Manuscript Em01-F@3 · Em01-F-v3.tex · lemma · 225e59af

Lemma: Response reduction

Substitution and the weighted comparison panel › Source incidence · lines 1973–2002 · proof included or linked

Em01-F@3#lem:response, Em01-F@3#eq:response

For $D_{0} = R, B$ ,

\int D_{0} (b - v) Q^{L, I, J} (v; a, z) d^{4} v = α, β \sum (- 1)^{∣ α ∣ + ∣ β ∣} \partial_{z}^{α} \partial_{a}^{β} [q_{α β}^{L, I, J} D_{0} (b - a) δ_{D}^{(4)} (a - z)] .

The incidence and response distributions exist, including at $a = z$ .

Proof. The map $(v, a, z) \mapsto (v - z, v - a)$ is a submersion. Adding $b - v$ provides an independent propagation variable. Thus the products are tensor products in independent coordinates. The incidence makes the integration locally proper on compact external test supports.

Move each delta derivative to its external variable, producing $(- 1)^{∣ α ∣ + ∣ β ∣}$ . The undifferentiated integration is $q_{α β} D_{0} (b - a) δ_{D}^{(4)} (a - z)$ .

Displayed formulas in this passage

Em01-F@3#eq:response · line 1976
$\int D_{0} (b - v) Q^{L, I, J} (v; a, z) d^{4} v = α, β \sum (- 1)^{∣ α ∣ + ∣ β ∣} \partial_{z}^{α} \partial_{a}^{β} [q_{α β}^{L, I, J} D_{0} (b - a) δ_{D}^{(4)} (a - z)] .$

Nearby in this section

section: Source incidence