By Kenji Ueno, Koji Shiga, Shigeyuki Morita, Toshikazu Sunada
This ebook will convey the sweetness and enjoyable of arithmetic to the study room. It deals severe arithmetic in a full of life, reader-friendly type. incorporated are routines and lots of figures illustrating the most options.
The first bankruptcy provides the geometry and topology of surfaces. between different subject matters, the authors talk about the Poincaré-Hopf theorem on serious issues of vector fields on surfaces and the Gauss-Bonnet theorem at the relation among curvature and topology (the Euler characteristic). the second one bankruptcy addresses a number of features of the idea that of size, together with the Peano curve and the Poincaré method. additionally addressed is the constitution of third-dimensional manifolds. particularly, it really is proved that the three-d sphere is the union of 2 doughnuts.
This is the 1st of 3 volumes originating from a sequence of lectures given by way of the authors at Kyoto collage (Japan).
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Additional info for A Mathematical Gift, 1: The Interplay Between Topology, Functions, Geometry, and Algebra
Then we attach an index - 1 to the label of any edge whose arrow points in the opposite direction. Applying this procedure to the rectangle of Fig. 2. we obtain the diagram shown in Fig. 3 This is the final form of the plane model of the torus T. 7-1 Fig. 3. Notice that the vertices of this rectangle represent a single point on the torus. 7. With a single cut we can open out this cylinder to form a rectangle. Then suitable labelling gives us the plane model of Κ shown in Fig. 4. Let us see how to get a space model of Κ from this plane model.
By convention. Oris identified with 5. 5 A compact surface Μ with the appropriate edges identified in R"(n>3 if the surface is orientable and n>4 if it is non-orientable) is called a space model of M. e. a generalization to η dimensions of the 'ordinary' 3-dimensionaI Euclidean space of everyday life. 6 THE CLASSIFICATION THEOREM Let us consider again our fundamental theorem in the light of our new definitions. 1 The Classification Theorem. (a) An orientable compact surface is homeomorphic to η Γ for some n > 0 .
The operations by which we perform this manipulation must be reversible and must be such that the word reached at each stage of the manipulation represents the same surface, up to homeomoφhism. The following operations satisfy these critreria. Operation 1. The cyclic permutation of the edges in a word This can be expressed by the identity \AB = ΒA where A and Β denote strings of edges. For example, O i f t i f l f ' 6 f ' and b^^a\bxa^^ represent the same surface. In fact they represent the same plane model up to rotation.
A Mathematical Gift, 1: The Interplay Between Topology, Functions, Geometry, and Algebra by Kenji Ueno, Koji Shiga, Shigeyuki Morita, Toshikazu Sunada