| import OAI.MathematicalPhysics.Elasticity.Transport |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open scoped BigOperators |
| namespace ElasticityAugmented |
| variable {A : Type*} [CommRing A] [Algebra ℂ A] |
| variable (D : Fin 3 → Derivation ℂ A A) |
| lemma leadingP_neg (θ : Fin 3 → ℂ) (m f : A) : leadingP D θ m (-f)=-leadingP D θ m f := by |
| simp only [leadingP,direction,map_neg,smul_neg,Finset.sum_neg_distrib] |
| ring |
| lemma normal_sub (θ : Fin 3 → ℂ) (u v : Fin 3 → A) : normal θ (u-v)=normal θ u-normal θ v := by |
| simp only [normal,Pi.sub_apply,smul_sub,Finset.sum_sub_distrib] |
| lemma normal_neg (θ : Fin 3 → ℂ) (u : Fin 3 → A) : normal θ (-u)=-normal θ u := by |
| simp only [normal,Pi.neg_apply,smul_neg,Finset.sum_neg_distrib] |
| |
| lemma normal_particular (θ n : Fin 3 → ℂ) (hn : ∑ i,θ i*n i=1) (q : A) : |
| normal θ (fun i => -(n i • q))=-q := by |
| simp only [normal,smul_neg,smul_smul,Finset.sum_neg_distrib] |
| rw [← Finset.sum_smul,hn,one_smul] |
| lemma exists_normalizing_vector (θ : Fin 3 → ℂ) (hθ : θ≠0) : ∃ n : Fin 3 → ℂ,∑ i,θ i*n i=1 := by |
| obtain ⟨i,hi⟩ := Function.ne_iff.mp hθ |
| change θ i≠0 at hi |
| refine ⟨Pi.single i (θ i)⁻¹,?_⟩ |
| simp [Pi.single_apply,hi] |
|
|
| lemma residual_source_tangent (θ n : Fin 3 → ℂ) (hθ : ∑ i,θ i*θ i=0) (hn : ∑ i,θ i*n i=1) |
| (lam m : A) (u : Fin 3 → A) (b : A) |
| (hc : normal θ (R D lam m u b)=leadingP D θ m (div D u-b)) : |
| normal θ (-R D lam m u b-leadingR D θ lam m (fun i => -(n i • (div D u-b))) 0)=0 := by |
| rw [normal_sub,normal_neg,leading_normal D θ hθ,normal_particular θ n hn,leadingP_neg,hc] |
| exact sub_self _ |
| end ElasticityAugmented |
|
|
| end |
| end |
| section |
| noncomputable section |
| open scoped BigOperators |
| namespace ElasticityFrame |
| open ElasticityAugmented |
|
|
| def tangentProjection (θ n : Fin 3 → ℂ) (hn : ∑ i,θ i*n i=1) : Amp →ₗ[ℂ] Fiber θ := |
| (LinearMap.id-(normal θ).smulRight (n,0)).codRestrict (fiber θ) (by |
| intro v |
| change normal θ (v-normal θ v • (n,0))=0 |
| rw [map_sub,map_smul] |
| have he : normal θ (n,0)=1 := hn |
| rw [he,smul_eq_mul,mul_one,sub_self]) |
| lemma tangentProjection_apply (θ n : Fin 3 → ℂ) (hn : ∑ i,θ i*n i=1) (v : Amp) : |
| ((tangentProjection θ n hn v : Fiber θ) : Amp)=v-normal θ v • (n,0) := rfl |
| lemma tangentProjection_of_normal (θ n : Fin 3 → ℂ) (hn : ∑ i,θ i*n i=1) |
| (v : Amp) (hv : normal θ v=0) : |
| ((tangentProjection θ n hn v : Fiber θ) : Amp)=v := by |
| rw [tangentProjection_apply,hv,zero_smul,sub_zero] |
| lemma tangentProjection_smooth (θ n : Fin 3 → ℂ) (hn : ∑ i,θ i*n i=1) |
| (p : Fin 3 → Smooth) (s : Smooth) : |
| ContDiff ℝ (⊤ : ℕ∞) (fun x => tangentProjection θ n hn |
| ((fun i => (p i : X → ℂ) x),(s : X → ℂ) x)) := by |
| exact ((tangentProjection θ n hn).toContinuousLinearMap.restrictScalars ℝ).contDiff.comp |
| (((contDiff_pi.mpr (fun i => smooth_coe (p i))).prodMk (smooth_coe s))) |
|
|
| lemma smooth_vector_forced (θ : Fin 3 → ℂ) (hθ : ∑ i,θ i*θ i=0) |
| (u : X → Fiber θ) (hu : ContDiff ℝ (⊤ : ℕ∞) u) (t : Smooth) (qq : Fin 3 → Smooth) |
| (f : Fiber θ) (x : X) |
| (he : fderiv ℝ u x (ElasticityParameter.characteristicPlane ElasticityParameter.basis θ 1)+ |
| Complex.I • fderiv ℝ u x (ElasticityParameter.characteristicPlane ElasticityParameter.basis θ Complex.I)= |
| connection θ hθ ((t : X → ℂ) x) (fun i => (qq i : X → ℂ) x) (u x)+f) : |
| ∀ i,(direction (A := Smooth) coord θ (vectorSmooth θ u hu i) : X → ℂ) x= |
| θ i*(scalarSmooth θ u hu : X → ℂ) x+(f : Amp).1 i := by |
| intro i |
| rw [vectorSmooth,projection_direction,he,map_add] |
| exact congrArg (fun z : ℂ => z+(f : Amp).1 i) |
| (connection_vector θ hθ ((t : X → ℂ) x) (fun j => (qq j : X → ℂ) x) (u x) i) |
| lemma smooth_scalar_forced (θ : Fin 3 → ℂ) (hθ : ∑ i,θ i*θ i=0) |
| (u : X → Fiber θ) (hu : ContDiff ℝ (⊤ : ℕ∞) u) (t : Smooth) (qq : Fin 3 → Smooth) |
| (f : Fiber θ) (x : X) |
| (he : fderiv ℝ u x (ElasticityParameter.characteristicPlane ElasticityParameter.basis θ 1)+ |
