Research proof index

The scalar factor in the odd-dimensional iterated Laplacian of the norm is nonzero.

The measure on Space d weighted by 1 / ‖x‖ ^ (d - 1). -/ def radialAngularMeasure {d : ℕ} : Measure (Space d) := volume.withDensity (fun x : Space d => ENNReal.ofReal (1 / ‖x‖ ^ (d - 1))) /-! ### A.1. Basic equalities -/ lemma radialAngularMeasure_eq_volume_withDensity {d : ℕ} : radialAngularMeasure = volume.withDensity (fun x : Space d => ENNReal.ofR...

The linear equivalence between Space d.succ and ℝ × Space d extracting the ith coordinate. -/ def slice {d} (i : Fin d.succ) : Space d.succ ≃L[ℝ] ℝ × Space d where toFun x := ⟨x i, ⟨fun j => x (Fin.succAbove i j)⟩⟩ invFun p := ⟨fun j => Fin.insertNthEquiv (fun _ => ℝ) i (p.fst, p.snd) j⟩ map_add' x y := by simp only [Nat.succ_eq_add_one, Prod.mk_add...

Derivatives along space coordinates and time commute. -/ lemma fderiv_time_commute_fderiv_space {M} [NormedAddCommGroup M] [NormedSpace ℝ M] (f : Time → Space d → M) (t dt : Time) (x dx : Space d) (hf : ContDiff ℝ 2 ↿f) : fderiv ℝ (fun t' => fderiv ℝ (fun x' => f t' x') x dx) t dt = fderiv ℝ (fun x' => fderiv ℝ (fun t' => f t' x') t dt) x dx := by trans f...

Curl and time derivative commute. -/ lemma time_deriv_curl_commute (fₜ : Time → Space → EuclideanSpace ℝ (Fin 3)) (t : Time) (x : Space) (hf : ContDiff ℝ 2 ↿fₜ) : ∂ₜ (fun t => (∇ ⨯ fₜ t) x) t = (∇ ⨯ fun x => (∂ₜ (fun t => fₜ t x) t)) x:= by ext i rw [← Time.deriv_euclid] · fin_cases i all_goals simp [curl] rw [Time.deriv_eq, fderiv_fun_sub] simp [← Time.d...

Curl and time derivative commute.