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

Proposition: Existence and uniqueness of the scalar assignment

Switching before variation › The scalar assignment · lines 609–627 · proof included or linked

VII-A-ST@1#prop:scalar

The expression (eq:scalar-coeff) uniquely solves

D_{g} [γ] ϕ_{g} = 0 (mod κ^{3})

with the stated leading data and zero positive-order data. It is defined for arbitrary smooth $h, k$ .

Proof. The two coefficient equations are

P_{m} u_{1, g} = - V_{g} [h] ϕ_{0}, P_{m} u_{2, g} = - V_{g} [h] u_{1, g} - W_{g} [h, k] ϕ_{0} .

Apply Lemma lem:Cauchy successively. Substituting the first solution gives (eq:scalar-coeff).

Displayed formulas in this passage

unlabeled · line 612
$D_{g} [γ] ϕ_{g} = 0 (mod κ^{3})$
unlabeled · line 621
$P_{m} u_{1, g} = - V_{g} [h] ϕ_{0}, P_{m} u_{2, g} = - V_{g} [h] u_{1, g} - W_{g} [h, k] ϕ_{0} .$