| Complex.I • fderiv ℝ u x (ElasticityParameter.characteristicPlane ElasticityParameter.basis θ Complex.I)= |
| connection θ hθ ((t : X → ℂ) x) (fun i => (qq i : X → ℂ) x) (u x)+f) : |
| (direction (A := Smooth) coord θ (scalarSmooth θ u hu) : X → ℂ) x= |
| ((t*scalarSmooth θ u hu+∑ i,qq i*vectorSmooth θ u hu i : Smooth) : X → ℂ) x+(f : Amp).2 := by |
| rw [scalarSmooth,projection_direction,he,map_add] |
| apply congrArg (fun z : ℂ => z+(f : Amp).2) |
| have h := connection_scalar θ hθ ((t : X → ℂ) x) (fun i => (qq i : X → ℂ) x) (u x) |
| simp only [Fin.sum_univ_three,Subalgebra.coe_add,Subalgebra.coe_mul,Pi.add_apply,Pi.mul_apply] |
| change ((connection θ hθ ((t : X → ℂ) x) (fun i => (qq i : X → ℂ) x) (u x) : Fiber θ) : Amp).2= |
| (t : X → ℂ) x*((u x : Fiber θ) : Amp).2+ |
| ((qq 0 : X → ℂ) x*((u x : Fiber θ) : Amp).1 0+ |
| (qq 1 : X → ℂ) x*((u x : Fiber θ) : Amp).1 1+ |
| (qq 2 : X → ℂ) x*((u x : Fiber θ) : Amp).1 2) |
| simpa only [Fin.sum_univ_three] using h |
| end ElasticityFrame |
|
|
| end |
| end |
| section |
| noncomputable section |
| open scoped BigOperators |
| namespace ElasticityAugmented |
|
|
| lemma transformed_scalar_exact (θ : Fin 3 → ℂ) (k r : Smooth) |
| (hr : ∀ x,(r : X → ℂ) x≠0) (a : Fin 3 → Smooth) (b : Smooth) |
| (hR : leadingR coord θ (k-r*r) (r*r) a b=0) : |
| leadingS coord θ (k-r*r) (r*r) a b= |
| 2*((k+r*r)*direction (A := Smooth) coord θ b+direction (A := Smooth) coord θ k*b+ |
| r*(2*thetaGamma θ r hr*changeS k r hr a b+ |
| ∑ i,direction (A := Smooth) coord θ (logarithmicGrad r hr i)*changeP r a i)) := by |
| have ht : (∑ i, (2*coord i (r*r)*direction (A := Smooth) coord θ (a i)+ |
| direction (A := Smooth) coord θ (coord i (r*r))*a i) : Smooth)= |
| r*(2*thetaGamma θ r hr*changeS k r hr a b+ |
| ∑ i, direction (A := Smooth) coord θ (logarithmicGrad r hr i)*changeP r a i) := by |
| simp_rw [transformed_gradient θ r hr a,normal_form_top θ k r hr a b hR] |
| rw [← Finset.mul_sum,Finset.sum_add_distrib] |
| congr 2 |
| simp only [thetaGamma,Finset.sum_mul,Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro i _ |
| simp only [Algebra.smul_def] |
| ring |
| rw [← ht] |
| simp only [leadingS,sub_add_cancel,Finset.sum_add_distrib,← Finset.mul_sum,mul_assoc] |
| ring |
|
|
| lemma scalar_residual_algebra (r k R K b db H J s : ℂ) (hr : r≠0) (hk : k≠0) |
| (hl : k+r*r≠0) (hB : 2*r*s+k*b+r*H=0) : |
| -(1/2 : ℂ)*((K/r-k*R/r^2)*b+k/r*db+J+(2*R/r)*s)- |
| ((r*r/(k+r*r))*(K/k-2*R/r)*s+ |
| (1/2 : ℂ)*(((r*r/(k+r*r))*(K/k)-(2*R/r)/2)*H-(r*r/(k+r*r))*J))= |
| (-k/(2*r*(k+r*r)))*((k+r*r)*db+K*b+r*(2*(2*R/r)*s+J)) := by |
| calc |
| _ = (-k/(2*r*(k+r*r)))*((k+r*r)*db+K*b+r*(2*(2*R/r)*s+J))+ |
| (R/(2*r^2)-r*K/(2*k*(k+r*r)))*(2*r*s+k*b+r*H) := by |
| have hp : k+r^2≠0 := by simpa only [pow_two] using hl |
| field_simp [hr,hk,hl,hp] |
| ring |
| _ = _ := by rw [hB] |
|
|
| lemma normal_form_bottom_residual (θ : Fin 3 → ℂ) (k r : Smooth) |
| (hr : ∀ x,(r : X → ℂ) x≠0) (hk : ∀ x,(k : X → ℂ) x≠0) |
| (hl : ∀ x,((k+r*r : Smooth) : X → ℂ) x≠0) (a : Fin 3 → Smooth) (b : Smooth) |
| (hR : leadingR coord θ (k-r*r) (r*r) a b=0) : |
| direction (A := Smooth) coord θ (changeS k r hr a b)- |
| (transportT θ k r hk hl*changeS k r hr a b+ |
| ∑ i,transportQ θ k r hr hk hl i*changeP r a i)= |
| -(1/4 : ℂ) • (k*smoothInv r hr*smoothInv (k+r*r) hl*leadingS coord θ (k-r*r) (r*r) a b) := by |
| rw [changeS_derivative θ k r hr a b hR,transportT_eq θ k r hr hk hl, |
| transportQ_contract,transformed_scalar_exact θ k r hr a b hR] |
| apply Subtype.ext |
| funext x |
| let R := (r : X → ℂ) x |
| let K := (k : X → ℂ) x |
| let DR := (direction (A := Smooth) coord θ r : X → ℂ) x |
| let DK := (direction (A := Smooth) coord θ k : X → ℂ) x |
| let B := (b : X → ℂ) x |
| let DB := (direction (A := Smooth) coord θ b : X → ℂ) x |
| let H := ((∑ i,logarithmicGrad r hr i*changeP r a i : Smooth) : X → ℂ) x |
| let J := ((∑ i,direction (A := Smooth) coord θ (logarithmicGrad r hr i)*changeP r a i : Smooth) : X → ℂ) x |
| let S := (changeS k r hr a b : X → ℂ) x |
| have he : S=-(1/2 : ℂ)*(K*R⁻¹*B+H) := rfl |
| have hB : 2*R*S+K*B+R*H=0 := by |
