Manuscript XX-A@1 · XX-A.tex · section · 3a586110

The complete inverse relational map

Complete raw and relational lapse through second order › The complete inverse relational map · lines 645–670

XX-A@1#eq:forward, XX-A@1#eq:alpha1, XX-A@1#eq:alpha2

The forward label map is

C_{s} (t, x) = (T_{s} (t, x), R_{s} (T_{s} (t, x), x)) .

Let its formal inverse be

α_{s} = id + s α_{1} + \frac{s ^{2}}{2} α_{2} .

First invert $R_{s}$ at fixed $τ$ , then insert the resulting spatial argument into $θ_{s}$ . This gives

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

Displayed formulas in this passage

XX-A@1#eq:forward · line 648
$C_{s} (t, x) = (T_{s} (t, x), R_{s} (T_{s} (t, x), x)) .$
XX-A@1#eq:alpha1 · line 660
$α_{1}^{0} α_{2}^{0} = u_{1}, = u_{2} - 2 Y_{1}^{i} \partial_{i} u_{1}, α_{1}^{i} α_{2}^{i} = - Y_{1}^{i}, = 2 Y_{1}^{j} \partial_{j} Y_{1}^{i} - Y_{2}^{i} .$
XX-A@1#eq:alpha2 · line 660
$α_{1}^{0} α_{2}^{0} = u_{1}, = u_{2} - 2 Y_{1}^{i} \partial_{i} u_{1}, α_{1}^{i} α_{2}^{i} = - Y_{1}^{i}, = 2 Y_{1}^{j} \partial_{j} Y_{1}^{i} - Y_{2}^{i} .$
unlabeled · line 655
$α_{s} = id + s α_{1} + \frac{s ^{2}}{2} α_{2} .$