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785c727 | 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 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 | import OAI.Geometry.EinsteinFour.AngularNormal
namespace OAI
/-! The directional-derivative part of the angular average. -/
namespace PositiveEinsteinFour.Angular
open Matrix MeasureTheory
noncomputable section
abbrev Gradient := Fin 4 → Mat3
def directional (Z : Gradient) (q : Directions) : Mat3 :=
∑ i, q.val i • Z i
def gradientSquareNorm (Z : Gradient) : ℝ := ∑ i, squareNorm (Z i)
private theorem integral_directional_entries (h : FollandMoments)
(Z : Fin 4 → Fin 3 → Fin 3 → ℝ) :
(∫ q : Directions, ∑ r, ∑ s, (∑ a, q.val a * Z a r s) ^ 2 ∂sphereMeasure) =
(∑ a, ∑ r, ∑ s, Z a r s ^ 2) / 4 := by
calc
_ = ∑ r, ∫ q : Directions, ∑ s, (∑ a, q.val a * Z a r s) ^ 2 ∂sphereMeasure :=
integral_sum h _ (fun _ => by fun_prop)
_ = ∑ r, ∑ s, ∫ q : Directions, (∑ a, q.val a * Z a r s) ^ 2 ∂sphereMeasure := by
apply Finset.sum_congr rfl
intro r _
exact integral_sum h _ (fun _ => by fun_prop)
_ = ∑ r, ∑ s, (∑ a, Z a r s ^ 2) / 4 := by
simp_rw [integral_directional_square h]
_ = _ := by
simp_rw [← Finset.sum_div]
congr 1
calc
_ = ∑ r, ∑ a, ∑ s, Z a r s ^ 2 := by
apply Finset.sum_congr rfl
intro r _
rw [Finset.sum_comm]
_ = _ := by rw [Finset.sum_comm]
theorem integral_directional_squareNorm (h : FollandMoments) (Z : Gradient) :
(∫ q, squareNorm (directional Z q) ∂sphereMeasure) = gradientSquareNorm Z / 4 := by
simpa only [squareNorm, directional, Matrix.sum_apply, Matrix.smul_apply,
smul_eq_mul, gradientSquareNorm] using!
integral_directional_entries h (fun a r s => Z a r s)
theorem continuous_directional (Z : Gradient) : Continuous (directional Z) := by
unfold directional
fun_prop
def normalDerivative (Z W : Gradient) (q : Directions) : Mat3 :=
normalDeviation (directional Z q) (directional W q) q
/-- The two individual terms are exact directional averages; the mixed
term is retained explicitly. -/
theorem integral_normalDerivative_squareNorm (h : FollandMoments) (Z W : Gradient) :
(∫ q, squareNorm (normalDerivative Z W q) ∂sphereMeasure) =
(gradientSquareNorm Z + gradientSquareNorm W) / 16 +
(∫ q, pairing (directional Z q) (rotated (directional W q) q)
∂sphereMeasure) / 2 := by
have hZ : Integrable (fun q => squareNorm (directional Z q)) sphereMeasure :=
continuous_integrable h (by
simp only [squareNorm, directional, Matrix.sum_apply, Matrix.smul_apply, smul_eq_mul]
fun_prop)
have hW : Integrable (fun q => squareNorm (directional W q)) sphereMeasure :=
continuous_integrable h (by
simp only [squareNorm, directional, Matrix.sum_apply, Matrix.smul_apply, smul_eq_mul]
fun_prop)
have hP : Integrable
(fun q => pairing (directional Z q) (rotated (directional W q) q)) sphereMeasure :=
continuous_integrable h (by
simp only [pairing, directional, rotated, Matrix.mul_apply, Matrix.transpose_apply,
Matrix.sum_apply, Matrix.smul_apply, smul_eq_mul]
simp only [mixingMatrix, mixing, quadratic]
fun_prop)
have he : (fun q => squareNorm (normalDerivative Z W q)) =
fun q => (squareNorm (directional Z q) + squareNorm (directional W q) +
2 * pairing (directional Z q) (rotated (directional W q) q)) / 4 := by
funext q
exact squareNorm_normalDeviation _ _ q
have hZW : Integrable (fun q =>
squareNorm (directional Z q) + squareNorm (directional W q)) sphereMeasure := hZ.add hW
rw [he, integral_div, integral_add hZW (hP.const_mul 2),
integral_add hZ hW, integral_const_mul,
integral_directional_squareNorm h Z, integral_directional_squareNorm h W]
ring
end
end PositiveEinsteinFour.Angular
end OAI
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