| rw [he] |
| have hn : R≠0 := hr x |
| field_simp [hn] |
| ring |
| have hgamma : (thetaGamma θ r hr : X → ℂ) x=2*DR/R := by |
| rw [thetaGamma_eq] |
| change 2*R⁻¹*DR=2*DR/R |
| ring |
| have ha := scalar_residual_algebra R K DR DK B DB H J S (hr x) (hk x) |
| (show K+R*R≠0 from hl x) hB |
| simp only [Subalgebra.coe_add,Subalgebra.coe_sub,Subalgebra.coe_mul,Subalgebra.coe_pow, |
| Subalgebra.coe_smul,Pi.add_apply,Pi.sub_apply,Pi.mul_apply,Pi.pow_apply,Pi.smul_apply, |
| smul_eq_mul,smoothInv_apply,hgamma] |
| have h2 : ((2 : Smooth) : X → ℂ) x=(2 : ℂ) := rfl |
| rw [h2] |
| change -(1/2 : ℂ)*((DK*R⁻¹-K*(R⁻¹)^2*DR)*B+K*R⁻¹*DB+J+(2*DR/R)*S)- |
| (R*R*(K+R*R)⁻¹*(K⁻¹*DK-2*DR/R)*S+ |
| (1/2 : ℂ)*((R*R*(K+R*R)⁻¹*K⁻¹*DK-(1/2 : ℂ)*(2*DR/R))*H-R*R*(K+R*R)⁻¹*J))= |
| -(1/4 : ℂ)*(K*R⁻¹*(K+R*R)⁻¹*(2*((K+R*R)*DB+DK*B+R*(2*(2*DR/R)*S+J)))) |
| convert ha using 1 < |
|
|
| /-- Both directions of the actual physical leading transport equivalence. -/ |
| theorem leading_equations_iff (θ : Fin 3 → ℂ) (k r : Smooth) |
| (hr : ∀ x,(r : X → ℂ) x≠0) (hk : ∀ x,(k : X → ℂ) x≠0) |
| (hl : ∀ x,((k+r*r : Smooth) : X → ℂ) x≠0) (a : Fin 3 → Smooth) (b : Smooth) : |
| (leadingR coord θ (k-r*r) (r*r) a b=0 ∧ leadingS coord θ (k-r*r) (r*r) a b=0) ↔ |
| (∀ i,direction (A := Smooth) coord θ (changeP r a i)=θ i • changeS k r hr a b) ∧ |
| direction (A := Smooth) coord θ (changeS k r hr a b)= |
| transportT θ k r hk hl*changeS k r hr a b+ |
| ∑ i,transportQ θ k r hr hk hl i*changeP r a i := by |
| constructor |
| · rintro ⟨hR,hS⟩ |
| exact ⟨normal_form_top θ k r hr a b hR,normal_form_bottom θ k r hr hk hl a b hR hS⟩ |
| · rintro ⟨hP,hT⟩ |
| have hR := (leadingR_iff θ k r hr a b).mpr hP |
| refine ⟨hR,?_⟩ |
| have hh := normal_form_bottom_residual θ k r hr hk hl a b hR |
| rw [hT,sub_self] at hh |
| apply Subtype.ext |
| funext x |
| have he := congrArg (fun f : Smooth => (f : X → ℂ) x) hh |
| simp only [Subalgebra.coe_smul,Subalgebra.coe_mul,Subalgebra.coe_zero,Pi.zero_apply,Pi.mul_apply, |
| Pi.smul_apply,smul_eq_mul,smoothInv_apply] at he |
| rw [← mul_assoc] at he |
| exact (mul_eq_zero.mp he.symm).resolve_left (by |
| apply mul_ne_zero |
| · norm_num |
| · exact mul_ne_zero (mul_ne_zero (hk x) (inv_ne_zero (hr x))) (inv_ne_zero (hl x))) |
| end ElasticityAugmented |
|
|
| end |
| end |
| section |
| noncomputable section |
| open scoped BigOperators |
| namespace ElasticityAugmented |
|
|
| lemma changeS_derivative_raw (θ : Fin 3 → ℂ) (k r : Smooth) |
| (hr : ∀ x,(r : X → ℂ) x≠0) (a : Fin 3 → Smooth) (b : Smooth) : |
| direction (A := Smooth) coord θ (changeS k r hr a b)= |
| -(1/2 : ℂ) • ((direction (A := Smooth) coord θ k*smoothInv r hr- |
| k*(smoothInv r hr)^2*direction (A := Smooth) coord θ r)*b+ |
| k*smoothInv r hr*direction (A := Smooth) coord θ b+ |
| (∑ i,direction (A := Smooth) coord θ (logarithmicGrad r hr i)*changeP r a i)+ |
| ∑ i,logarithmicGrad r hr i*direction (A := Smooth) coord θ (changeP r a i)) := by |
| conv_lhs => rw [changeS,direction_smul,direction_add] |
| simp only [direction_sum,direction_mul,direction_inv,Finset.sum_add_distrib] |
| congr 1 |
| ring |
|
|
| lemma transformed_scalar_raw (θ : Fin 3 → ℂ) (k r : Smooth) |
| (hr : ∀ x,(r : X → ℂ) x≠0) (a : Fin 3 → Smooth) (b : Smooth) : |
| leadingS coord θ (k-r*r) (r*r) a b= |
| 2*((k+r*r)*direction (A := Smooth) coord θ b+direction (A := Smooth) coord θ k*b+ |
| r*(2*(∑ i,logarithmicGrad r hr i*direction (A := Smooth) coord θ (changeP r a i))+ |
| ∑ i,direction (A := Smooth) coord θ (logarithmicGrad r hr i)*changeP r a i)) := by |
| have ht : (∑ i, (2*coord i (r*r)*direction (A := Smooth) coord θ (a i)+ |
| direction (A := Smooth) coord θ (coord i (r*r))*a i) : Smooth)= |
| r*(2*(∑ i,logarithmicGrad r hr i*direction (A := Smooth) coord θ (changeP r a i))+ |
| ∑ i, direction (A := Smooth) coord θ (logarithmicGrad r hr i)*changeP r a i) := by |
| simp_rw [transformed_gradient θ r hr a] |
| rw [← Finset.mul_sum,Finset.sum_add_distrib] |
| simp only [Finset.mul_sum,mul_assoc] |
| rw [← ht] |
