Manuscript VI@1 · VI.tex · section · b5b60afb

The observable

Finite-time harmonic coordinates and second-order descent › The observable · lines 1045–1068

VI@1#eq:inverse-coordinate, VI@1#eq:observable

The formal inverse coordinate map is

Y_{γ}^{μ} (X) = X^{μ} - κ w^{μ} (X) + κ^{2} (w^{ν} \partial_{ν} w^{μ} - v^{μ}) (X) mod κ^{3} .

The formula follows by substituting it into $X_{γ} (Y_{γ} (X))$ .

For $f \in C_{c}^{\infty} ((0, T) \times R^{3}; R)$ , define

O_{f} [γ, ϕ] = \int f (X) ϕ (Y_{γ} (X)) d^{4} X mod κ^{3} .

This is a formal coordinate-space smearing. Its definition does not assume that a numerical nonlinear coordinate map exists globally.

Displayed formulas in this passage

VI@1#eq:inverse-coordinate · line 1048
$Y_{γ}^{μ} (X) = X^{μ} - κ w^{μ} (X) + κ^{2} (w^{ν} \partial_{ν} w^{μ} - v^{μ}) (X) mod κ^{3} .$
VI@1#eq:observable · line 1059
$O_{f} [γ, ϕ] = \int f (X) ϕ (Y_{γ} (X)) d^{4} X mod κ^{3} .$