By Kenji Ueno, Koji Shiga, Shigeyuki Morita, Toshikazu Sunada
This booklet will convey the wonder and enjoyable of arithmetic to the school room. It bargains severe arithmetic in a full of life, reader-friendly sort. incorporated are routines and plenty of figures illustrating the most techniques.
The first bankruptcy provides the geometry and topology of surfaces. between different themes, the authors speak about the Poincaré-Hopf theorem on severe 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 elements of the idea that of measurement, together with the Peano curve and the Poincaré technique. additionally addressed is the constitution of third-dimensional manifolds. particularly, it's 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 via the authors at Kyoto collage (Japan).
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Additional resources for A Mathematical Gift, 1: The Interplay Between Topology, Functions, Geometry, and Algebra
27) This implies, in particular, the following relation. Suppose that the vector field X is Lipschitz. ). For any y ∈ N , t ∈ R, write expy tX = γ(t) ∈ N , where γ is the solution of γ(0) = y dγ = X(γ) . 27) is that for every map u from Ω to N , L(x, expu tX, d(expu tX)) = L(x, u, du) . t. 27). 1 Let X be a Lipschitz tangent vector field on N , which is an infinitesimal symmetry for L. 29) or equivalently, using the coordinates (x1 , . . , xm ) on Ω such that dµ = ρ(x)dx1 . . dxm , m ∂ α ∂x α=1 ρ(x)X i (u) ∂L (x, u, du) ∂Aiα = 0.
Dxm = Ω Xi +t Ω ∂φ ∂L (x, u, du)dx1 . . dxm + o(t) . 28), L(x, u + tφX(u), du + tφd(X(u))) = L(x, u, du) + o(t), and so ∂φ ∂L (x, u, du)) α dx1 . . dxm + o(t) . 32), we obtain (X i L(u + tφX + o(t)) = L(u) + Ω ∂φ ∂L (X i (x, u, du))dx1 . . 30). As an example of applying this result, let us consider the case of harmonic maps. We have L(x, y, A) and ρ(x) 1 αβ (x)hij (y)Aiα Ajβ 2g 1 αβ (x) Aα , Aβ , 2g = = = √ det g dx1 . . dxm . e. if X is a Killing vector field. Such fields are characterized by the fact that LX h = 0, where L is the Lie derivative.
It is an integer which represents the number of times the point g(x, y) goes around S 1 , when (x, y) goes once around S 1 . We will see that if u ∈ H 1 (B 2 , S 1 ), then the degree of u|∂B 2 is necessarily equal to 0. Since for g(x, y) = (x, y), we have deg(g) = 1, this implies that Hg1 (B 2 , S 1 ) is empty. 61) ∂B 2 where α = u1 du2 − u2 du1 . This is a well-known formula in the case where u is of class C 2 . 61) are continuous functionals over H 1 (B 2 , R2 ) 1 and H 2 (∂B 2 , R2 ), respectively.
A Mathematical Gift, 1: The Interplay Between Topology, Functions, Geometry, and Algebra by Kenji Ueno, Koji Shiga, Shigeyuki Morita, Toshikazu Sunada