| simp only [leadingS,sub_add_cancel,Finset.sum_add_distrib,← Finset.mul_sum,mul_assoc] |
| ring |
|
|
| lemma contraction_at (θ : Fin 3 → ℂ) (k r : Smooth) |
| (hr : ∀ x,(r : X → ℂ) x≠0) (a : Fin 3 → Smooth) (b : Smooth) (x : X) |
| (hP : ∀ i,(direction (A := Smooth) coord θ (changeP r a i) : X → ℂ) x= |
| θ i*(changeS k r hr a b : X → ℂ) x) : |
| ((∑ i,logarithmicGrad r hr i*direction (A := Smooth) coord θ (changeP r a i) : Smooth) : X → ℂ) x= |
| (thetaGamma θ r hr : X → ℂ) x*(changeS k r hr a b : X → ℂ) x := by |
| simp only [thetaGamma,Fin.sum_univ_three,Subalgebra.coe_add,Subalgebra.coe_mul, |
| Subalgebra.coe_smul,Pi.add_apply,Pi.mul_apply,Pi.smul_apply,smul_eq_mul,hP] |
| ring |
|
|
| lemma changeS_derivative_at (θ : Fin 3 → ℂ) (k r : Smooth) |
| (hr : ∀ x,(r : X → ℂ) x≠0) (a : Fin 3 → Smooth) (b : Smooth) (x : X) |
| (hP : ∀ i,(direction (A := Smooth) coord θ (changeP r a i) : X → ℂ) x= |
| θ i*(changeS k r hr a b : X → ℂ) x) : |
| (direction (A := Smooth) coord θ (changeS k r hr a b) : X → ℂ) x= |
| ((-(1/2 : ℂ) • ((direction (A := Smooth) coord θ k*smoothInv r hr- |
| k*(smoothInv r hr)^2*direction (A := Smooth) coord θ r)*b+ |
| k*smoothInv r hr*direction (A := Smooth) coord θ b+ |
| (∑ i,direction (A := Smooth) coord θ (logarithmicGrad r hr i)*changeP r a i)+ |
| thetaGamma θ r hr*changeS k r hr a b) : Smooth) : X → ℂ) x := by |
| rw [changeS_derivative_raw] |
| simp only [Subalgebra.coe_smul,Subalgebra.coe_add,Subalgebra.coe_mul,Pi.smul_apply,Pi.add_apply,Pi.mul_apply] |
| rw [contraction_at θ k r hr a b x hP] |
|
|
| lemma transformed_scalar_at (θ : Fin 3 → ℂ) (k r : Smooth) |
| (hr : ∀ x,(r : X → ℂ) x≠0) (a : Fin 3 → Smooth) (b : Smooth) (x : X) |
| (hP : ∀ i,(direction (A := Smooth) coord θ (changeP r a i) : X → ℂ) x= |
| θ i*(changeS k r hr a b : X → ℂ) x) : |
| ((leadingS coord θ (k-r*r) (r*r) a b : Smooth) : X → ℂ) x= |
| ((2*((k+r*r)*direction (A := Smooth) coord θ b+direction (A := Smooth) coord θ k*b+ |
| r*(2*thetaGamma θ r hr*changeS k r hr a b+ |
| ∑ i,direction (A := Smooth) coord θ (logarithmicGrad r hr i)*changeP r a i)) : Smooth) : X → ℂ) x := by |
| rw [transformed_scalar_raw θ k r hr a b] |
| simp only [Subalgebra.coe_add,Subalgebra.coe_mul,Pi.add_apply,Pi.mul_apply] |
| rw [contraction_at θ k r hr a b x hP] |
| ring |
|
|
| lemma normal_form_bottom_residual_at (θ : Fin 3 → ℂ) (k r : Smooth) |
| (hr : ∀ x,(r : X → ℂ) x≠0) (hk : ∀ x,(k : X → ℂ) x≠0) |
| (hl : ∀ x,((k+r*r : Smooth) : X → ℂ) x≠0) (a : Fin 3 → Smooth) (b : Smooth) |
| (x : X) (hP : ∀ i,(direction (A := Smooth) coord θ (changeP r a i) : X → ℂ) x= |
| θ i*(changeS k r hr a b : X → ℂ) x) : |
| ((direction (A := Smooth) coord θ (changeS k r hr a b)- |
| (transportT θ k r hk hl*changeS k r hr a b+ |
| ∑ i,transportQ θ k r hr hk hl i*changeP r a i) : Smooth) : X → ℂ) x= |
| ((-(1/4 : ℂ) • (k*smoothInv r hr*smoothInv (k+r*r) hl*leadingS coord θ (k-r*r) (r*r) a b) : Smooth) : X → ℂ) x := by |
| rw [transportT_eq θ k r hr hk hl,transportQ_contract] |
| simp only [Subalgebra.coe_sub,Subalgebra.coe_smul,Subalgebra.coe_mul,Pi.sub_apply, |
| Pi.smul_apply,Pi.mul_apply] |
| rw [changeS_derivative_at θ k r hr a b x hP,transformed_scalar_at θ k r hr a b x hP] |
| let R := (r : X → ℂ) x |
| let K := (k : X → ℂ) x |
| let DR := (direction (A := Smooth) coord θ r : X → ℂ) x |
| let DK := (direction (A := Smooth) coord θ k : X → ℂ) x |
| let B := (b : X → ℂ) x |
| let DB := (direction (A := Smooth) coord θ b : X → ℂ) x |
| let H := ((∑ i,logarithmicGrad r hr i*changeP r a i : Smooth) : X → ℂ) x |
| let J := ((∑ i,direction (A := Smooth) coord θ (logarithmicGrad r hr i)*changeP r a i : Smooth) : X → ℂ) x |
| let S := (changeS k r hr a b : X → ℂ) x |
| have he : S=-(1/2 : ℂ)*(K*R⁻¹*B+H) := rfl |
| have hB : 2*R*S+K*B+R*H=0 := by |
| rw [he] |
| have hn : R≠0 := hr x |
| field_simp [hn] |
| ring |
| have hgamma : (thetaGamma θ r hr : X → ℂ) x=2*DR/R := by |
| rw [thetaGamma_eq] |
| change 2*R⁻¹*DR=2*DR/R |
| ring |
| have ha := scalar_residual_algebra R K DR DK B DB H J S (hr x) (hk x) |
| (show K+R*R≠0 from hl x) hB |
