1402054378

# 33 fundamental property the theorem for the product

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Unformatted text preview: . Example of G final parametric manifold G final parametric manifold is visualised by a 2D surface in our 3D space for a particular case, too simple to be of practical use, but one that allows the general case of a dimension n rank 3 manifold to be pictured. Given the Ω perturbation as the most common for an object, reflected by a Ginit metric tensor with a dimension n = 2 and rank 1 ( b 2 − a ⋅ c = 0 ), with a b Ginit = b c 28 P. Serré A. Rivière and A. Clé , ment the G final manifold will then represent all 2-dimension, rank 1 tensors. X G final = Y q2 a b 1 + q1 q3 Y 1 + q1 ⋅ = ⋅ Z q3 1 + q4 b c q2 1 + q4 By identifying the 2 members of the equation, we obtain a parametric representation (see Figure 1) with a 4-parameter manifold: X = a + 2 ⋅ a ⋅ q1 + 2 ⋅ b ⋅ q2 + 2 ⋅ b ⋅ q1 ⋅ q2 + a ⋅ q12 + c ⋅ q2 2 Y = b + a ⋅ q3 + b ⋅ q1 + b ⋅ q4 + c ⋅ q2 + a ⋅ q1 ⋅ q3 + b ⋅ q1 ⋅ q4 + b ⋅ q2 ⋅ q3 + c ⋅ q2 ⋅ q4 2 2 Z = c + 2 ⋅ b ⋅ q3 + 2 ⋅ c ⋅ q4 + 2 ⋅ b ⋅ q3 ⋅ q4 + a ⋅ q3 + c ⋅ q4 This is the...
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