| import OAI.Analysis.BernoulliCone.PositiveQuotient |
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| namespace OAI |
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| universe uE |
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| section |
| noncomputable section |
| open Set Filter |
| open scoped Topology ContDiff |
| namespace Bernoulli.ConeCalculus |
| variable {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] |
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| lemma euler_identity {f : E → ℝ} {x : E} (hf : DifferentiableAt ℝ f x) |
| (hom : ∀ r : ℝ, 0 < r → ∀ y, f (r • y) = r*f y) : |
| fderiv ℝ f x x = f x := by |
| have hc : HasDerivAt (fun r : ℝ => f (r • x)) (fderiv ℝ f x x) 1 := by |
| have hf1 : HasFDerivAt f (fderiv ℝ f x) ((fun r : ℝ => r • x) 1) := by simpa using hf.hasFDerivAt |
| convert! hf1.comp_hasDerivAt 1 ((hasDerivAt_id (1:ℝ)).smul_const x) using 1 |
| simp |
| have he : (fun r : ℝ => f (r • x)) =ᶠ[𝓝 1] (fun r => r*f x) := by |
| filter_upwards [eventually_gt_nhds (by norm_num : (0:ℝ)<1)] with r hr |
| exact hom r hr x |
| apply hc.unique |
| simpa using (((hasDerivAt_id (1:ℝ)).mul_const (f x)).congr_of_eventuallyEq he) |
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| / |
| phase. No Hessian or angular equation is assumed. -/ |
| lemma hessian_radial {f : E → ℝ} {O : Set E} (hO : IsOpen O) |
| (hf : ContDiffOn ℝ ∞ f O) |
| (hom : ∀ r : ℝ, 0 < r → ∀ y, f (r • y) = r*f y) |
| {x : E} (hx : x∈O) (v : E) : |
| fderiv ℝ (fderiv ℝ f) x v x = 0 := by |
| have hs := hf.contDiffAt (hO.mem_nhds hx) |
| have hd := hs.differentiableAt (by simp) |
| have hD : DifferentiableAt ℝ (fderiv ℝ f) x := |
| (hs.fderiv_right (m:=1) (by change (↑(2 : ℕ∞) : WithTop ℕ∞) ≤ ↑(⊤ : ℕ∞); exact WithTop.coe_le_coe.mpr le_top)).differentiableAt (by simp) |
| have he : (fun y => fderiv ℝ f y y) =ᶠ[𝓝 x] f := by |
| filter_upwards [hO.mem_nhds hx] with y hy |
| exact euler_identity ((hf.contDiffAt (hO.mem_nhds hy)).differentiableAt (by simp)) hom |
| have hh := (hD.hasFDerivAt.clm_apply (hasFDerivAt_id x)).congr_of_eventuallyEq he.symm |
| have heq := congrArg (fun A : E →L[ℝ] ℝ => A v) (hh.unique hd.hasFDerivAt) |
| simp only [add_apply,ContinuousLinearMap.coe_comp, |
| Function.comp_apply,ContinuousLinearMap.id_apply,ContinuousLinearMap.flip_apply,id_eq] at heq |
| linarith |
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| variable {d : ℕ} |
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| lemma laplacian_trace (f : EuclideanSpace ℝ (Fin d) → ℝ) (x : EuclideanSpace ℝ (Fin d)) : |
| Laplacian.laplacian f x = ∑ i, fderiv ℝ (fderiv ℝ f) x |
| (EuclideanSpace.basisFun (Fin d) ℝ i) (EuclideanSpace.basisFun (Fin d) ℝ i) := by |
| rw [InnerProductSpace.laplacian_eq_iteratedFDeriv_orthonormalBasis f |
| (EuclideanSpace.basisFun (Fin d) ℝ)] |
| simp [iteratedFDeriv_two_apply] |
| end Bernoulli.ConeCalculus |
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| end |
| end |
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| end OAI |
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