| simp only [Subalgebra.coe_add,Subalgebra.coe_sub,Subalgebra.coe_mul,Subalgebra.coe_pow, |
| Subalgebra.coe_smul,Pi.add_apply,Pi.sub_apply,Pi.mul_apply,Pi.pow_apply,Pi.smul_apply, |
| smul_eq_mul,smoothInv_apply,hgamma] |
| have h2 : ((2 : Smooth) : X → ℂ) x=(2 : ℂ) := rfl |
| rw [h2] |
| change -(1/2 : ℂ)*((DK*R⁻¹-K*(R⁻¹)^2*DR)*B+K*R⁻¹*DB+J+(2*DR/R)*S)- |
| (R*R*(K+R*R)⁻¹*(K⁻¹*DK-2*DR/R)*S+ |
| (1/2 : ℂ)*((R*R*(K+R*R)⁻¹*K⁻¹*DK-(1/2 : ℂ)*(2*DR/R))*H-R*R*(K+R*R)⁻¹*J))= |
| -(1/4 : ℂ)*(K*R⁻¹*(K+R*R)⁻¹*(2*((K+R*R)*DB+DK*B+R*(2*(2*DR/R)*S+J)))) |
| convert ha using 1 < |
|
|
| end ElasticityAugmented |
|
|
| end |
| end |
| section |
| noncomputable section |
| open scoped BigOperators |
| namespace ElasticityAugmented |
|
|
| lemma leadingR_contract (θ : Fin 3 → ℂ) (k r : Smooth) |
| (hr : ∀ x,(r : X → ℂ) x≠0) (a : Fin 3 → Smooth) (b : Smooth) : |
| (∑ i,logarithmicGrad r hr i*leadingR coord θ (k-r*r) (r*r) a b i)= |
| 2*r*((∑ i,logarithmicGrad r hr i*direction (A := Smooth) coord θ (changeP r a i))- |
| thetaGamma θ r hr*changeS k r hr a b) := by |
| simp_rw [leadingR_change θ k r hr a b] |
| simp only [thetaGamma,mul_sub,Finset.mul_sum,Finset.sum_mul,Finset.sum_sub_distrib, |
| Algebra.smul_def] |
| congr 1 < |
|
|
| lemma scalar_forced_residual (r k R K b db H J G s : ℂ) (hr : r≠0) (hk : k≠0) |
| (hl : k+r*r≠0) (hB : 2*r*s+k*b+r*H=0) : |
| -(1/2 : ℂ)*((K/r-k*R/r^2)*b+k/r*db+J+G)- |
| ((r*r/(k+r*r))*(K/k-2*R/r)*s+ |
| (1/2 : ℂ)*(((r*r/(k+r*r))*(K/k)-(2*R/r)/2)*H-(r*r/(k+r*r))*J))= |
| (-k/(2*r*(k+r*r)))*((k+r*r)*db+K*b+r*(2*G+J))+ |
| ((k-r*r)/(2*(k+r*r)))*(G-(2*R/r)*s) := by |
| have ha := scalar_residual_algebra r k R K b db H J s hr hk hl hB |
| calc |
| _ = (-(1/2 : ℂ)*((K/r-k*R/r^2)*b+k/r*db+J+(2*R/r)*s)- |
| ((r*r/(k+r*r))*(K/k-2*R/r)*s+ |
| (1/2 : ℂ)*(((r*r/(k+r*r))*(K/k)-(2*R/r)/2)*H-(r*r/(k+r*r))*J)))- |
| (1/2 : ℂ)*(G-(2*R/r)*s) := by ring |
| _ = (-k/(2*r*(k+r*r)))*((k+r*r)*db+K*b+r*(2*(2*R/r)*s+J))- |
| (1/2 : ℂ)*(G-(2*R/r)*s) := by rw [ha] |
| _ = _ := by |
| have hp : k+r^2≠0 := by simpa only [pow_two] using hl |
| field_simp [hr,hl,hp] |
| ring |
|
|
| /-- Exact forced transport intertwining, with no homogeneous leading-equation |
| premise. This is the source term needed for the physical compatible recursion. -/ |
| lemma normal_form_forced_residual (θ : Fin 3 → ℂ) (k r : Smooth) |
| (hr : ∀ x,(r : X → ℂ) x≠0) (hk : ∀ x,(k : X → ℂ) x≠0) |
| (hl : ∀ x,((k+r*r : Smooth) : X → ℂ) x≠0) (a : Fin 3 → Smooth) (b : Smooth) : |
| direction (A := Smooth) coord θ (changeS k r hr a b)- |
| (transportT θ k r hk hl*changeS k r hr a b+ |
| ∑ i,transportQ θ k r hr hk hl i*changeP r a i)= |
| -(1/4 : ℂ) • (k*smoothInv r hr*smoothInv (k+r*r) hl*leadingS coord θ (k-r*r) (r*r) a b)+ |
| (1/4 : ℂ) • ((k-r*r)*smoothInv r hr*smoothInv (k+r*r) hl* |
| ∑ i,logarithmicGrad r hr i*leadingR coord θ (k-r*r) (r*r) a b i) := by |
| rw [changeS_derivative_raw θ k r hr a b,transportT_eq θ k r hr hk hl, |
| transportQ_contract,transformed_scalar_raw θ k r hr a b,leadingR_contract] |
| apply Subtype.ext |
| funext x |
| let R := (r : X → ℂ) x |
| let K := (k : X → ℂ) x |
| let DR := (direction (A := Smooth) coord θ r : X → ℂ) x |
| let DK := (direction (A := Smooth) coord θ k : X → ℂ) x |
| let B := (b : X → ℂ) x |
| let DB := (direction (A := Smooth) coord θ b : X → ℂ) x |
| let H := ((∑ i,logarithmicGrad r hr i*changeP r a i : Smooth) : X → ℂ) x |
| let J := ((∑ i,direction (A := Smooth) coord θ (logarithmicGrad r hr i)*changeP r a i : Smooth) : X → ℂ) x |
| let G := ((∑ i,logarithmicGrad r hr i*direction (A := Smooth) coord θ (changeP r a i) : Smooth) : X → ℂ) x |
| let S := (changeS k r hr a b : X → ℂ) x |
| have he : S=-(1/2 : ℂ)*(K*R⁻¹*B+H) := rfl |
| have hB : 2*R*S+K*B+R*H=0 := by |
| rw [he] |
| have hn : R≠0 := hr x |
| field_simp [hn] |
| ring |
| have hgamma : (thetaGamma θ r hr : X → ℂ) x=2*DR/R := by |
| rw [thetaGamma_eq] |
| change 2*R⁻¹*DR=2*DR/R |
| ring |
| have ha := scalar_forced_residual R K DR DK B DB H J G S (hr x) (hk x) |
| (show K+R*R≠0 from hl x) hB |
| simp only [Subalgebra.coe_add,Subalgebra.coe_sub,Subalgebra.coe_mul,Subalgebra.coe_pow, |
