Product rules for the space and phase cutoffs #
The spatial cutoff and normal-phase cutoff are differentiated before the lower-time factor is added.
theorem
ESS.bu_spatialSecondPartial_mul_full
{a b : CKN.Foundation.Parabolic.Vec3 × ℝ → ℝ}
(ha : ContDiff ℝ (↑⊤) a)
(hb : ContDiff ℝ (↑⊤) b)
(j : Fin 3)
(z : CKN.Foundation.Parabolic.ParabolicPoint)
:
CKN.spatialSecondPartial (fun (q : CKN.Foundation.Parabolic.ParabolicPoint) => a q * b q) j j z = CKN.spatialSecondPartial a j j z * b z + 2 * CKN.spatialPartial a j z * CKN.spatialPartial b j z + a z * CKN.spatialSecondPartial b j j z
The diagonal Hessian of a product has the two cross terms of the ordinary product rule.