Manuscript X-A-conv-ST@1 · X-A-conv-ST.tex · section · 8ca47088

Density transport and the nonlocal kernel

Support, kernels, and the source-specific chart certificate › Density transport and the nonlocal kernel · lines 1314–1357

X-A-conv-ST@1#eq:transportpairing, X-A-conv-ST@1#eq:transportkernel

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Let $V = ι (L)$ , let $χ = ι^{- 1} : V \to L$ , and let

J_{χ} (y) = ∣ det D χ (y) ∣.

Here $χ$ denotes the inverse chart only in this subsection; it is unrelated to the switch profile.

For a completion vector field $v$ , denote its transported field by $v^{ι} = ι_{*} v$ . For a scalar $s$ , write $s^{ι} = s \circ χ$ . Then

c_{ε} = \int_{V} (f \circ χ) (y) (ξ_{*}^{ι} - (S_{ε} ξ_{*})^{ι})^{a} (y) \cdot \partial_{y^{a}} χ^{1} (y) J_{χ} (y) d^{4} y,

whenever the witness certificate holds.

The transported convolution is the two-point operator

(S_{ε}^{ι} v^{ι})^{a} (y) = \frac{\partial ι ^{a}}{\partial x ^{μ}}_{x = χ (y)} \int_{V} ϱ_{ε} (χ (y) - χ (y^{'})) \cdot \frac{\partial χ ^{μ}}{\partial y ^{' b}} (y^{'}) (v^{ι})^{b} (y^{'}) J_{χ} (y^{'}) d^{4} y^{'} .

For compact chart-supported inputs, their pullbacks are extended by zero outside the chart. For the full calculation, certificate (SC-2) guarantees that the relevant convolution inputs are contained there.

Displayed formulas in this passage

X-A-conv-ST@1#eq:transportpairing · line 1328
$c_{ε} = \int_{V} (f \circ χ) (y) (ξ_{*}^{ι} - (S_{ε} ξ_{*})^{ι})^{a} (y) \cdot \partial_{y^{a}} χ^{1} (y) J_{χ} (y) d^{4} y,$
X-A-conv-ST@1#eq:transportkernel · line 1341
$(S_{ε}^{ι} v^{ι})^{a} (y) = \frac{\partial ι ^{a}}{\partial x ^{μ}}_{x = χ (y)} \int_{V} ϱ_{ε} (χ (y) - χ (y^{'})) \cdot \frac{\partial χ ^{μ}}{\partial y ^{' b}} (y^{'}) (v^{ι})^{b} (y^{'}) J_{χ} (y^{'}) d^{4} y^{'} .$
unlabeled · line 1318
$J_{χ} (y) = ∣ det D χ (y) ∣.$

Density transport and the nonlocal kernel

Displayed formulas in this passage

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