| import OAI.Geometry.EinsteinFour.AngularNormal |
|
|
| namespace OAI |
|
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| /-! The directional-derivative part of the angular average. -/ |
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| namespace PositiveEinsteinFour.Angular |
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|
| open Matrix MeasureTheory |
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| noncomputable section |
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|
| abbrev Gradient := Fin 4 → Mat3 |
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| def directional (Z : Gradient) (q : Directions) : Mat3 := |
| ∑ i, q.val i • Z i |
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| def gradientSquareNorm (Z : Gradient) : ℝ := ∑ i, squareNorm (Z i) |
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| 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] |
|
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| 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) |
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|
| theorem continuous_directional (Z : Gradient) : Continuous (directional Z) := by |
| unfold directional |
| fun_prop |
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|
| 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 |
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| end |
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| end PositiveEinsteinFour.Angular |
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| end OAI |
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