Research proof index

Distributions on Time × Space d from distributions on Space d. These distributions are constant in time. -/ def constantTime {M : Type} [NormedAddCommGroup M] [NormedSpace ℝ M] {d : ℕ} : ((Space d) →d[ℝ] M) →ₗ[ℝ] (Time × Space d) →d[ℝ] M where toFun f := f ∘L timeIntegralSchwartz map_add' f g := by ext η simp map_smul' c f := by ext η simp lemma const...

Distributions on Time × Space d from distributions on Space d. These distributions are constant in time. -/ def constantTime {M : Type} [NormedAddCommGroup M] [NormedSpace ℝ M] {d : ℕ} : ((Space d) →d[ℝ] M) →ₗ[ℝ] (Time × Space d) →d[ℝ] M where toFun f := f ∘L timeIntegralSchwartz map_add' f g := by ext η simp map_smul' c f := by ext η simp lemma const...

Second derivatives distribute coordinate-wise over subtraction (two components for curl). -/ lemma deriv_coord_2nd_sub (f : Space → EuclideanSpace ℝ (Fin 3)) (hf : ContDiff ℝ 2 f) : ∂[u] (fun x => ∂[v] (fun x => f x w) x - ∂[w] (fun x => f x v) x) = (∂[u] (∂[v] (fun x => f x w))) - (∂[u] (∂[w] (fun x => f x v))) := by repeat rw [deriv_eq_fderiv_fun] ext x...

The homotopy operator is defined as ∫ t in 0..1, (t • x) ⨯ f (t • x). This is the integrand of that function. -/ private noncomputable def homotopyOperatorIntegrand (f : Space → EuclideanSpace ℝ (Fin 3)) : Space → ℝ → EuclideanSpace ℝ (Fin 3) := fun x t => (t • basis.repr x) ⨯ₑ₃ f (t • x) private lemma homotopyOperatorIntegrand_eq (f : Space → Euclidean...

The spatial-derivative in terms of the derivative of functions between manifolds with the manifold structure Space.manifoldStructure d.

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...