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

Lemma: Boundary representatives matter after weighting

Density weighting is not source weighting › Two different specifications · lines 581–603 · proof included or linked

XII-B@1#lem:boundary, XII-B@1#eq:boundaryrepresentative

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

EL (ℓ_{3}^{'}) = EL (ℓ_{3}),

but

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

Proof. Euler variation annihilates a total divergence under compact variations. Use

σ_{ε} \partial_{μ} B^{μ} = \partial_{μ} (σ_{ε} B^{μ}) - (\partial_{μ} σ_{ε}) B^{μ}

and apply that fact.

Displayed formulas in this passage

XII-B@1#eq:boundaryrepresentative · line 588
$EL (σ_{ε} ℓ_{3}^{'}) - EL (σ_{ε} ℓ_{3}) = - EL ((\partial_{μ} σ_{ε}) B^{μ}) .$
unlabeled · line 584
$EL (ℓ_{3}^{'}) = EL (ℓ_{3}),$
unlabeled · line 598
$σ_{ε} \partial_{μ} B^{μ} = \partial_{μ} (σ_{ε} B^{μ}) - (\partial_{μ} σ_{ε}) B^{μ}$