D Wirtinger Dir d Wirtinger Anti Dir comm
Physlib.Wirtinger.dWirtingerDir_dWirtingerAntiDir_comm
Project documentation
Schwarz's theorem for the directional Wirtinger operators: on a C² field f the holomorphic and anti-holomorphic directional derivatives commute in any two directions, ∂_v ∂̄_w f = ∂̄_w ∂_v f. The commutation adds no analytic input. By the §G bridge each order expands into a real-linear combination of the second real Fréchet derivative `fderiv ℝ...
Exact Lean statement
theorem dWirtingerDir_dWirtingerAntiDir_comm (hf2 : ContDiffAt ℝ 2 f u) (v w : V) :
dWirtingerDir (fun p => dWirtingerAntiDir f w p) v u
= dWirtingerAntiDir (fun p => dWirtingerDir f v p) w uFormal artifact
Lean source
theorem dWirtingerDir_dWirtingerAntiDir_comm (hf2 : ContDiffAt ℝ 2 f u) (v w : V) : dWirtingerDir (fun p => dWirtingerAntiDir f w p) v u = dWirtingerAntiDir (fun p => dWirtingerDir f v p) w u := by have hf' : DifferentiableAt ℝ (fderiv ℝ f) u := (hf2.fderiv_right (m := 1) le_rfl).differentiableAt one_ne_zero have hsymm : IsSymmSndFDerivAt ℝ f u := hf2.isSymmSndFDerivAt (by simp) rw [dWirtingerDir_apply, dWirtingerAntiDir_apply] simp only [fderiv_dWirtingerAntiDir hf', fderiv_dWirtingerDir hf', hsymm.eq] ring- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Mathematics/Calculus/Wirtinger/Basic.lean:754-762
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