Manuscript X-D-ST@1 · X-D-ST.tex · section · e0cf8fdb

Limiting bare and completed tests

Propagation and the fixed-support convergence theorem › Limiting bare and completed tests · lines 1049–1071

X-D-ST@1#eq:limiting-tests

Set

B_{0}^{μ} = w_{0}^{ν} \partial_{ν} w_{0}^{μ} - v_{0}^{μ},

and define

c_{1, 0} c_{2, 0} d_{1, 0} d_{2, 0} = \partial_{μ} (f w_{0}^{μ}), = - \partial_{μ} (f B_{0}^{μ}) + \frac{1}{2} \partial_{μ} \partial_{ν} (f w_{0}^{μ} w_{0}^{ν}), = c_{1, 0} - L_{1} E_{A} f, = c_{2, 0} - L_{1} E_{A} c_{1, 0} - L_{2} E_{A} f + L_{1} E_{A} L_{1} E_{A} f .

The products with $f$ extend smoothly by zero to the completion, because $f$ is supported away from the initial surface.

Displayed formulas in this passage

X-D-ST@1#eq:limiting-tests · line 1056
$c_{1, 0} c_{2, 0} d_{1, 0} d_{2, 0} = \partial_{μ} (f w_{0}^{μ}), = - \partial_{μ} (f B_{0}^{μ}) + \frac{1}{2} \partial_{μ} \partial_{ν} (f w_{0}^{μ} w_{0}^{ν}), = c_{1, 0} - L_{1} E_{A} f, = c_{2, 0} - L_{1} E_{A} c_{1, 0} - L_{2} E_{A} f + L_{1} E_{A} L_{1} E_{A} f .$
unlabeled · line 1052
$B_{0}^{μ} = w_{0}^{ν} \partial_{ν} w_{0}^{μ} - v_{0}^{μ},$