By Kenji Ueno, Koji Shiga, Shigeyuki Morita
This booklet will deliver the wonder and enjoyable of arithmetic to the school room. It bargains critical arithmetic in a full of life, reader-friendly sort. integrated are routines and lots of figures illustrating the most options.
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 numerous elements of the concept that of measurement, together with the Peano curve and the PoincarÃ© process. additionally addressed is the constitution of third-dimensional manifolds. specifically, it really is proved that the third-dimensional sphere is the union of 2 doughnuts.
This is the 1st of 3 volumes originating from a sequence of lectures given by means of the authors at Kyoto college (Japan).
Read Online or Download A Mathematical Gift I: The Interplay Between Topology, Functions, Geometry, and Algebra (Mathematical World, Volume 19) PDF
Similar topology books
Nonlinear research is a extensive, interdisciplinary box characterised by means of a striking mix of research, topology, and purposes. Its strategies and methods give you the instruments for constructing extra practical and exact versions for numerous phenomena encountered in fields starting from engineering and chemistry to economics and biology.
Within the Nineteen Fifties, Eilenberg and Steenrod awarded their recognized characterization of homology conception through seven axioms. just a little later, it was once came upon that conserving simply the 1st six of those axioms (all other than the situation at the "homology" of the point), you can actually receive many different fascinating platforms of algebraic invariants of topological manifolds, akin to $K$-theory, cobordisms, and others.
The writing bears the marks of authority of a mathematician who was once actively thinking about developing the topic. many of the papers spoke of are not less than two decades previous yet this displays the time whilst the information have been demonstrated and one imagines that the placement should be diversified within the moment quantity.
This court cases is a set of articles on Topology and Teichmuller areas. detailed emphasis is being wear the common Teichmuller area, the topology of moduli of algebraic curves, the gap of representations of discrete teams, Kleinian teams and Dehn filling deformations, the geometry of Riemann surfaces, and a few comparable themes.
- Elements of set theory vol 1
- Topological and Symbolic Dynamics
- Topology now
- Riemann Surfaces and Algebraic Curves : A First Course in Hurwitz Theory
Additional resources for A Mathematical Gift I: The Interplay Between Topology, Functions, Geometry, and Algebra (Mathematical World, Volume 19)
Deduce that , if R lies on the line I with equation 2 ax + by= 0, then the mid-point of PQ also lies on I. 21 X and Yare two points on a rectangular hyperbola. The tangents at X and Y to the hyperbola meet one of its asymptotes at A and C respectively, and meet the other asymptote at Band D respectively. Prove that AD and BC are both parallel to XY and are equidistant from it. 22 The gradient m of the chord PQ of the rectangular hyperbola is constant and positive. Show that for all positions of PQ, there are two fixed points through which the circle on PQ as diameter passes.
Prove that the distance of Q from the centre of the ellipse does not exceed (a 2 - b 2 )/a. (b) A ladder 12 m long slides in a vertical plane with its ends in contact with a vertical wall and a horizontal floor. Find the locus of a point on the ladder 4 m from its foot. Prove that the tangent to an ellipse at any point bisects one pair of angles between the lines joining that point to the foci of the ellipse. j3. The distance of a point P on this ellipse from one focus S is f a, and the tangent at P to the ellipse cuts the major axis at T .
8), its equation becomes xy = c2 , C > 0, and we can y then use (ct, cit) as parametric coordinates. , and the tangent at t x t == y _:. t = -~(x t ct) == t 2 y + X = 2ct. 3) v x Fig. P1P2P3 also lies on the curve. 26 Coordinate Geometry and Complex Numbers The line through P2 perpendicular to PI P3 == y c - - = t 1t 3(X - ct 2)· t2 The line through P 3 perpendicular to PI P2 These lines meet where (I 1) c - - - t2 t3 c . =t 1X(t3 - t 2)=X=--t 1t 213 By substitution, Y = - ct 1 t 2 t 3' Hence the orthocentre, the meet of the altitudes of 6P1 P2 P3 , lies on the curve xy = c 2 .
A Mathematical Gift I: The Interplay Between Topology, Functions, Geometry, and Algebra (Mathematical World, Volume 19) by Kenji Ueno, Koji Shiga, Shigeyuki Morita