Manuscript X-D@3 · X-D-v3.tex · section · b7774529

Limiting tests and the rate theorem

Propagation and convergence › Limiting tests and the rate theorem · lines 950–969

X-D@3#eq:limit-bare

Define

B_{0}^{μ} = X_{1, 0}^{ν} \partial_{ν} X_{1, 0}^{μ} - X_{2, 0}^{μ},

and

c_{1, 0} c_{2, 0} = \partial_{μ} (f X_{1, 0}^{μ}), = - \partial_{μ} (f B_{0}^{μ}) + \frac{1}{2} \partial_{μ} \partial_{ν} (f X_{1, 0}^{μ} X_{1, 0}^{ν}) .

They are smooth compact tests on $M$ , because $f$ vanishes near $Σ$ . Define $d_{j, 0}$ by (eq:limiting-completed).

Displayed formulas in this passage

X-D@3#eq:limit-bare · line 958
$c_{1, 0} c_{2, 0} = \partial_{μ} (f X_{1, 0}^{μ}), = - \partial_{μ} (f B_{0}^{μ}) + \frac{1}{2} \partial_{μ} \partial_{ν} (f X_{1, 0}^{μ} X_{1, 0}^{ν}) .$
unlabeled · line 953
$B_{0}^{μ} = X_{1, 0}^{ν} \partial_{ν} X_{1, 0}^{μ} - X_{2, 0}^{μ},$