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a458723 | 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 | import Mathlib
import OAI.Analysis.Conductivity.Variational.SynchronizedChartWaves
import OAI.Analysis.Conductivity.Geometry.BoxIteration
namespace OAI
noncomputable section
namespace ScalarConductivity
open Set MeasureTheory
def boxCoordinates : Coord3 ≃L[ℝ] Box3 :=
(show Coord3 ≃ₗ[ℝ] Box3 from
{ toFun := fun x => ((x 0,x 1),x 2)
invFun := fun p => ![p.1.1,p.1.2,p.2]
left_inv := by intro x; ext i; fin_cases i <;> rfl
right_inv := by intro p; rfl
map_add' := by intro x y; rfl
map_smul' := by intro r x; rfl }).toContinuousLinearEquiv
lemma boxCoordinates_apply (x : Coord3) : boxCoordinates x=((x 0,x 1),x 2) := rfl
lemma boxCoordinates_symm_apply (p : Box3) : boxCoordinates.symm p=![p.1.1,p.1.2,p.2] := rfl
lemma boxCoordinates_volume : MeasurePreserving boxCoordinates := by
exact (volume_preserving_prodAssoc.symm (MeasurableEquiv.prodAssoc : Box3≃ᵐ ℝ×(ℝ×ℝ))).comp
threeArrow_volume
lemma boxCoordinates_integral (f : Box3 → ℝ) :
(∫ x,f (boxCoordinates x))=(∫ p,f p) :=
boxCoordinates_volume.integral_comp boxCoordinates.toHomeomorph.measurableEmbedding f
lemma boxCoordinates_integral_symm (f : Coord3 → ℝ) :
(∫ p,f (boxCoordinates.symm p))=(∫ x,f x) := by
have h := boxCoordinates_integral (fun p => f (boxCoordinates.symm p))
simpa only [boxCoordinates.symm_apply_apply] using h.symm
lemma boxCoordinates_fderiv (f : Box3 → ℝ) (hf : Differentiable ℝ f)
(x v : Coord3) :
fderiv ℝ (f∘boxCoordinates) x v=fderiv ℝ f (boxCoordinates x) (boxCoordinates v) := by
rw [fderiv_comp x (hf _) boxCoordinates.differentiableAt]
have he : fderiv ℝ (fun x => boxCoordinates x) x=boxCoordinates.toContinuousLinearMap :=
boxCoordinates.toContinuousLinearMap.hasFDerivAt.fderiv
rw [he]
rfl
end ScalarConductivity
end
end OAI
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