Manuscript Em02-D@4 · Em02-D-v4.tex · section · fcf2fac7

Ambient source jet

Fixed model, root, and localization › Ambient source jet · lines 315–363

Em02-D@4#eq:source-reduction, Em02-D@4#eq:relative

The sparse component data are

A^{00, 2} = A^{00}_{2} = D^{00, 2, 2} = 1, P^{L}_{M} = δ^{L}_{M}, b_{1} = b_{2} = 1.

Every undisplayed $A$ or $D$ entry vanishes.

For $j = 1, 2$ , define the source incidence

Q^{j, j, j} (v; a, z) = δ^{(4)} (v - z) δ^{(4)} (v - a) + \partial_{v^{1}} δ^{(4)} (v - z) δ^{(4)} (v - a) .

The products are tensor-product pullbacks in independent incidence coordinates.

For $E = R$ or $B$ ,

\int E (b - v) Q^{j, j, j} (v; a, z) d^{4} v = E (b - a) δ^{(4)} (a - z) - \partial_{z^{1}} [E (b - a) δ^{(4)} (a - z)] .

The derivative is transferred before incidence restriction. The identity can be checked in independent propagation and incidence coordinates; their pushforward is proper on compact joint tests.

Introduce

S = b - a, r = z - a, x = y - a, H_{r} = δ^{(4)} (r) - \partial_{r^{1}} δ^{(4)} (r) .

Then

⟨ H_{r}, ψ ⟩ = ψ (0) + \partial_{r^{1}} ψ (0) .

Displayed formulas in this passage

Em02-D@4#eq:source-reduction · line 338
$\int E (b - v) Q^{j, j, j} (v; a, z) d^{4} v = E (b - a) δ^{(4)} (a - z) - \partial_{z^{1}} [E (b - a) δ^{(4)} (a - z)] .$
Em02-D@4#eq:relative · line 352
$S = b - a, r = z - a, x = y - a, H_{r} = δ^{(4)} (r) - \partial_{r^{1}} δ^{(4)} (r) .$
unlabeled · line 318
$A^{00, 2} = A^{00}_{2} = D^{00, 2, 2} = 1, P^{L}_{M} = δ^{L}_{M}, b_{1} = b_{2} = 1.$
unlabeled · line 327
$Q^{j, j, j} (v; a, z) = δ^{(4)} (v - z) δ^{(4)} (v - a) + \partial_{v^{1}} δ^{(4)} (v - z) δ^{(4)} (v - a) .$
unlabeled · line 359
$⟨ H_{r}, ψ ⟩ = ψ (0) + \partial_{r^{1}} ψ (0) .$