106 lines
3.3 KiB
Plaintext
106 lines
3.3 KiB
Plaintext
import Mathlib.Data.Fin.Tuple.Basic
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import Mathlib.Analysis.Complex.Basic
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import Mathlib.Analysis.Complex.TaylorSeries
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import Mathlib.Analysis.Calculus.LineDeriv.Basic
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import Mathlib.Analysis.Calculus.ContDiff.Basic
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import Mathlib.Analysis.Calculus.ContDiff.Defs
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import Mathlib.Analysis.Calculus.FDeriv.Basic
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import Mathlib.Analysis.Calculus.FDeriv.Comp
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import Mathlib.Analysis.Calculus.FDeriv.Linear
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import Mathlib.Analysis.Calculus.FDeriv.Symmetric
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noncomputable def Real.partialDeriv : ℂ → (ℂ → ℂ) → (ℂ → ℂ) :=
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fun v ↦ (fun f ↦ (fun w ↦ fderiv ℝ f w v))
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theorem partialDeriv_smul {f : ℂ → ℂ} {a v : ℂ} (h : Differentiable ℝ f) : Real.partialDeriv v (a • f) = a • Real.partialDeriv v f := by
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unfold Real.partialDeriv
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have : a • f = fun y ↦ a • f y := by rfl
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rw [this]
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conv =>
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left
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intro w
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rw [fderiv_const_smul (h w)]
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theorem partialDeriv_add {f₁ f₂ : ℂ → ℂ} {v : ℂ} (h₁ : Differentiable ℝ f₁) (h₂ : Differentiable ℝ f₂) : Real.partialDeriv v (f₁ + f₂) = (Real.partialDeriv v f₁) + (Real.partialDeriv v f₂) := by
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unfold Real.partialDeriv
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have : f₁ + f₂ = fun y ↦ f₁ y + f₂ y := by rfl
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rw [this]
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conv =>
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left
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intro w
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left
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rw [fderiv_add (h₁ w) (h₂ w)]
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theorem partialDeriv_compLin {f : ℂ → ℂ} {l : ℂ →L[ℝ] ℂ} {v : ℂ} (h : Differentiable ℝ f) : Real.partialDeriv v (l ∘ f) = l ∘ Real.partialDeriv v f := by
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unfold Real.partialDeriv
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conv =>
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left
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intro w
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left
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rw [fderiv.comp w (ContinuousLinearMap.differentiableAt l) (h w)]
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simp
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rfl
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theorem partialDeriv_contDiff {n : ℕ} {f : ℂ → ℂ} (h : ContDiff ℝ (n + 1) f) : ∀ v : ℂ, ContDiff ℝ n (Real.partialDeriv v f) := by
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unfold Real.partialDeriv
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intro v
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let A := (contDiff_succ_iff_fderiv.1 h).right
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simp at A
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have : (fun w => (fderiv ℝ f w) v) = (fun f => f v) ∘ (fun w => (fderiv ℝ f w)) := by
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rfl
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rw [this]
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refine ContDiff.comp ?hg A
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refine ContDiff.of_succ ?hg.h
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refine ContDiff.clm_apply ?hg.h.hf ?hg.h.hg
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exact contDiff_id
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exact contDiff_const
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lemma partialDeriv_fderiv {f : ℂ → ℂ} (hf : ContDiff ℝ 2 f) (z a b : ℂ) :
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fderiv ℝ (fderiv ℝ f) z b a = Real.partialDeriv b (Real.partialDeriv a f) z := by
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unfold Real.partialDeriv
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rw [fderiv_clm_apply]
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· simp
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· exact (contDiff_succ_iff_fderiv.1 hf).2.differentiable le_rfl z
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· simp
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theorem partialDeriv_comm {f : ℂ → ℂ} (h : ContDiff ℝ 2 f) :
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∀ v₁ v₂ : ℂ, Real.partialDeriv v₁ (Real.partialDeriv v₂ f) = Real.partialDeriv v₂ (Real.partialDeriv v₁ f) := by
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intro v₁ v₂
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funext z
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have derivSymm :
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(fderiv ℝ (fun w => fderiv ℝ f w) z) v₁ v₂ = (fderiv ℝ (fun w => fderiv ℝ f w) z) v₂ v₁ := by
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let f' := fderiv ℝ f
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have h₀ : ∀ y, HasFDerivAt f (f' y) y := by
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intro y
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exact DifferentiableAt.hasFDerivAt ((h.differentiable one_le_two) y)
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let f'' := (fderiv ℝ f' z)
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have h₁ : HasFDerivAt f' f'' z := by
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apply DifferentiableAt.hasFDerivAt
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apply (contDiff_succ_iff_fderiv.1 h).right.differentiable (Submonoid.oneLE.proof_2 ℕ∞)
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apply second_derivative_symmetric h₀ h₁ v₁ v₂
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rw [← partialDeriv_fderiv h z v₂ v₁]
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rw [derivSymm]
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rw [partialDeriv_fderiv h z v₁ v₂]
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