Manuscript VII-I@1 · VII-I.tex · section · 907e96ab

Route B1b: a divergence-form density preparation

Route B1b: a divergence-form density preparation · lines 757–803

VII-I@1#eq:prepared-density, VII-I@1#eq:ramp-coefficients

Choose a smooth ramp

σ_{ε} (t) = χ (t / ε), χ = 0 for t \leq 0, χ = 1 for t \geq 1, 0 \leq χ \leq 1.

It is flat at the endpoints.

Define a coordinate density

A_{ϵ}^{α β} [q] = η^{α β} + σ_{ε} (- q q^{α β} - η^{α β}) .

The prepared coordinates solve

\partial_{α} (A_{ϵ}^{α β} [q] \partial_{β} X_{q, ϵ}^{μ}) = 0

with the same coordinate-time Cauchy data as (eq:coordinates).

The coefficients are

w_{ϵ}^{μ} [h] v_{ϵ}^{μ} [h, k] = - E^{R}_{0} \partial_{α} (σ_{ε} a_{1}^{α μ} [h]), = - E^{R}_{0} [\partial_{α} (σ_{ε} (a_{1}^{α μ} [k] + a_{2}^{α μ} [h])) + \partial_{α} (σ_{ε} a_{1}^{α β} [h] \partial_{β} w_{ϵ}^{μ} [h])] .

The resulting functional $O_{f, ϵ}^{D}$ is defined by formal inverse-coordinate composition, not by assigning a finite value to one graph.

Displayed formulas in this passage

VII-I@1#eq:prepared-density · line 770
$A_{ϵ}^{α β} [q] = η^{α β} + σ_{ε} (- q q^{α β} - η^{α β}) .$
VII-I@1#eq:ramp-coefficients · line 786
$w_{ϵ}^{μ} [h] v_{ϵ}^{μ} [h, k] = - E^{R}_{0} \partial_{α} (σ_{ε} a_{1}^{α μ} [h]), = - E^{R}_{0} [\partial_{α} (σ_{ε} (a_{1}^{α μ} [k] + a_{2}^{α μ} [h])) + \partial_{α} (σ_{ε} a_{1}^{α β} [h] \partial_{β} w_{ϵ}^{μ} [h])] .$
unlabeled · line 761
$σ_{ε} (t) = χ (t / ε), χ = 0 for t \leq 0, χ = 1 for t \geq 1, 0 \leq χ \leq 1.$
unlabeled · line 777
$\partial_{α} (A_{ϵ}^{α β} [q] \partial_{β} X_{q, ϵ}^{μ}) = 0$