| Subalgebra.coe_smul,Pi.add_apply,Pi.sub_apply,Pi.mul_apply,Pi.pow_apply,Pi.smul_apply, |
| smul_eq_mul,smoothInv_apply,hgamma] |
| have h2 : ((2 : Smooth) : X → ℂ) x=(2 : ℂ) := rfl |
| rw [h2] |
| change -(1/2 : ℂ)*((DK*R⁻¹-K*(R⁻¹)^2*DR)*B+K*R⁻¹*DB+J+G)- |
| (R*R*(K+R*R)⁻¹*(K⁻¹*DK-2*DR/R)*S+ |
| (1/2 : ℂ)*((R*R*(K+R*R)⁻¹*K⁻¹*DK-(1/2 : ℂ)*(2*DR/R))*H-R*R*(K+R*R)⁻¹*J))= |
| -(1/4 : ℂ)*(K*R⁻¹*(K+R*R)⁻¹*(2*((K+R*R)*DB+DK*B+R*(2*G+J))))+ |
| (1/4 : ℂ)*((K-R*R)*R⁻¹*(K+R*R)⁻¹*(2*R*(G-(2*DR/R)*S))) |
| have hR : R≠0 := hr x |
| have hL : K+R^2≠0 := by |
| simpa only [pow_two] using (show K+R*R≠0 from hl x) |
| convert ha using 1 < |
| field_simp [hR,hL] |
| all_goals norm_num |
| end ElasticityAugmented |
|
|
| end |
| end |
| section |
| noncomputable section |
| open scoped BigOperators |
| namespace ElasticityAugmented |
|
|
| def topSource (r : Smooth) (hr : ∀ x,(r : X → ℂ) x≠0) (H : Fin 3 → Smooth) : Fin 3 → Smooth := |
| fun i => (1/2 : ℂ) • (smoothInv r hr*H i) |
| def scalarSource (k r : Smooth) (hr : ∀ x,(r : X → ℂ) x≠0) |
| (hl : ∀ x,((k+r*r : Smooth) : X → ℂ) x≠0) (H : Fin 3 → Smooth) (K : Smooth) : Smooth := |
| -(1/4 : ℂ) • (k*smoothInv r hr*smoothInv (k+r*r) hl*K)+ |
| (1/4 : ℂ) • ((k-r*r)*smoothInv r hr*smoothInv (k+r*r) hl* |
| ∑ i,logarithmicGrad r hr i*H i) |
|
|
| lemma forced_equations_at (θ : Fin 3 → ℂ) (k r : Smooth) |
| (hr : ∀ x,(r : X → ℂ) x≠0) (hk : ∀ x,(k : X → ℂ) x≠0) |
| (hl : ∀ x,((k+r*r : Smooth) : X → ℂ) x≠0) |
| (H : Fin 3 → Smooth) (K : Smooth) (a : Fin 3 → Smooth) (b : Smooth) (x : X) |
| (hp : ∀ i,(direction (A := Smooth) coord θ (changeP r a i) : X → ℂ) x= |
| θ i*(changeS k r hr a b : X → ℂ) x+(topSource r hr H i : X → ℂ) x) |
| (hs : (direction (A := Smooth) coord θ (changeS k r hr a b) : X → ℂ) x= |
| ((transportT θ k r hk hl*changeS k r hr a b+ |
| ∑ i,transportQ θ k r hr hk hl i*changeP r a i : Smooth) : X → ℂ) x+ |
| (scalarSource k r hr hl H K : X → ℂ) x) : |
| (∀ i,((leadingR coord θ (k-r*r) (r*r) a b i : Smooth) : X → ℂ) x=(H i : X → ℂ) x) ∧ |
| ((leadingS coord θ (k-r*r) (r*r) a b : Smooth) : X → ℂ) x=(K : X → ℂ) x := by |
| have hR (i) : ((leadingR coord θ (k-r*r) (r*r) a b i : Smooth) : X → ℂ) x=(H i : X → ℂ) x := by |
| have h := congrArg (fun f : Smooth => (f : X → ℂ) x) (leadingR_change θ k r hr a b i) |
| change ((leadingR coord θ (k-r*r) (r*r) a b i : Smooth) : X → ℂ) x= |
| 2*(r : X → ℂ) x*((direction (A := Smooth) coord θ (changeP r a i) : X → ℂ) x- |
| θ i*(changeS k r hr a b : X → ℂ) x) at h |
| rw [h,hp i] |
| change 2*(r : X → ℂ) x*(θ i*(changeS k r hr a b : X → ℂ) x+ |
| (1/2 : ℂ)*(((r : X → ℂ) x)⁻¹*(H i : X → ℂ) x)-θ i*(changeS k r hr a b : X → ℂ) x)=_ |
| field_simp [hr x] |
| ring |
| refine ⟨hR,?_⟩ |
| have h := congrArg (fun f : Smooth => (f : X → ℂ) x) |
| (normal_form_forced_residual θ k r hr hk hl a b) |
| change (direction (A := Smooth) coord θ (changeS k r hr a b) : X → ℂ) x- |
| ((transportT θ k r hk hl*changeS k r hr a b+ |
| ∑ i,transportQ θ k r hr hk hl i*changeP r a i : Smooth) : X → ℂ) x=_ at h |
| rw [hs,add_sub_cancel_left] at h |
| simp only [scalarSource,Subalgebra.coe_add,Subalgebra.coe_mul,Subalgebra.coe_sub, |
| Subalgebra.coe_smul,Pi.add_apply,Pi.mul_apply,Pi.sub_apply,Pi.smul_apply, |
| smul_eq_mul,smoothInv_apply,Fin.sum_univ_three,hR] at h |
| have hc : -(1/4 : ℂ)*(k : X → ℂ) x*((r : X → ℂ) x)⁻¹* |
| (((k : X → ℂ) x+(r : X → ℂ) x*(r : X → ℂ) x)⁻¹)≠0 := by |
| exact mul_ne_zero (mul_ne_zero (mul_ne_zero (by norm_num) (hk x)) (inv_ne_zero (hr x))) |
| (inv_ne_zero (hl x)) |
| apply (mul_left_cancel₀ hc) |
| linear_combination -h |
| end ElasticityAugmented |
|
|
| end |
| end |
| section |
| noncomputable section |
| open scoped BigOperators |
| namespace ElasticityAugmented |
|
|
| theorem leading_equations_at_iff (θ : Fin 3 → ℂ) (k r : Smooth) |
| (hr : ∀ x,(r : X → ℂ) x≠0) (hk : ∀ x,(k : X → ℂ) x≠0) |
| (hl : ∀ x,((k+r*r : Smooth) : X → ℂ) x≠0) (a : Fin 3 → Smooth) (b : Smooth) (x : X) : |
| ((∀ i,((leadingR coord θ (k-r*r) (r*r) a b i : Smooth) : X → ℂ) x=0) ∧ |
