Manuscript X-A-conv-ST@1 · X-A-conv-ST.tex · proposition · 8ca47088

Proposition: First-order defect

Strict positivity and the evaluated classical obstruction › Relational expansion and insertion sign · lines 691–713 · proof included or linked

X-A-conv-ST@1#prop:firstdefect, X-A-conv-ST@1#eq:firstdefect

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With

δ_{ξ} h = K ξ + κ L_{ξ} h, δ_{ξ} ϕ = κ X_{ξ} ϕ,

the coefficient of $κ$ in the variation is

[κ] δ_{ξ} O = \int f X_{ξ - S_{ε} ξ} ϕ .

Proof. The zeroth coefficient varies by $κ \int f X_{ξ} ϕ$ . The leading variation of $w$ is $w [K ξ] = S_{ε} ξ$ , so the first insertion contributes $- κ \int f X_{S_{ε} ξ} ϕ$ . Their sum is (eq:firstdefect).

Displayed formulas in this passage

X-A-conv-ST@1#eq:firstdefect · line 699
$[κ] δ_{ξ} O = \int f X_{ξ - S_{ε} ξ} ϕ .$
unlabeled · line 694
$δ_{ξ} h = K ξ + κ L_{ξ} h, δ_{ξ} ϕ = κ X_{ξ} ϕ,$

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section: Relational expansion and insertion sign