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

Coordinate coefficients

Finite-time harmonic coordinates and second-order descent › Coordinate coefficients · lines 930–950

VI@1#eq:coordinates, VI@1#eq:harmonic-coordinate-data

Define four scalar coordinate functions, with an external coordinate label $μ$ , by

X_{γ}^{μ} (x) = x^{μ} + κ w^{μ} (x) + κ^{2} v^{μ} (x) mod κ^{3} .

They are required to satisfy

□_{γ} X_{γ}^{μ} = 0 mod κ^{3}, X_{γ}^{μ} ∣_{t = 0} = x^{μ} ∣_{t = 0}, \partial_{t} X_{γ}^{μ} ∣_{t = 0} = δ^{μ}_{0} .

Displayed formulas in this passage

VI@1#eq:coordinates · line 934
$X_{γ}^{μ} (x) = x^{μ} + κ w^{μ} (x) + κ^{2} v^{μ} (x) mod κ^{3} .$
VI@1#eq:harmonic-coordinate-data · line 942
$□_{γ} X_{γ}^{μ} = 0 mod κ^{3}, X_{γ}^{μ} ∣_{t = 0} = x^{μ} ∣_{t = 0}, \partial_{t} X_{γ}^{μ} ∣_{t = 0} = δ^{μ}_{0} .$