Manuscript VII-A-ST@1 · VII-A-ST.tex · proposition · c39a67a3

Proposition: Harmonic Cauchy construction

Unswitched harmonic labels and the relational coefficients · lines 654–678 · proof included or linked

VII-A-ST@1#prop:coordinates, VII-A-ST@1#eq:coordinates

The unique coefficients are

w^{μ} [h] v^{μ} [h, k] = E_{0} C^{μ} [h], = E_{0} (C^{μ} [k] - \partial_{α} B^{α μ} (h) - Q_{h}^{(0)} w^{μ} [h]) .

Proof. At first order the equation is $□_{η} w^{μ} = C^{μ} [h]$ . At second order it is

□_{η} v^{μ} = C^{μ} [k] - \partial_{α} B^{α μ} (h) - Q_{h}^{(0)} w^{μ} .

Both have zero data, so Lemma lem:Cauchy applies.

Displayed formulas in this passage

VII-A-ST@1#eq:coordinates · line 657
$w^{μ} [h] v^{μ} [h, k] = E_{0} C^{μ} [h], = E_{0} (C^{μ} [k] - \partial_{α} B^{α μ} (h) - Q_{h}^{(0)} w^{μ} [h]) .$
unlabeled · line 672
$□_{η} v^{μ} = C^{μ} [k] - \partial_{α} B^{α μ} (h) - Q_{h}^{(0)} w^{μ} .$