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

Proposition: Exact retained infinitesimal variation

The retained second-order defects › Infinitesimal ledger · lines 1057–1093 · proof included or linked

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

The variation is

δ_{ξ} O = κ \int f X_{ξ - b} ϕ + κ^{2} \int f (X_{X_{b - ξ} w} ϕ - X_{R_{2, ξ}} ϕ + X_{w} X_{b - ξ} ϕ) + O (κ^{3}) .

Proof. The leading variation of $w$ is $b$ , while

δ_{ξ}^{(1)} w = W T_{g, ε} (L_{ξ} h), δ_{ξ}^{(0)} v = 2 V (H, K b) .

Thus their sum is $d_{2}$ . Differentiating $J$ gives

D J (w) [b] = X_{w} X_{b} + X_{X_{b} w} .

The second-order integrand from (eq:Ocoefficients) is consequently

- X_{d_{2}} ϕ + X_{w} X_{b} ϕ + X_{X_{b} w} ϕ - X_{w} X_{ξ} ϕ .

Using $d_{2} = R_{2, ξ} + X_{ξ} w$ yields (eq:seconddefect).

Displayed formulas in this passage

X-A-conv-ST@1#eq:seconddefect · line 1060
$δ_{ξ} O = κ \int f X_{ξ - b} ϕ + κ^{2} \int f (X_{X_{b - ξ} w} ϕ - X_{R_{2, ξ}} ϕ + X_{w} X_{b - ξ} ϕ) + O (κ^{3}) .$
unlabeled · line 1076
$δ_{ξ}^{(1)} w = W T_{g, ε} (L_{ξ} h), δ_{ξ}^{(0)} v = 2 V (H, K b) .$
unlabeled · line 1083
$D J (w) [b] = X_{w} X_{b} + X_{X_{b} w} .$
unlabeled · line 1088
$- X_{d_{2}} ϕ + X_{w} X_{b} ϕ + X_{X_{b} w} ϕ - X_{w} X_{ξ} ϕ .$