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Analytic geometry and principles of algebra by Alexander Ziwet, Louis Allen Hopkins PDF

By Alexander Ziwet, Louis Allen Hopkins

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For r some r ≥ 2, assume that mappings Θnr ∈ C 3 ( s=1 Bs ; E3 ), n ≥ 0, have been found that satisfy r (∇Θnr )T ∇Θnr = Cn in Bs , s=1 r lim n→∞ Θnr − Θ 2,K = 0 for all K Bs . 7-1 shows that there exist vectors cn ∈ E3 and matrices Qn ∈ O3 , n ≥ 0, such that r n Bs ∩ Br+1 . Θr+1 (x) = cn + Qn Θnr (x) for all x ∈ s=1 Then an argument similar to that used in part (ii) shows that limn→∞ Qn = I and limn→∞ cn = 0, and an argument similar to that used in part (iii) (note that the ball Br+1 may intersect more than one of the balls Bs , 1 ≤ s ≤ r) r shows that the mappings Θnr+1 ∈ C 3 ( s=1 Bs ; E3 ), n ≥ 0, defined by r Θnr+1 (x) := Θnr (x) for all x ∈ Bs , s=1 n Θnr+1 (x) := (Qn )T (Θr (x) − cn ) for all x ∈ Br+1 , satisfy r lim n→∞ Θnr+1 − Θ 3,K = 0 for all K Bs .

In order that the three values ζj (1) found by solving the above Cauchy problem for a given integer ∈ {1, 2, 3} be acceptable candidates for the three unknown values F j (x1 ), they must be of course independent of the path chosen for joining x0 to x1 . So, let γ 0 ∈ C 1 ([0, 1]; R3 ) and γ 1 ∈ C 1 ([0, 1]; R3 ) be two paths joining x0 to x1 in Ω. , such that G(·, 0) = γ 0 , G(·, 1) = γ 1 , G(t, λ) ∈ Ω for all 0 ≤ t ≤ 1, 0 ≤ λ ≤ 1, G(0, λ) = x0 and G(1, λ) = x1 for all 0 ≤ λ ≤ 1, and smooth enough in the sense that G ∈ C 1 ([0, 1] × [0, 1]; R3 ) and ∂ ∂G ∂ ∂G = ∈ C 0 ([0, 1] × [0, 1]; R3 ).

46 Three-dimensional differential geometry [Ch. 1 Proof. The proof is broken into four parts, numbered (i) to (iv). The first part is a preliminary result about matrices (for convenience, it is stated here for matrices of order three, but it holds as well for matrices of arbitrary order). (i) Let matrices An ∈ M3 , n ≥ 0, satisfy lim (An )T An = I. n→∞ Then there exist matrices Qn ∈ O3 , n ≥ 0, that satisfy lim Qn An = I. n→∞ Since the set O3 is compact, there exist matrices Qn ∈ O3 , n ≥ 0, such that |Qn An − I| = inf 3 |RAn − I|.

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Analytic geometry and principles of algebra by Alexander Ziwet, Louis Allen Hopkins


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