By Kenji Ueno, Koji Shiga, Shigeyuki Morita

This e-book will carry the sweetness and enjoyable of arithmetic to the school room. It deals severe arithmetic in a full of life, reader-friendly sort. incorporated are routines and lots of figures illustrating the most thoughts.

The first bankruptcy provides the geometry and topology of surfaces. between different issues, the authors talk about the PoincarÃ©-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 variety of features of the concept that of measurement, together with the Peano curve and the PoincarÃ© method. additionally addressed is the constitution of 3-dimensional manifolds. specifically, it really is proved that the 3-dimensional sphere is the union of 2 doughnuts.

This is the 1st of 3 volumes originating from a sequence of lectures given through the authors at Kyoto collage (Japan).

**Read Online or Download A Mathematical Gift I: The Interplay Between Topology, Functions, Geometry, and Algebra (Mathematical World, Volume 19) PDF**

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**Extra info for A Mathematical Gift I: The Interplay Between Topology, Functions, Geometry, and Algebra (Mathematical World, Volume 19)**

**Sample text**

It was a stroke of luck (or genius) that Lie decided to look at infinitesimal elements, because it enabled him to prove simplicity for whole infinite families of Lie algebras in one fell swoop. ) Around 1885 Lie proved results so general that they cover all but a finite number of simple Lie algebras—namely, those of the exceptional groups mentioned at the end of Chapter 1 (see Hawkins [2000], pp. 92–98). In the avalanche of Lie’s results, the special case of so(3) and SO(3) seems to have gone unnoticed.

Also, the group operation of SO(3) corresponds to quaternion multiplication, because if one rotation is induced by conjugation by t1 , and another by conjugation by t2 , then conjugation by t1 t2 induces the product rotation (first rotation followed by the second). Of course, we multiply pairs ±t of quaternions by the rule (±t1 )(±t2 ) = ±t1t2 . We therefore identify SO(3) with the group RP3 of unit quaternion pairs ±t under this product operation. The map ϕ : SU(2) → SO(3) defined by ϕ (t) = {±t} is a 2-to-1 homomorphism, because the two elements t and −t of SU(2) go to the single pair ±t in SO(3).

5. As is clear from the figure, the tetrahedron is mapped into itself by two types of rotation: • A 1/2 turn about each line through the centers of opposite edges. • A 1/3 turn about each line through a vertex and the opposite face center. 3 Show that there are 11 distinct rotations among these two types. What rotation accounts for the 12th position of the tetrahedron? 5. Remember that a rotation about axis u through angle θ corresponds to the quaternion pair ±q, where q = cos θ θ + u sin . 4 Show that the identity, and the three 1/2 turns, correspond to the four quaternion pairs ±1, ±i, ±j, ±k.