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Additional resources for An introduction to differential geometry with applications to elasticity (lecture notes)
The issue of uniqueness reduces in this case to ﬁnding Θ ∈ C 1 (Ω; E3 ) such that ∇Θ(x)T ∇Θ(x) = I for all x ∈ Ω. Parts (i) to (iii) are devoted to solving these equations. (i) We ﬁrst establish that a mapping Θ ∈ C 1 (Ω; E3 ) that satisﬁes ∇Θ(x)T ∇Θ(x) = I for all x ∈ Ω is locally an isometry: Given any point x0 ∈ Ω, there exists an open neighborhood V of x0 contained in Ω such that |Θ(y) − Θ(x)| = |y − x| for all x, y ∈ V. Let B be an open ball centered at x0 and contained in Ω. , Schwartz ) can be applied.
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|. R∈O n We assert that the matrices Q deﬁned in this fashion satisfy limn→∞ Qn An = I. For otherwise, there would exist a subsequence (Qp )p≥0 of the sequence (Qn )n≥0 and δ > 0 such that |Qp Ap − I| = inf 3 |RAp − I| ≥ δ for all p ≥ 0. R∈O Since lim |Ap | = lim p→∞ ρ((Ap )T Ap ) = p→∞ ρ(I) = 1, the sequence (Ap )p≥0 is bounded.
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|. R∈O n We assert that the matrices Q deﬁned in this fashion satisfy limn→∞ Qn An = I.