D Wirtinger Anti Coord comp holomorphic apply
Physlib.Wirtinger.dWirtingerAntiCoord_comp_holomorphic_apply
Plain-language statement
The single-term coordinate chain rule for a holomorphic outer g, anti-holomorphic version, pointwise at u: ∂̄_I (g ∘ f) = deriv g (f u) · ∂̄_I f. As in dWirtingerCoord_comp_holomorphic_apply, the ∂g/∂f̄ channel vanishes and ∂g/∂f collapses to deriv g (f u).
Exact Lean statement
lemma dWirtingerAntiCoord_comp_holomorphic_apply {u : (ι → ℂ)}
(hg : DifferentiableAt ℂ g (f u)) (hf : DifferentiableAt ℝ f u) (I : ι) :
dWirtingerAntiCoord (fun v => g (f v)) I u =
deriv g (f u) * dWirtingerAntiCoord f I uFormal artifact
Lean source
lemma dWirtingerAntiCoord_comp_holomorphic_apply {u : (ι → ℂ)} (hg : DifferentiableAt ℂ g (f u)) (hf : DifferentiableAt ℝ f u) (I : ι) : dWirtingerAntiCoord (fun v => g (f v)) I u = deriv g (f u) * dWirtingerAntiCoord f I u := by rw [dWirtingerAntiCoord_comp_apply (hg.restrictScalars ℝ) hf I, dWirtingerDir_eq_of_clinear (clinear_of_holomorphic hg 1), DifferentiableAt.fderiv_restrictScalars ℝ hg, ContinuousLinearMap.coe_restrictScalars', fderiv_apply_one_eq_deriv, dWirtingerAntiDir_eq_zero_of_clinear (clinear_of_holomorphic hg 1), zero_mul, add_zero]- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Mathematics/Calculus/Wirtinger/Coordinate.lean:605-613
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