Research proof index

The space derivative of a distribution dependent on time and space. -/ noncomputable def distSpaceDeriv {M d} [NormedAddCommGroup M] [NormedSpace ℝ M] (i : Fin d) : ((Time × Space d) →d[ℝ] M) →ₗ[ℝ] (Time × Space d) →d[ℝ] M where toFun f := let ev : (Time × Space d →L[ℝ] M) →L[ℝ] M := { toFun v := v (0, basis i) map_add' v1 v2 := by simp only [root.add_a...

The space derivative of a distribution dependent on time and space. -/ noncomputable def distSpaceDeriv {M d} [NormedAddCommGroup M] [NormedSpace ℝ M] (i : Fin d) : ((Time × Space d) →d[ℝ] M) →ₗ[ℝ] (Time × Space d) →d[ℝ] M where toFun f := let ev : (Time × Space d →L[ℝ] M) →L[ℝ] M := { toFun v := v (0, basis i) map_add' v1 v2 := by simp only [root.add_a...

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...

The spatial projection is nonexpanding for the product metric. -/ lemma dist_space_le (tx ty : TimeAndSpace d) : dist (space tx) (space ty) ≤ dist tx ty := by rw [Prod.dist_eq] exact le_max_right _ _ end TimeAndSpace namespace Space /-! ### A.2. Space and time derivatives in terms of curried functions

Derivatives along space coordinates and time commute.

A power series which is differentiable everywhere, and in the limit as n → ∞ tends to ‖x‖. -/ def normPowerSeries {d} : ℕ → Space d → ℝ := fun n x => √(‖x‖ ^ 2 + 1/(n + 1)) lemma normPowerSeries_eq (n : ℕ) : normPowerSeries (d := d) n = fun x => √(‖x‖ ^ 2 + 1/(n + 1)) := rfl lemma normPowerSeries_eq_rpow {d} (n : ℕ) : normPowerSeries (d := d) n = fun...