Part II: The legacy of trigonometric functions: Introduction Trigonometric functions and infinite series Elliptic functions Intersection of geometry and algebra: Introduction The Poncelet closure theorem The Poncelet theorem for circles The Poncelet theorem in the world of complex numbers Proof of the Poncelet theorem using plane geometry Conclusion Part III: 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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- ID: 9780821838594
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