| ((leadingS coord θ (k-r*r) (r*r) a b : Smooth) : X → ℂ) x=0) ↔ |
| (∀ i,(direction (A := Smooth) coord θ (changeP r a i) : X → ℂ) x= |
| θ i*(changeS k r hr a b : X → ℂ) x) ∧ |
| (direction (A := Smooth) coord θ (changeS k r hr a b) : X → ℂ) x= |
| ((transportT θ k r hk hl*changeS k r hr a b+ |
| ∑ i,transportQ θ k r hr hk hl i*changeP r a i : Smooth) : X → ℂ) x := by |
| have hR (i) := congrArg (fun f : Smooth => (f : X → ℂ) x) (leadingR_change θ k r hr a b i) |
| change ∀ i, ((leadingR coord θ (k-r*r) (r*r) a b i : Smooth) : X → ℂ) x= |
| 2*(r : X → ℂ) x*((direction (A := Smooth) coord θ (changeP r a i) : X → ℂ) x- |
| θ i*(changeS k r hr a b : X → ℂ) x) at hR |
| have hn : (2 : ℂ)*(r : X → ℂ) x≠0 := mul_ne_zero (by norm_num) (hr x) |
| constructor |
| · rintro ⟨hp,hs⟩ |
| have hP (i) : (direction (A := Smooth) coord θ (changeP r a i) : X → ℂ) x= |
| θ i*(changeS k r hr a b : X → ℂ) x := by |
| have hh := (hR i).symm.trans (hp i) |
| exact sub_eq_zero.mp ((mul_eq_zero.mp hh).resolve_left hn) |
| refine ⟨hP,?_⟩ |
| have hh := normal_form_bottom_residual_at θ k r hr hk hl a b x hP |
| simp only [Subalgebra.coe_sub,Pi.sub_apply,Subalgebra.coe_smul,Subalgebra.coe_mul, |
| Pi.smul_apply,Pi.mul_apply,smul_eq_mul,hs,mul_zero] at hh |
| exact sub_eq_zero.mp hh |
| · rintro ⟨hP,hS⟩ |
| refine ⟨fun i => ?_,?_⟩ |
| · rw [hR i,hP i,sub_self,mul_zero] |
| · have hh := normal_form_bottom_residual_at θ k r hr hk hl a b x hP |
| simp only [Subalgebra.coe_sub,Pi.sub_apply,hS,sub_self,Subalgebra.coe_smul, |
| Subalgebra.coe_mul,Pi.smul_apply,Pi.mul_apply,smul_eq_mul,smoothInv_apply] at hh |
| rw [← mul_assoc] at hh |
| exact (mul_eq_zero.mp hh.symm).resolve_left (by |
| apply mul_ne_zero |
| · norm_num |
| · exact mul_ne_zero (mul_ne_zero (hk x) (inv_ne_zero (hr x))) (inv_ne_zero (hl x))) |
| end ElasticityAugmented |
|
|
| end |
| end |
| section |
| noncomputable section |
| open scoped BigOperators |
| namespace ElasticityAugmented |
|
|
| lemma normal_inverseP (θ : Fin 3 → ℂ) (r : Smooth) (hr : ∀ x,(r : X → ℂ) x≠0) |
| (p : Fin 3 → Smooth) : normal θ (inverseP r hr p)=smoothInv r hr*normal θ p := by |
| simp only [normal,inverseP,Finset.mul_sum,Algebra.smul_def] |
| exact Finset.sum_congr rfl (fun _ _ => by ring) |
|
|
| /-- Actual physical leading amplitudes, with three independent transformed |
| columns and both augmented leading equations on any prescribed compact region. |
| The equations are local: no global transport equation is assumed. -/ |
| theorem exists_physical_leading_frame (θ : Fin 3 → ℂ) (hθ : ∑ i,θ i*θ i=0) (hne : θ≠0) |
| (k r : Smooth) (hr : ∀ x,(r : X → ℂ) x≠0) (hk : ∀ x,(k : X → ℂ) x≠0) |
| (hl : ∀ x,((k+r*r : Smooth) : X → ℂ) x≠0) |
| {C : Set X} (hC : IsCompact C) : |
| ∃ a : Fin 3 → Fin 3 → Smooth, ∃ b : Fin 3 → Smooth, |
| ∃ B : X → Module.Basis (Fin 3) ℂ (ElasticityFrame.Fiber θ), |
| (∀ ν,normal θ (a ν)=0) ∧ |
| (∀ ν x, x∈C → |
| (∀ i,((leadingR coord θ (k-r*r) (r*r) (a ν) (b ν) i : Smooth) : X → ℂ) x=0) ∧ |
| ((leadingS coord θ (k-r*r) (r*r) (a ν) (b ν) : Smooth) : X → ℂ) x=0) ∧ |
| ∀ ν x, ((B x ν : ElasticityFrame.Fiber θ) : ElasticityFrame.Amp)= |
| ((fun i => (changeP r (a ν) i : X → ℂ) x),(changeS k r hr (a ν) (b ν) : X → ℂ) x) := by |
| let t := transportT θ k r hk hl |
| let q := transportQ θ k r hr hk hl |
| obtain ⟨B,hB,he⟩ := ElasticityFrame.exists_constraint_columns θ hθ hne |
| (t : X → ℂ) (fun i => (q i : X → ℂ)) (smooth_coe t) (fun i => smooth_coe (q i)) hC |
| let p (ν : Fin 3) := ElasticityFrame.vectorSmooth θ (fun x => B x ν) (hB ν) |
| let s (ν : Fin 3) := ElasticityFrame.scalarSmooth θ (fun x => B x ν) (hB ν) |
| let a (ν : Fin 3) := inverseP r hr (p ν) |
| let b (ν : Fin 3) := inverseS k r hr hk (p ν) (s ν) |
| refine ⟨a,b,B,?_,?_,?_⟩ |
| · intro ν |
| rw [show a ν=inverseP r hr (p ν) from rfl,normal_inverseP, |
| ElasticityFrame.smooth_normal θ (fun x => B x ν) (hB ν),mul_zero] |
| · intro ν x hx |
| apply (leading_equations_at_iff θ k r hr hk hl (a ν) (b ν) x).mpr |
| have hp := ElasticityFrame.smooth_vector_equation θ hθ (fun x => B x ν) (hB ν) t q x (he ν x hx) |
