A Mathematical Gift III: The Interplay Between Topology, by Kenji Ueno, Koji Shiga, Shigeyuki Morita

By Kenji Ueno, Koji Shiga, Shigeyuki Morita

This publication will convey the sweetness and enjoyable of arithmetic to the school room. It deals critical arithmetic in a full of life, reader-friendly type. integrated are workouts and plenty of figures illustrating the most recommendations.
The first bankruptcy provides the geometry and topology of surfaces. between different themes, the authors speak 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 quite a few features of the idea that of size, together with the Peano curve and the Poincaré process. additionally addressed is the constitution of 3-dimensional manifolds. particularly, 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 chain of lectures given by way of the authors at Kyoto collage (Japan).

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Extra info for A Mathematical Gift III: The Interplay Between Topology, Functions, Geometry, and Algebra (Mathematical World, Volume 23)

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The problem is to find the extremal values of this function on the constraint space G given by a number of equations gi (x1 , . . , xn ) = 0, i = 1, . . , k. The LMM says that the points at which extrema of f may occur are contained in the space of solutions of ∂L ∂L = 0, i = 1, . . , n; = 0, j = 1, . . 60) where L is the Lagrange multiplier function defined by k L := L(x, Λ) = f (x) + λi gi (x). 61) i=1 ∂L When we equate the partial derivatives ∂λ to zero, we get back the constraint space i itself.

This is what we need in the following theorem. 2 Recall that a metric space X is complete if every Cauchy sequence in X is convergent in X. 2 Inverse Function Theorem (IFT): Let E ⊂ Rn be an open set, 0 ∈ E and let f ∈ C 1 (E, Rn ) be such that Df (0) is invertible. Then there exist open neighborhoods U of 0 and V of f (0) such that (i) f : U → V is a bijection; (ii) g = f −1 : V → U is differentiable. (iii) D(f −1 ) is continuous on V. Proof: (i) Put A = Df (0) and consider fˆ = A−1 ◦ f. Then fˆ ∈ (E; Rn ) and D(fˆ)(0) = Id.

There exists a unique point y ∈ X such that φ(y) = y. Proof: Start with any point x0 . Define x1 = φ(x0 ), x2 = φ(x1 ), . . , xn = φ(xn−1 ), . . Observe that d(xn+1 , xn ) ≤ cn d(x1 , x0 ) for some 0 < c < 1. Since that given > 0 we can find n0 such that for m > n > n0 : n cn < ∞ it follows m−1 d(xm , xn ) ≤ ck d(x1 , x0 ) < d(x1 , x0 ). k=n Therefore {xn } is a Cauchy sequence. Since X is complete, this sequence has a limit point y ∈ X. Also observe that any contraction is a continuous function.

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