| import Mathlib |
| import OAI.Analysis.Conductivity.Flux.PotentialCurl |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Real Set Filter Topology MeasureTheory Matrix |
|
|
| lemma direction_scalar_cos {f : ℝ → ℝ} (hf : Differentiable ℝ f) |
| (r : ℝ) (i a : Fin 3) (x : Coord3) : |
| direction (Pi.single i 1) (fun y : Coord3 => f (y 0)*cos (r*y a)) x = |
| f (x 0)*(-r*sin (r*x a)*(if a=i then 1 else 0)) + |
| cos (r*x a)*(deriv f (x 0)*(if (0:Fin 3)=i then 1 else 0)) := by |
| rw [direction_mul (by fun_prop) (by fun_prop),direction_cos_coord,direction_scalar_coord hf] |
|
|
| def crossingValue3 (L : ℝ) (a b : Fin 3) (x : Coord3) : ℝ := |
| crossingValue L (x 0) (x a) (x b) |
| |
| lemma crossingValue3_diff (L : ℝ) (a b : Fin 3) : Differentiable ℝ (crossingValue3 L a b) := by |
| have hf := (crossingFirst_smooth L).differentiable (by simp) |
| have hg := (crossingSecond_smooth L).differentiable (by simp) |
| unfold crossingValue3 crossingValue |
| fun_prop |
|
|
| lemma crossingValue3_direction (L : ℝ) (a b i : Fin 3) (x : Coord3) : |
| direction (Pi.single i 1) (crossingValue3 L a b) x = |
| (if i=0 then crossingCurrentZ L (x 0) (x a) (x b) else 0) + |
| (if i=a then -crossingFirst L (x 0)*sin (x a) else 0) + |
| (if i=b then -2*crossingSecond L (x 0)*sin (2*x b) else 0) := by |
| have hf := (crossingFirst_smooth L).differentiable (by simp) |
| have hg := (crossingSecond_smooth L).differentiable (by simp) |
| have he : crossingValue3 L a b = fun x => |
| crossingFirst L (x 0)*cos (1*x a)+crossingSecond L (x 0)*cos (2*x b) := by |
| funext x |
| rw [he,direction_add (by fun_prop) (by fun_prop),direction_scalar_cos hf,direction_scalar_cos hg] |
| simp only [one_mul,crossingCurrentZ] |
| simp_rw [@eq_comm _ a i, @eq_comm _ b i, @eq_comm _ (0 : Fin 3) i] |
| split_ifs < |
|
|
| def crossingTensor (L : ℝ) (a b : Fin 3) (x : Coord3) : Matrix (Fin 3) (Fin 3) ℝ := |
| fun i j => |
| (if i=0 ∧ j=0 then 1 else 0) + |
| (if i=a ∧ j=a then crossingXX L (x 0) (x a) (x b) else 0) + |
| (if i=b ∧ j=b then crossingYY L (x 0) (x a) (x b) else 0) + |
| (if i=a ∧ j=b then crossingXY L (x 0) (x a) (x b) else 0) + |
| (if i=b ∧ j=a then crossingXY L (x 0) (x a) (x b) else 0) |
|
|
| lemma crossingTensor_mulVec (L : ℝ) (a b : Fin 3) (x v : Coord3) (i : Fin 3) : |
| (crossingTensor L a b x *ᵥ v) i = |
| (if i=0 then v 0 else 0) + |
| (if i=a then crossingXX L (x 0) (x a) (x b)*v a+crossingXY L (x 0) (x a) (x b)*v b else 0) + |
| (if i=b then crossingXY L (x 0) (x a) (x b)*v a+crossingYY L (x 0) (x a) (x b)*v b else 0) := by |
| simp only [Matrix.mulVec,dotProduct,crossingTensor,add_mul,Finset.sum_add_distrib, |
| ite_and,ite_mul,zero_mul] |
| simp only [Finset.sum_ite_irrel,Finset.sum_ite_eq',Finset.mem_univ,ite_true,Finset.sum_const_zero,one_mul] |
| split_ifs <;> ring |
| |
| lemma crossingTensor_constitution {L : ℝ} (hL : 0<L) {a b : Fin 3} |
| (ha : a≠0) (hb : b≠0) (hab : a≠b) (x : Coord3) : |
| crossingTensor L a b x *ᵥ (fun i => direction (Pi.single i 1) (crossingValue3 L a b) x) = |
| potentialCurl (crossingPotential L a b) x := by |
| ext i |
| rw [crossingTensor_mulVec,crossingPotential_curl L ha hb hab] |
| simp only [crossingValue3_direction,ha,hb,Ne.symm ha,Ne.symm hb,hab,Ne.symm hab, |
| ite_true,ite_false,zero_add,add_zero] |
| obtain ⟨hfx,hfy⟩ := crossing_current_constitution hL (x 0) (x a) (x b) |
| rw [←hfx,←hfy] |
| |
| end ScalarConductivity |
| |
| end |
| |
| end OAI |
| |
