Manuscript XI-B@1 · XI-B.tex · section · 063b989e

General and conditional formulas

The ramp layer dressing › General and conditional formulas · lines 1058–1081

XI-B@1#eq:Xgeneral

The density coefficient inherited from X-A is

a_{1}^{μν} [γ] = \frac{1}{2} η^{μν} tr_{η} γ - γ^{μν} .

Its divergence gives exactly the de Donder combination:

C^{μ} [γ] = - \partial_{α} a_{1}^{α μ} [γ] = D^{μ} γ .

The regulated first coordinate coefficient, with zero initial data, is therefore

X_{1}^{μ} [γ] = E_{0}^{R} (σ_{ε} D^{μ} γ - σ_{ε}^{'} a_{1}^{0 μ} [γ]) .

This is the divergence-form ramp prescription, not a multiplied-equation replacement.

Displayed formulas in this passage

XI-B@1#eq:Xgeneral · line 1073
$X_{1}^{μ} [γ] = E_{0}^{R} (σ_{ε} D^{μ} γ - σ_{ε}^{'} a_{1}^{0 μ} [γ]) .$
unlabeled · line 1061
$a_{1}^{μν} [γ] = \frac{1}{2} η^{μν} tr_{η} γ - γ^{μν} .$
unlabeled · line 1066
$C^{μ} [γ] = - \partial_{α} a_{1}^{α μ} [γ] = D^{μ} γ .$