Project documentation
Second derivatives distribute coordinate-wise over subtraction (two components for curl). -/ lemma deriv_coord_2nd_sub (f : Space → EuclideanSpace ℝ (Fin 3)) (hf : ContDiff ℝ 2 f) : ∂[u] (fun x => ∂[v] (fun x => f x w) x - ∂[w] (fun x => f x v) x) = (∂[u] (∂[v] (fun x => f x w))) - (∂[u] (∂[w] (fun x => f x v))) := by repeat rw [deriv_eq_fderiv_fun] ext x...
Exact Lean statement
lemma curl_of_curl (f : Space → EuclideanSpace ℝ (Fin 3)) (hf : ContDiff ℝ 2 f) :
∇ ⨯ (∇ ⨯ f) = ∇ (∇ ⬝ f) - Δᵥ fFormal artifact
Lean source
lemma curl_of_curl (f : Space → EuclideanSpace ℝ (Fin 3)) (hf : ContDiff ℝ 2 f) : ∇ ⨯ (∇ ⨯ f) = ∇ (∇ ⬝ f) - Δᵥ f := by unfold laplacianVec laplacian div grad curl Finset.sum simp only [Fin.isValue, Fin.univ_val_map, List.ofFn_succ, Fin.succ_zero_eq_one, Fin.succ_one_eq_two, List.ofFn_zero, Multiset.sum_coe, List.sum_cons, List.sum_nil, add_zero] ext x i fin_cases i <;> · simp only [Fin.reduceFinMk, Fin.isValue, Fin.reduceAdd, zero_add, Pi.sub_apply, PiLp.sub_apply] rw [deriv_coord_2nd_sub, deriv_coord_2nd_sub] simp only [Fin.isValue, Pi.sub_apply] rw [deriv_coord_2nd_add] rw [deriv_commute fun x => f x 0, deriv_commute fun x => f x 1, deriv_commute fun x => f x 2] simp only [Fin.isValue, Pi.add_apply] ring repeat fun_prop- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/SpaceAndTime/Space/Derivatives/Curl.lean:235-250
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