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

Proposition: Response distribution and adjoint

Exact response and seed assembly › Response assembly · lines 638–681 · proof included or linked

Em02-D@4#prop:response, Em02-D@4#eq:response-adjoint

The second-component response sum is

K_{resp}^{22} = - D (f D R) δ^{(4)} (r) - [D (A_{S} R) + f R] H_{r} .

Its exact action is

⟨ K_{resp}^{22}, M ⟩ = \int d^{4} a d R (S) [- f D^{2} M + f c_{S} D M - f M + A_{S} D \partial_{r^{1}} M - f \partial_{r^{1}} M]_{r = 0} .

Proof. The separate actions are

⟨ K_{6}^{22}, M ⟩ ⟨ K_{8}^{22}, M ⟩ ⟨ K_{9 d}^{22}, M ⟩ ⟨ K_{9 m}^{22}, M ⟩ = - \int d^{4} a d R (S) [f D^{2} M + (D f) D M]_{r = 0}, = \int d^{4} a d R (S) [f c_{S} D (M + \partial_{r^{1}} M)]_{r = 0}, = \int d^{4} a d R (S) [D (M + \partial_{r^{1}} M)]_{r = 0}, = - \int d^{4} a d R (S) [f (M + \partial_{r^{1}} M)]_{r = 0} .

For row $6$ , transfer the outer derivative first and the line derivative second:

⟨ - D (f D R), M ⟩ = ⟨ D R, f D M ⟩ = - ⟨ R, f D^{2} M + (D f) D M ⟩ .

Here $D f = 1$ . Summation proves (eq:response-adjoint). In particular, the $- D M$ term from row $6$ cancels the $+ D M$ term from row $9 d$ .

Displayed formulas in this passage

Em02-D@4#eq:response-adjoint · line 647
$⟨ K_{resp}^{22}, M ⟩ = \int d^{4} a d R (S) [- f D^{2} M + f c_{S} D M - f M + A_{S} D \partial_{r^{1}} M - f \partial_{r^{1}} M]_{r = 0} .$
unlabeled · line 661
$⟨ K_{6}^{22}, M ⟩ ⟨ K_{8}^{22}, M ⟩ ⟨ K_{9 d}^{22}, M ⟩ ⟨ K_{9 m}^{22}, M ⟩ = - \int d^{4} a d R (S) [f D^{2} M + (D f) D M]_{r = 0}, = \int d^{4} a d R (S) [f c_{S} D (M + \partial_{r^{1}} M)]_{r = 0}, = \int d^{4} a d R (S) [D (M + \partial_{r^{1}} M)]_{r = 0}, = - \int d^{4} a d R (S) [f (M + \partial_{r^{1}} M)]_{r = 0} .$
unlabeled · line 641
$K_{resp}^{22} = - D (f D R) δ^{(4)} (r) - [D (A_{S} R) + f R] H_{r} .$
unlabeled · line 673
$⟨ - D (f D R), M ⟩ = ⟨ D R, f D M ⟩ = - ⟨ R, f D^{2} M + (D f) D M ⟩ .$