Manuscript I-B-VP-1@1 · I-B-VP-1-v1.tex · corollary · 6d700be9

Corollary: Computed defects and an admissible correction

Simultaneous extensions and the exact Ward repair › The direct cutoff prescription need not obey the Ward equations · lines 816–862 · proof included or linked

I-B-VP-1@1#cor:repair, I-B-VP-1@1#eq:stressdefect, I-B-VP-1@1#eq:tracedefect, I-B-VP-1@1#eq:currentrawconserved, I-B-VP-1@1#eq:stressrepair, I-B-VP-1@1#eq:currentrepair

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The preliminary lift has

D_{j} T_{L}^{c} t T_{L}^{c} - 3 q Δ (n F_{L}) d J_{L}^{c} = - \frac{b σ _{α}}{16} \partial_{j} Δ δ_{W}, = 3 k σ_{α} Δ δ_{W}, = 0.

The corrections

(T_{L} - T_{L}^{c})^{A}_{B} (T_{L} - T_{L}^{c})^{i}_{j} (J_{L} - J_{L}^{c})^{i} = - \frac{a σ _{α}}{4} δ^{A}_{B} Δ δ_{W}, = b σ_{α} (\frac{7}{16} δ^{i}_{j} Δ - \frac{3}{8} \partial^{i} \partial_{j}) δ_{W}, = \frac{j σ _{α}}{4 α ^{2}} ϵ^{i}_{k} \partial_{k} δ_{W}

give exactly Theorem thm:extension.

Proof. The divergence follows from (eq:rawdefects). The raw trace is

2 (a - b) V_{L}^{c} = 12 k V_{L}^{c} .

Equation (eq:convertV) makes $3 q Δ (n F_{L}) = 12 k V_{L}^{c} - 3 k σ_{α} Δ δ_{W}$ , proving (eq:tracedefect). The vector in (eq:convertQ) differs from a gradient by a delta derivative, so its rotation by $ϵ$ is conserved. The corrections are the three conversions multiplied by their regular coefficients.

Displayed formulas in this passage

I-B-VP-1@1#eq:stressdefect · line 819
$D_{j} T_{L}^{c} t T_{L}^{c} - 3 q Δ (n F_{L}) d J_{L}^{c} = - \frac{b σ _{α}}{16} \partial_{j} Δ δ_{W}, = 3 k σ_{α} Δ δ_{W}, = 0.$
I-B-VP-1@1#eq:tracedefect · line 819
$D_{j} T_{L}^{c} t T_{L}^{c} - 3 q Δ (n F_{L}) d J_{L}^{c} = - \frac{b σ _{α}}{16} \partial_{j} Δ δ_{W}, = 3 k σ_{α} Δ δ_{W}, = 0.$
I-B-VP-1@1#eq:currentrawconserved · line 819
$D_{j} T_{L}^{c} t T_{L}^{c} - 3 q Δ (n F_{L}) d J_{L}^{c} = - \frac{b σ _{α}}{16} \partial_{j} Δ δ_{W}, = 3 k σ_{α} Δ δ_{W}, = 0.$
I-B-VP-1@1#eq:stressrepair · line 831
$(T_{L} - T_{L}^{c})^{A}_{B} (T_{L} - T_{L}^{c})^{i}_{j} (J_{L} - J_{L}^{c})^{i} = - \frac{a σ _{α}}{4} δ^{A}_{B} Δ δ_{W}, = b σ_{α} (\frac{7}{16} δ^{i}_{j} Δ - \frac{3}{8} \partial^{i} \partial_{j}) δ_{W}, = \frac{j σ _{α}}{4 α ^{2}} ϵ^{i}_{k} \partial_{k} δ_{W}$
I-B-VP-1@1#eq:currentrepair · line 831
$(T_{L} - T_{L}^{c})^{A}_{B} (T_{L} - T_{L}^{c})^{i}_{j} (J_{L} - J_{L}^{c})^{i} = - \frac{a σ _{α}}{4} δ^{A}_{B} Δ δ_{W}, = b σ_{α} (\frac{7}{16} δ^{i}_{j} Δ - \frac{3}{8} \partial^{i} \partial_{j}) δ_{W}, = \frac{j σ _{α}}{4 α ^{2}} ϵ^{i}_{k} \partial_{k} δ_{W}$
unlabeled · line 852
$2 (a - b) V_{L}^{c} = 12 k V_{L}^{c} .$

Corollary: Computed defects and an admissible correction

Displayed formulas in this passage

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