Manuscript X-C-ST@1 · X-C-ST.tex · lemma · b679954e

Lemma: Coordinate defect coefficients

Formal duality and the two defects › The coordinate defect · lines 1316–1355 · proof included or linked

X-C-ST@1#lem:theta, X-C-ST@1#eq:theta

The coordinate defect has compact coefficients with $supp θ_{j} \subset K$ , and

θ_{1} θ_{2} = \partial_{μ} (f (w [K ξ] - ξ)^{μ}), = c_{2} [h^{'}, k^{'}] - c_{2} [h, k] - D_{ξ} c_{1} [h] - (D_{ζ} + \frac{1}{2} D_{ξ}^{2}) f .

For $F \in G_{p}$ , $Θ_{F} = 0$ .

Proof. Expansion of (eq:defects) gives (eq:theta). Formal pullback coefficients are differential operators and do not enlarge coefficient supports. This proves the compact-support statement.

On $G_{p}$ , substitute the inherited coordinate laws (eq:coordinate-laws). They give

c_{1} [h^{'}] - c_{1} [h] = D_{ξ} f, c_{2} [h^{'}, k^{'}] - c_{2} [h, k] = D_{ξ} c_{1} [h] + (D_{ζ} + \frac{1}{2} D_{ξ}^{2}) f .

For example, with $B = w \cdot \partial w - v$ ,

B^{'} = B + w \cdot \partial ξ + \frac{1}{2} ξ \cdot \partial ξ - ζ .

Insertion in (eq:bare), followed by compact integration by parts, gives the second test identity. Both defect coefficients vanish.

Displayed formulas in this passage

X-C-ST@1#eq:theta · line 1320
$θ_{1} θ_{2} = \partial_{μ} (f (w [K ξ] - ξ)^{μ}), = c_{2} [h^{'}, k^{'}] - c_{2} [h, k] - D_{ξ} c_{1} [h] - (D_{ζ} + \frac{1}{2} D_{ξ}^{2}) f .$
unlabeled · line 1341
$c_{1} [h^{'}] - c_{1} [h] = D_{ξ} f, c_{2} [h^{'}, k^{'}] - c_{2} [h, k] = D_{ξ} c_{1} [h] + (D_{ζ} + \frac{1}{2} D_{ξ}^{2}) f .$
unlabeled · line 1348
$B^{'} = B + w \cdot \partial ξ + \frac{1}{2} ξ \cdot \partial ξ - ζ .$

Nearby in this section

section: The coordinate defect