Manuscript Em01-B@3 · Em01-B-v3.tex · section · 69189992

Source-coefficient reproducibility

Complete branch–vertex inventory › Source-coefficient reproducibility · lines 2112–2149

The analytic theorem is stated for the fixed smooth arrays $q_{α β}^{L, I, J}$ , rather than pretending that their numerical components were independently expanded here.

For the designated physical source, these arrays may be recovered algorithmically from its local differential expression

(J_{D}^{'} [G] h)^{L} (v) = ∣ α ∣ + ∣ β ∣ \leq 2 \sum q_{α β}^{L, I, J} (v) \partial^{α} G_{J} (v) \partial^{β} h_{I} (v) .

At a fixed point $v$ , choose polynomial jet tests

G_{J} (x) = e_{J} \frac{( x - v ) ^{α}}{α !}, h_{I} (x) = e_{I} \frac{( x - v ) ^{β}}{β !},

multiplied by a cutoff equal to one near $v$ . Evaluation of $J_{D}^{'} [G] h$ at $v$ isolates the stated coefficient. This preserves derivatives of the frozen switch in the actual Euler source.

The designated source is inherited as

J_{D} = EL_{γ} (σ_{ε} ℓ_{3}), ℓ_{3} = - 2 [ρ^{3}] (- det (η + ρ γ) R [η + ρ γ]),

with its fixed curvature and unintegrated-density representative. A numerical evaluation of the physical coefficients requires that fixed source data; the theorems do not invent or modify it.

Displayed formulas in this passage

unlabeled · line 2121
$(J_{D}^{'} [G] h)^{L} (v) = ∣ α ∣ + ∣ β ∣ \leq 2 \sum q_{α β}^{L, I, J} (v) \partial^{α} G_{J} (v) \partial^{β} h_{I} (v) .$
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$G_{J} (x) = e_{J} \frac{( x - v ) ^{α}}{α !}, h_{I} (x) = e_{I} \frac{( x - v ) ^{β}}{β !},$
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$J_{D} = EL_{γ} (σ_{ε} ℓ_{3}), ℓ_{3} = - 2 [ρ^{3}] (- det (η + ρ γ) R [η + ρ γ]),$