Research proof index

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...

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...

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...

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...

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...