Manuscript XVIII-B@2 · XVIII-B-v2.tex · section · 27ed4f4b

Exact transformation formula

Prepared response and its calculated residual › Exact transformation formula · lines 1357–1374

XVIII-B@2#eq:Bres, XVIII-B@2#eq:productrule

For a compact smooth vector field $ξ$ , put

B_{ε} (ξ)^{ab} = (\partial^{a} σ_{ε}) ξ^{b} + (\partial^{b} σ_{ε}) ξ^{a} .

It is compact and supported in the preparation layer where the ramp derivative meets $supp ξ$ . The product rule gives

σ_{ε} K ξ = K (σ_{ε} ξ) - B_{ε} (ξ) .

Displayed formulas in this passage

XVIII-B@2#eq:Bres · line 1360
$B_{ε} (ξ)^{ab} = (\partial^{a} σ_{ε}) ξ^{b} + (\partial^{b} σ_{ε}) ξ^{a} .$
XVIII-B@2#eq:productrule · line 1370
$σ_{ε} K ξ = K (σ_{ε} ξ) - B_{ε} (ξ) .$