Manuscript XVI-B@1 · XVI-B.tex · proposition · 6a0abffd

Proposition: Weighted divergence residual

Weighted representatives and the boundary question · lines 677–708 · proof included or linked

XVI-B@1#prop:boundary, XVI-B@1#eq:boundaryrepresentative

For local densities related by

ℓ_{3}^{'} = ℓ_{3} + \partial_{μ} B^{μ},

one has

EL (ℓ_{3}^{'}) = EL (ℓ_{3}),

whereas

EL (σ ℓ_{3}^{'}) - EL (σ ℓ_{3}) = - EL ((\partial_{μ} σ) B^{μ}) .

Proof. A total divergence has zero bulk Euler derivative under compact interior variations: its variation is again a divergence whose integral has no boundary contribution.

Use the product rule

σ \partial_{μ} B^{μ} = \partial_{μ} (σ B^{μ}) - (\partial_{μ} σ) B^{μ} .

Apply the linear bulk Euler derivative. The first term on the right is a total divergence and contributes zero. This proves (eq:boundaryrepresentative).

Displayed formulas in this passage

XVI-B@1#eq:boundaryrepresentative · line 688
$EL (σ ℓ_{3}^{'}) - EL (σ ℓ_{3}) = - EL ((\partial_{μ} σ) B^{μ}) .$
unlabeled · line 680
$ℓ_{3}^{'} = ℓ_{3} + \partial_{μ} B^{μ},$
unlabeled · line 684
$EL (ℓ_{3}^{'}) = EL (ℓ_{3}),$
unlabeled · line 701
$σ \partial_{μ} B^{μ} = \partial_{μ} (σ B^{μ}) - (\partial_{μ} σ) B^{μ} .$

Proposition: Weighted divergence residual

Displayed formulas in this passage

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