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

By George Salmon

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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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