Manuscript XX-A@1 · XX-A.tex · section · 3a586110

The linear observable and Hessian must be coupled

Classical gauge identity and rod-detection tests › The linear observable and Hessian must be coupled · lines 823–842

XX-A@1#eq:observable-jet

Let $z = (h, φ)$ , $ℓ = (k, ψ)$ . Define

F_{1, f} (z) = \int f n_{1} [z], F_{2, f} (z, z) = \int f n_{2} [z, 0],

and polarize $F_{2, f}$ bilinearly. The complete expansion is

O_{f} (s) = O_{0, f} + s F_{1, f} (z) + \frac{s ^{2}}{2} (F_{1, f} (ℓ) + F_{2, f} (z, z)) .

For a compact vector field $ξ$ , put

K ξ = (L_{ξ} g ˉ, L_{ξ} ϕ ˉ) .

Displayed formulas in this passage

XX-A@1#eq:observable-jet · line 831
$O_{f} (s) = O_{0, f} + s F_{1, f} (z) + \frac{s ^{2}}{2} (F_{1, f} (ℓ) + F_{2, f} (z, z)) .$
unlabeled · line 826
$F_{1, f} (z) = \int f n_{1} [z], F_{2, f} (z, z) = \int f n_{2} [z, 0],$
unlabeled · line 839
$K ξ = (L_{ξ} g ˉ, L_{ξ} ϕ ˉ) .$