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

Proposition: Actual coordinate coefficients

Stage B: Cauchy solution and regulator preflight › Auxiliary coordinates through second order · lines 573–601 · proof included or linked

X-A-conv-ST@1#prop:coordinates, X-A-conv-ST@1#eq:w, X-A-conv-ST@1#eq:v

The unique coefficients of (eq:harmonic) are

w^{μ} [h] v^{μ} [h, k] = E_{0} C^{μ} H, = E_{0} C^{μ} J - E_{0} [\partial_{α} a_{2} (H)^{α μ} + \partial_{α} (a_{1} (H)^{α β} \partial_{β} w^{μ})] .

Their values and first time derivatives vanish at $Σ$ .

Proof. Insert the displayed density expansion into (eq:harmonic). The coefficients of $κ$ and $κ^{2}$ give respectively

□_{η} w^{μ} = C^{μ} H

and the source displayed in (eq:v). Lemma lem:Cauchy gives the formulas and successive uniqueness.

Displayed formulas in this passage

X-A-conv-ST@1#eq:w · line 576
$w^{μ} [h] v^{μ} [h, k] = E_{0} C^{μ} H, = E_{0} C^{μ} J - E_{0} [\partial_{α} a_{2} (H)^{α μ} + \partial_{α} (a_{1} (H)^{α β} \partial_{β} w^{μ})] .$
X-A-conv-ST@1#eq:v · line 576
$w^{μ} [h] v^{μ} [h, k] = E_{0} C^{μ} H, = E_{0} C^{μ} J - E_{0} [\partial_{α} a_{2} (H)^{α μ} + \partial_{α} (a_{1} (H)^{α β} \partial_{β} w^{μ})] .$
unlabeled · line 595
$□_{η} w^{μ} = C^{μ} H$