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

Infinitesimal ledger

The retained second-order defects › Infinitesimal ledger · lines 1013–1056

Let

W (A) = E_{0} C A,

and let $a_{2} (A, B)$ denote the symmetric polarization

a_{2} (A, B) = \frac{1}{2} (a_{2} (A + B) - a_{2} (A) - a_{2} (B)) .

Define

V^{μ} (A, B) = - E_{0} [\partial_{α} a_{2} (A, B)^{α μ} + \frac{1}{2} \partial_{α} (a_{1} (A)^{α β} \partial_{β} W (B)^{μ} + a_{1} (B)^{α β} \partial_{β} W (A)^{μ})] .

Then

w = W (H), v = W (J) + V (H, H) .

For the bare functional $k = 0$ , write $δ_{ξ} = δ_{ξ}^{(0)} + κ δ_{ξ}^{(1)}$ , where

δ_{ξ}^{(0)} h = K ξ, δ_{ξ}^{(0)} ϕ = 0, δ_{ξ}^{(1)} h = L_{ξ} h, δ_{ξ}^{(1)} ϕ = X_{ξ} ϕ .

Set

b = S_{ε} ξ, d_{2} = W T_{g, ε} (L_{ξ} h) + 2 V (H, K b), R_{2, ξ} = d_{2} - X_{ξ} w .

Displayed formulas in this passage

unlabeled · line 1016
$W (A) = E_{0} C A,$
unlabeled · line 1020
$a_{2} (A, B) = \frac{1}{2} (a_{2} (A + B) - a_{2} (A) - a_{2} (B)) .$
unlabeled · line 1024
$V^{μ} (A, B) = - E_{0} [\partial_{α} a_{2} (A, B)^{α μ} + \frac{1}{2} \partial_{α} (a_{1} (A)^{α β} \partial_{β} W (B)^{μ} + a_{1} (B)^{α β} \partial_{β} W (A)^{μ})] .$
unlabeled · line 1036
$w = W (H), v = W (J) + V (H, H) .$
unlabeled · line 1044
$δ_{ξ}^{(0)} h = K ξ, δ_{ξ}^{(0)} ϕ = 0, δ_{ξ}^{(1)} h = L_{ξ} h, δ_{ξ}^{(1)} ϕ = X_{ξ} ϕ .$
unlabeled · line 1051
$b = S_{ε} ξ, d_{2} = W T_{g, ε} (L_{ξ} h) + 2 V (H, K b), R_{2, ξ} = d_{2} - X_{ξ} w .$