Download PDF by George Salmon: A treatise on the analytic geometry of three dimensions

By George Salmon

Quantity: 2 writer: London Longmans, eco-friendly matters: Geometry, Analytic -- reliable Surfaces Notes: this can be an OCR reprint. there is typos or lacking textual content. There aren't any illustrations or indexes. if you happen to purchase the final Books variation of this booklet you get unfastened trial entry to Million-Books.com the place you could choose between greater than 1000000 books at no cost. it's also possible to preview the booklet there.

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Extra info for A treatise on the analytic geometry of three dimensions

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2012 by Taylor & Francis Group, LLC ✐ ✐ ✐ ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 18 — #28 18 ✐ ✐ 1. Preliminaries Proof. 1, Theorem 1], there is θ ∈ Lip(Rn ; Rm ) such that √ θ(x) = φ(x) for each x ∈ E and Lip θ ≤ m Lip φ. As there is nothing to prove otherwise, assume c := �φ�L∞ (E;Rm ) belongs to R+ . Define γ : Rm → Rm by � y if |y| > c, c |y| γ(y) := y if |y| ≤ c. Since Lip γ = 1 and |γ|L∞ (Rm ;Rm ) ≤ c, the composition ψ := γ ◦ θ is the desired extension of φ. 2 by showing that Lip ψ = Lip φ.

Fortunately, there is a suitable measure-theoretic analogue of the topological boundary. For any E ⊂ Rn , the sets � � � � �E ∩ B(x, r)� n � =0 , ext∗ E := x ∈ R : lim �� r→0 B(x, r)� int∗ E := ext∗ (Rn − E), cl∗ E := Rn − ext∗ E ∂∗ E := ∂∗ E − int∗ E, 49 ✐ © 2012 by Taylor & Francis Group, LLC ✐ ✐ ✐ ✐ ✐ “Book˙2011” — 2012/2/26 — 9:58 — page 50 — #60 50 ✐ ✐ 4. Perimeter depend only on the equivalence class of E. We call them, respectively, the essential exterior , essential interior , essential closure, and essential boundary of E.

Passing to subsequences, still denoted by {xk } and {yk }, we obtain the limit points x = lim xk and y = lim yk in K. The continuity of φ implies � � � � ∞ > �φ(x) − φ(y)� = lim�φ(xk ) − φ(yk )� ≥ lim sup k|xk − yk |, and consequently x = y. Since φ is Lipschitz in a neighborhood of x, there is 0 < c < ∞ such that for all sufficiently large k, � � c|xk − yk | ≥ �φ(xk ) − φ(yk )� > k|xk − yk | > 0. A contradiction follows. 2. Let E ⊂ Rn and φ ∈ Lip(E; Rm ). There is a map ψ ∈ Lip(Rn ; Rm ) such that ψ(x) = φ(x) for each x ∈ E, √ Lip ψ ≤ m Lip φ, and �ψ�L∞ (Rn ;Rm ) ≤ �φ�L∞ (E;Rm ) .

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A treatise on the analytic geometry of three dimensions by George Salmon


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