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110f4f3 | 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 52 53 54 55 56 57 58 59 60 61 | import OAI.Analysis.BernoulliCone.PositiveQuotient
namespace OAI
universe uE
section
noncomputable section
open Set Filter
open scoped Topology ContDiff
namespace Bernoulli.ConeCalculus
variable {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E]
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)
/-- Euler's homogeneous identity differentiated on the actual open smooth
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
variable {d : ℕ}
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
end
end
end OAI
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