Manuscript XII-B@1 · XII-B.tex · lemma · 229bc869

Lemma: Weighted Euler expression

Density weighting is not source weighting › A local Euler convention · lines 507–543 · proof included or linked

XII-B@1#lem:weighted, XII-B@1#eq:weightedEuler, XII-B@1#eq:layerEuler

For a fixed smooth scalar function $σ$ ,

EL_{a} (σ ℓ) = σ EL_{a} (ℓ) + D_{σ, a} [ℓ],

where

D_{σ, a} [ℓ] = ∣ I ∣ \leq r \sum 0 < J \leq I \sum (- 1)^{∣ I ∣} (J I) (\partial^{J} σ) D_{I - J} (\frac{\partial ℓ}{\partial ψ _{I} ^{a}}) .

For a time-dependent ramp, all terms in $D_{σ, a}$ contain a positive-order time derivative of $σ$ .

Proof. Since $σ$ is independent of the field jets,

\frac{\partial ( σ ℓ )}{\partial ψ _{I} ^{a}} = σ \frac{\partial ℓ}{\partial ψ _{I} ^{a}} .

Apply the multi-index Leibniz rule to every term of (eq:Euler). The terms with $J = 0$ give $σ EL_{a} (ℓ)$ . The others give (eq:layerEuler). Spatial derivatives of a time-only weight vanish.

Displayed formulas in this passage

XII-B@1#eq:weightedEuler · line 510
$EL_{a} (σ ℓ) = σ EL_{a} (ℓ) + D_{σ, a} [ℓ],$
XII-B@1#eq:layerEuler · line 516
$D_{σ, a} [ℓ] = ∣ I ∣ \leq r \sum 0 < J \leq I \sum (- 1)^{∣ I ∣} (J I) (\partial^{J} σ) D_{I - J} (\frac{\partial ℓ}{\partial ψ _{I} ^{a}}) .$
unlabeled · line 534
$\frac{\partial ( σ ℓ )}{\partial ψ _{I} ^{a}} = σ \frac{\partial ℓ}{\partial ψ _{I} ^{a}} .$

Lemma: Weighted Euler expression

Displayed formulas in this passage

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