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...

The distributional Laplacian of ‖x‖ ^ (2 - d) is (2 - d) * d * volume (Metric.ball 0 1) times the Dirac delta at the origin. For d ≥ 3 this ‖x‖ ^ (2 - d) is the (singular) fundamental solution of the Laplacian, and for d = 1 it is ‖x‖. When d = 2 the exponent vanishes, so the identity collapses to the trivial Δᵈ 1 = 0; the genuine two-dime...

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...

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...

A constructive form of the statement that if the curl of a function is zero, then it is equal to the grad of another function. In the context of e.g. electromagnetism the potential given by this lemma corresponds to that defined by the work done to move a unit charge from the origin to the point x along a straight line.