Manuscript X-A-conv@1 · X-A-conv.tex · section · c76fffa3

Switch derivatives inside this prescription

Cauchy operators and the regulated coordinate problem › Switch derivatives inside this prescription · lines 966–1030

X-A-conv@1#eq:dH, X-A-conv@1#eq:CHswitch, X-A-conv@1#eq:da2, X-A-conv@1#eq:da1w

Convolution commutes with constant-coefficient derivatives. Therefore

\partial_{α} H_{μν} C^{μ} (H) = S_{ε} (g \partial_{α} h_{μν} + (\partial_{α} g) h_{μν}), = S_{ε} (g C^{μ} (h) + (\partial_{ν} g) [h^{ν μ} - \frac{1}{2} η^{ν μ} tr h]) .

The same identities hold with $J, k$ .

For clarity, the nonlinear divergence in (eq:v) is

\partial_{α} a_{2} (H)^{α μ} = (\partial_{α} H^{α}_{ρ}) H^{ρ μ} + H^{α}_{ρ} \partial_{α} H^{ρ μ} - \frac{1}{2} (\partial_{α} tr H) H^{α μ} - \frac{1}{2} (tr H) \partial_{α} H^{α μ} + η^{α μ} (\frac{1}{4} (tr H) \partial_{α} tr H - \frac{1}{2} H_{ρ σ} \partial_{α} H^{ρ σ}),

and

\partial_{α} (a_{1} (H)^{α β} \partial_{β} w^{μ}) = (\frac{1}{2} η^{α β} \partial_{α} tr H - \partial_{α} H^{α β}) \partial_{β} w^{μ} + a_{1} (H)^{α β} \partial_{α} \partial_{β} w^{μ} .

Equations (eq:dH)–(eq:da1w) retain the actual switch-derivative terms. No $\partial^{2} g$ term occurs in these displayed coefficient sources. Such terms would occur in some further differentiated presentations, but no such presentation is used here.

In particular,

a_{2} (S_{ε} (g h))

is not replaced by $g a_{2} (h)$ , by $S_{ε} (g a_{2} (h))$ , or by $a_{2} (g h)$ . Those are different prescriptions.

Displayed formulas in this passage

X-A-conv@1#eq:dH · line 969
$\partial_{α} H_{μν} C^{μ} (H) = S_{ε} (g \partial_{α} h_{μν} + (\partial_{α} g) h_{μν}), = S_{ε} (g C^{μ} (h) + (\partial_{ν} g) [h^{ν μ} - \frac{1}{2} η^{ν μ} tr h]) .$
X-A-conv@1#eq:CHswitch · line 969
$\partial_{α} H_{μν} C^{μ} (H) = S_{ε} (g \partial_{α} h_{μν} + (\partial_{α} g) h_{μν}), = S_{ε} (g C^{μ} (h) + (\partial_{ν} g) [h^{ν μ} - \frac{1}{2} η^{ν μ} tr h]) .$
X-A-conv@1#eq:da2 · line 987
$\partial_{α} a_{2} (H)^{α μ} = (\partial_{α} H^{α}_{ρ}) H^{ρ μ} + H^{α}_{ρ} \partial_{α} H^{ρ μ} - \frac{1}{2} (\partial_{α} tr H) H^{α μ} - \frac{1}{2} (tr H) \partial_{α} H^{α μ} + η^{α μ} (\frac{1}{4} (tr H) \partial_{α} tr H - \frac{1}{2} H_{ρ σ} \partial_{α} H^{ρ σ}),$
X-A-conv@1#eq:da1w · line 1004
$\partial_{α} (a_{1} (H)^{α β} \partial_{β} w^{μ}) = (\frac{1}{2} η^{α β} \partial_{α} tr H - \partial_{α} H^{α β}) \partial_{β} w^{μ} + a_{1} (H)^{α β} \partial_{α} \partial_{β} w^{μ} .$
unlabeled · line 1023
$a_{2} (S_{ε} (g h))$