CK-12 Geometry by CK-12 Foundation

By CK-12 Foundation

CK-12’s Geometry - moment version is a transparent presentation of the necessities of geometry for the highschool pupil. subject matters contain: Proofs, Triangles, Quadrilaterals, Similarity, Perimeter & region, quantity, and alterations. quantity 1 contains the 1st 6 chapters: fundamentals of Geometry, Reasoning and facts, Parallel and Perpendicular traces, Triangles and Congruence, Relationships with Triangles, and Polygons and Quadrilaterals.

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8] An immersion as a function of its metric tensor 47 (iv) It remains to iterate the procedure described in parts (ii) and (iii). 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 .

Viewed as a system of partial differential equations, together with initial values at x0 , with respect to the matrix field (gij ) : Ω → M3 , the above system can have at most one solution in the space C 2 (Ω; M3 ). To see this, let x1 ∈ Ω be distinct from x0 and let γ ∈ C 1 ([0, 1]; R3 ) be any path joining x0 to x1 in Ω, as in part (ii). Then the nine functions gij (γ(t)), 0 ≤ t ≤ 1, satisfy a Cauchy problem for a linear system of nine ordinary differential equations and this system has at most one solution.

In Ω. To see this, let for instance Ω be an open ball centered at the origin in R3 , let Θ(x) = (x1 x22 , x2 , x3 ) and let Θ(x) = Θ(x) if x2 ≥ 0 and Θ(x) = (−x1 x22 , −x2 , x3 ) if x2 < 0 (this counterexample was kindly communicated to the author by Herv´e Le Dret). (6) If a mapping Θ ∈ C 1 (Ω; E3 ) satisfies det ∇Θ > 0 in Ω, then Θ is an immersion. Conversely, if Ω is a connected open set and Θ ∈ C 1 (Ω; E3 ) is an immersion, then either det ∇Θ > 0 in Ω or det ∇Θ < 0 in Ω. 7-3 is simply intended to fix ideas (a similar result clearly holds under the other assumption).

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