| 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 |
| right_inv := by intro p |
| map_add' := by intro x y; rfl |
| map_smul' := by intro r x |
|
|
| 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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|
