Deriv comm of curl zero
Space.deriv_comm_of_curl_zero
Project documentation
The homotopy operator is defined as ∫ t in 0..1, (t • x) ⨯ f (t • x). This is the integrand of that function. -/ private noncomputable def homotopyOperatorIntegrand (f : Space → EuclideanSpace ℝ (Fin 3)) : Space → ℝ → EuclideanSpace ℝ (Fin 3) := fun x t => (t • basis.repr x) ⨯ₑ₃ f (t • x) private lemma homotopyOperatorIntegrand_eq (f : Space → Euclidean...
Exact Lean statement
lemma deriv_comm_of_curl_zero (f : Space → EuclideanSpace ℝ (Fin 3)) (hf : Differentiable ℝ f)
(hcurl : curl f = 0) (x : Space) (i j : Fin 3) :
∂[i] f x j = ∂[j] f x iFormal artifact
Lean source
lemma deriv_comm_of_curl_zero (f : Space → EuclideanSpace ℝ (Fin 3)) (hf : Differentiable ℝ f) (hcurl : curl f = 0) (x : Space) (i j : Fin 3) : ∂[i] f x j = ∂[j] f x i := by fin_cases i <;> fin_cases j any_goals rfl all_goals simp only [Fin.reduceFinMk, Fin.isValue, Fin.zero_eta] rw [← deriv_euclid (by fun_prop), ← deriv_euclid (by fun_prop)] have hcurl' (i : Fin 3) : curl f x i = 0 := by simp [hcurl] · specialize hcurl' 2 simp [curl] at hcurl' linear_combination (norm := ring_nf) hcurl' · specialize hcurl' 1 simp [curl] at hcurl' linear_combination (norm := ring_nf) -hcurl' · specialize hcurl' 2 simp [curl] at hcurl' linear_combination (norm := ring_nf) -hcurl' · specialize hcurl' 0 simp [curl] at hcurl' linear_combination (norm := ring_nf) hcurl' · specialize hcurl' 1 simp [curl] at hcurl' linear_combination (norm := ring_nf) hcurl' · specialize hcurl' 0 simp [curl] at hcurl' linear_combination (norm := ring_nf) -hcurl'- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/SpaceAndTime/Space/Derivatives/Curl.lean:557-583
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