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

Lemma: Inverse-map coefficients

Optional original-rod lapse attachment › The second-order composition formula · lines 1755–1788 · proof included or linked

I-B@1#lem:inverse-jets

Writing

α_{ε} = id + ε α_{1} + \frac{ε ^{2}}{2} α_{2} (mod ε^{3}),

one obtains

α_{1}^{0} = u_{1}, α_{1}^{i} = - Y_{1}^{i},

and

α_{2}^{0} = u_{2} - 2 Y_{1}^{i} \partial_{i} u_{1}, α_{2}^{i} = 2 Y_{1}^{j} \partial_{j} Y_{1}^{i} - Y_{2}^{i} .

Proof. Substitution into $R_{ε} (τ, S_{ε} (τ, y)) = y$ gives

S_{ε} = y - ε Y_{1} + \frac{ε ^{2}}{2} (2 Y_{1}^{j} \partial_{j} Y_{1} - Y_{2}) .

Expanding $θ_{ε} (τ, S_{ε} (τ, y))$ gives the time component, including $- 2 Y_{1}^{i} \partial_{i} u_{1}$ .

Displayed formulas in this passage

unlabeled · line 1758
$α_{ε} = id + ε α_{1} + \frac{ε ^{2}}{2} α_{2} (mod ε^{3}),$
unlabeled · line 1765
$α_{1}^{0} = u_{1}, α_{1}^{i} = - Y_{1}^{i},$
unlabeled · line 1769
$α_{2}^{0} = u_{2} - 2 Y_{1}^{i} \partial_{i} u_{1}, α_{2}^{i} = 2 Y_{1}^{j} \partial_{j} Y_{1}^{i} - Y_{2}^{i} .$
unlabeled · line 1778
$S_{ε} = y - ε Y_{1} + \frac{ε ^{2}}{2} (2 Y_{1}^{j} \partial_{j} Y_{1} - Y_{2}) .$