| have hs := ElasticityFrame.smooth_scalar_equation θ hθ (fun x => B x ν) (hB ν) t q x (he ν x hx) |
| change (∀ i,(direction (A := Smooth) coord θ |
| (changeP r (inverseP r hr (p ν)) i) : X → ℂ) x= |
| θ i*(changeS k r hr (inverseP r hr (p ν)) (inverseS k r hr hk (p ν) (s ν)) : X → ℂ) x) ∧ _ |
| rw [change_inverseP,change_inverseS] |
| exact ⟨hp,hs⟩ |
| · intro ν x |
| change ((B x ν : ElasticityFrame.Fiber θ) : ElasticityFrame.Amp)= |
| ((fun i => (changeP r (inverseP r hr (p ν)) i : X → ℂ) x), |
| (changeS k r hr (inverseP r hr (p ν)) (inverseS k r hr hk (p ν) (s ν)) : X → ℂ) x) |
| rw [change_inverseP,change_inverseS] |
| rfl |
| end ElasticityAugmented |
|
|
| end |
| end |
| section |
| noncomputable section |
| open scoped BigOperators |
| namespace ElasticityAugmented |
|
|
| lemma normal_topSource (θ : Fin 3 → ℂ) (r : Smooth) (hr : ∀ x,(r : X → ℂ) x≠0) |
| (H : Fin 3 → Smooth) : normal θ (topSource r hr H)=(1/2 : ℂ) • (smoothInv r hr*normal θ H) := by |
| simp only [normal,topSource,Algebra.smul_def,Finset.mul_sum] |
| exact Finset.sum_congr rfl (fun _ _ => by ring) |
|
|
| /-- Smooth physical leading equations with arbitrary tangent vector forcing. |
| Solvability is proved from actual planar transport, not assumed. Forcing need |
| only be tangent on the compact region on which the equations are requested. -/ |
| theorem exists_physical_forced_transport (θ : Fin 3 → ℂ) (hθ : ∑ i,θ i*θ i=0) (hne : θ≠0) |
| (k r : Smooth) (hr : ∀ x,(r : X → ℂ) x≠0) (hk : ∀ x,(k : X → ℂ) x≠0) |
| (hl : ∀ x,((k+r*r : Smooth) : X → ℂ) x≠0) |
| (H : Fin 3 → Smooth) (K : Smooth) {C : Set X} (hC : IsCompact C) |
| (hH : ∀ x∈C,((normal θ H : Smooth) : X → ℂ) x=0) : |
| ∃ a : Fin 3 → Smooth, ∃ b : Smooth, normal θ a=0 ∧ |
| ∀ x∈C, (∀ i,((leadingR coord θ (k-r*r) (r*r) a b i : Smooth) : X → ℂ) x=(H i : X → ℂ) x) ∧ |
| ((leadingS coord θ (k-r*r) (r*r) a b : Smooth) : X → ℂ) x=(K : X → ℂ) x := by |
| obtain ⟨n,hn⟩ := exists_normalizing_vector θ hne |
| let Fp := topSource r hr H |
| let Fs := scalarSource k r hr hl H K |
| let f : X → ElasticityFrame.Fiber θ := fun x => ElasticityFrame.tangentProjection θ n hn |
| ((fun i => (Fp i : X → ℂ) x),(Fs : X → ℂ) x) |
| have hf : ContDiff ℝ (⊤ : ℕ∞) f := ElasticityFrame.tangentProjection_smooth θ n hn Fp Fs |
| let t := transportT θ k r hk hl |
| let q := transportQ θ k r hr hk hl |
| obtain ⟨u,hu,he⟩ := ElasticityFrame.exists_constraint_transport θ hθ hne |
| (t : X → ℂ) (fun i => (q i : X → ℂ)) (smooth_coe t) (fun i => smooth_coe (q i)) f hf hC |
| let p := ElasticityFrame.vectorSmooth θ u hu |
| let s := ElasticityFrame.scalarSmooth θ u hu |
| let a := inverseP r hr p |
| let b := inverseS k r hr hk p s |
| refine ⟨a,b,?_,?_⟩ |
| · rw [show a=inverseP r hr p from rfl,normal_inverseP, |
| ElasticityFrame.smooth_normal θ u hu,mul_zero] |
| · intro x hx |
| have hF : ((f x : ElasticityFrame.Fiber θ) : ElasticityFrame.Amp)= |
| ((fun i => (Fp i : X → ℂ) x),(Fs : X → ℂ) x) := by |
| apply ElasticityFrame.tangentProjection_of_normal θ n hn |
| have h := congrArg (fun v : Smooth => (v : X → ℂ) x) (normal_topSource θ r hr H) |
| change ElasticityFrame.normal θ ((fun i => (Fp i : X → ℂ) x),(Fs : X → ℂ) x)= |
| (1/2 : ℂ)*(((r : X → ℂ) x)⁻¹*((normal θ H : Smooth) : X → ℂ) x) at h |
| rw [h,hH x hx,mul_zero,mul_zero] |
| have hp := ElasticityFrame.smooth_vector_forced θ hθ u hu t q (f x) x (he x hx) |
| have hs := ElasticityFrame.smooth_scalar_forced θ hθ u hu t q (f x) x (he x hx) |
| rw [hF] at hp hs |
| apply forced_equations_at θ k r hr hk hl H K a b x |
| · change ∀ i,(direction (A := Smooth) coord θ (changeP r (inverseP r hr p) i) : X → ℂ) x= |
| θ i*(changeS k r hr (inverseP r hr p) (inverseS k r hr hk p s) : X → ℂ) x+_ |
| rw [change_inverseP,change_inverseS] |
| exact hp |
| · change (direction (A := Smooth) coord θ |
| (changeS k r hr (inverseP r hr p) (inverseS k r hr hk p s)) : X → ℂ) x=_ |
| rw [change_inverseS,change_inverseP] |
| exact hs |
| end ElasticityAugmented |
|
|
| end |
| end |
|
|
| end OAI |
|
|
