By Kenji Ueno, Koji Shiga, Shigeyuki Morita

ISBN-10: 0821832824

ISBN-13: 9780821832820

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**

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

**Example text**

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

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