openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Detector /CylinderSpatialCalculus.lean
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Scalar.PlaneCylinderCalculus | |
| namespace OAI | |
| /-! Spatial derivatives on a closed time slab agree with ordinary | |
| derivatives of each full-space slice, including at time zero. -/ | |
| noncomputable section | |
| namespace ForcedComputation.VelocityDetector | |
| open Set | |
| open scoped ContDiff | |
| variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| [NormedAddCommGroup F] [NormedSpace ℝ F] | |
| theorem cylinder_spatial_fderiv_eq {T : ℝ} {U : ℝ × E → F} | |
| (hU : ContDiffOn ℝ ∞ U (Icc 0 T ×ˢ univ)) | |
| {t : ℝ} (ht : t ∈ Icc 0 T) (x : E) : | |
| fderiv ℝ (fun z => U (t, z)) x = | |
| (fderivWithin ℝ U (Icc 0 T ×ˢ univ) (t, x)).comp | |
| (ContinuousLinearMap.inr ℝ ℝ E) := by | |
| have hd := (hU.differentiableOn (by simp) (t, x) ⟨ht, mem_univ x⟩).hasFDerivWithinAt | |
| have hi := (hasFDerivAt_const (𝕜 := ℝ) t x).prodMk (hasFDerivAt_id (𝕜 := ℝ) x) | |
| have hc := hd.comp_hasFDerivAt x hi | |
| (Filter.Eventually.of_forall (fun z => ⟨ht, mem_univ z⟩)) | |
| convert hc.fderiv using 1 | |
| · rfl | |
| · ext z | |
| rfl | |
| theorem cylinder_spatial_fderiv_smooth {T : ℝ} (hT : 0 < T) {U : ℝ × E → F} | |
| (hU : ContDiffOn ℝ ∞ U (Icc 0 T ×ˢ univ)) : | |
| ContDiffOn ℝ ∞ (fun y : ℝ × E => fderiv ℝ (fun z => U (y.1, z)) y.2) | |
| (Icc 0 T ×ˢ univ) := by | |
| have hS : UniqueDiffOn ℝ (Icc 0 T ×ˢ (univ : Set E)) := | |
| (uniqueDiffOn_Icc hT).prod uniqueDiffOn_univ | |
| have hf : ContDiffOn ℝ ∞ (fderivWithin ℝ U (Icc 0 T ×ˢ univ)) | |
| (Icc 0 T ×ˢ univ) := hU.fderivWithin hS (by simp) | |
| have hh := hf.clm_comp | |
| (contDiffOn_const (c := ContinuousLinearMap.inr ℝ ℝ E)) | |
| apply hh.congr | |
| rintro ⟨t, x⟩ hx | |
| exact cylinder_spatial_fderiv_eq hU hx.1 x | |
| theorem cylinder_second_spatial_fderiv_smooth {T : ℝ} (hT : 0 < T) | |
| {U : ℝ × E → F} (hU : ContDiffOn ℝ ∞ U (Icc 0 T ×ˢ univ)) : | |
| ContDiffOn ℝ ∞ | |
| (fun y : ℝ × E => fderiv ℝ (fderiv ℝ (fun z => U (y.1, z))) y.2) | |
| (Icc 0 T ×ˢ univ) := | |
| cylinder_spatial_fderiv_smooth hT (cylinder_spatial_fderiv_smooth hT hU) | |
| end ForcedComputation.VelocityDetector | |
| end | |
| end OAI | |