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2fca478 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 | import OAI.Geometry.ProjectionBody.GraphChart
import Mathlib.Analysis.InnerProductSpace.NormDet
import Mathlib.LinearAlgebra.Matrix.SchurComplement
namespace OAI
noncomputable section
open scoped RealInnerProductSpace
namespace ProjectionCounterexample
/-- The Jacobian of a graph chart is the length of its unnormalized normal. -/
theorem graphLinear_normDet {n : ℕ} (i : Fin (n + 1)) (a : E n) :
(graphLinear i a).normDet = ‖graphNormal i a‖ := by
classical
let b := EuclideanSpace.basisFun (Fin n) ℝ
have hGram : Matrix.gram ℝ (fun j => graphLinear i a (b j)) =
1 + Matrix.replicateCol Unit (fun j => a j) *
Matrix.replicateRow Unit (fun j => a j) := by
ext j k
rw [Matrix.gram_apply, graphLinear_inner]
simp [b, EuclideanSpace.basisFun_apply, eq_comm,
EuclideanSpace.inner_single_right, Matrix.add_apply, Matrix.one_apply,
Matrix.mul_apply, Matrix.replicateCol_apply, Matrix.replicateRow_apply]
have hsquare : (graphLinear i a).normDet ^ 2 = ‖graphNormal i a‖ ^ 2 := by
calc
(graphLinear i a).normDet ^ 2 =
(Matrix.gram ℝ (fun j => graphLinear i a (b j))).det :=
(graphLinear i a).normDet_sq_eq_det_gram b
_ = ‖graphNormal i a‖ ^ 2 := by
rw [hGram, Matrix.det_one_add_replicateCol_mul_replicateRow,
graphNormal_norm_sq, EuclideanSpace.real_norm_sq_eq]
simp [dotProduct, pow_two]
exact (sq_eq_sq₀ (graphLinear i a).normDet_nonneg (norm_nonneg _)).mp hsquare
end ProjectionCounterexample
end
end OAI
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