Manuscript I-B@1 · I-B-v1.tex · proposition · 5357688f

Proposition: Gauge identity under the stated covariance contract

Optional original-rod lapse attachment › The gauge composition identity · lines 1848–1874 · proof included or linked

I-B@1#prop:gauge

Suppose a coordinate change $F$ acts on the supplied objects so that

N_{ε}^{'} = N_{ε} \circ F, C_{ε}^{'} = C_{ε} \circ F .

Then

α_{ε}^{'} = F^{- 1} \circ α_{ε}, n_{ε}^{'} = n_{ε} .

The same statement holds coefficientwise modulo $ε^{3}$ .

Proof. Invert the second covariance identity, then compose:

N_{ε}^{'} \circ α_{ε}^{'} = N_{ε} \circ F \circ F^{- 1} \circ α_{ε} = N_{ε} \circ α_{ε} .

Formal composition gives the jet statement.

Displayed formulas in this passage

unlabeled · line 1851
$N_{ε}^{'} = N_{ε} \circ F, C_{ε}^{'} = C_{ε} \circ F .$
unlabeled · line 1856
$α_{ε}^{'} = F^{- 1} \circ α_{ε}, n_{ε}^{'} = n_{ε} .$
unlabeled · line 1866
$N_{ε}^{'} \circ α_{ε}^{'} = N_{ε} \circ F \circ F^{- 1} \circ α_{ε} = N_{ε} \circ α_{ε} .$