Manuscript XVI-K@1 · XVI-K.tex · section · aa7553bb

Self-contained C 2 source definition

The selected source and differentiated graph kernels › Self-contained C 2 source definition · lines 704–739

XVI-K@1#eq:C2, XVI-K@1#eq:source

For a symmetric matrix $H$ , evaluate $u_{0}, u_{1}, u_{2}, b_{1}, b_{2}$ on $H$ , and introduce the formal polynomial

U (υ) = u_{0} + υ u_{1} + υ^{2} u_{2} .

Define

C_{2}^{μν α β} (H) = [υ^{2}] (1 + υ b_{1} + υ^{2} b_{2}) (U^{μν} (υ) U^{α β} (υ) - U^{μ (α} (υ) U^{β) ν} (υ)) .

Coefficient extraction in (eq:C2) is a finite polynomial operation; it is the retained XII-C coefficient.

For a symmetric covariant jet $Σ$ , put

Q (H; Σ)^{μν} s (X, Y; Σ) = 2 C_{2}^{μν α β} (H) Σ_{α β}, = \frac{1}{2} (Q (X + Y; Σ) - Q (X; Σ) - Q (Y; Σ)) .

This polarization has no derivatives on $X, Y$ . The actual jet is always $Σ (y) = \partial\partial σ_{ε} (y)$ .

Displayed formulas in this passage

XVI-K@1#eq:C2 · line 712
$C_{2}^{μν α β} (H) = [υ^{2}] (1 + υ b_{1} + υ^{2} b_{2}) (U^{μν} (υ) U^{α β} (υ) - U^{μ (α} (υ) U^{β) ν} (υ)) .$
XVI-K@1#eq:source · line 727
$Q (H; Σ)^{μν} s (X, Y; Σ) = 2 C_{2}^{μν α β} (H) Σ_{α β}, = \frac{1}{2} (Q (X + Y; Σ) - Q (X; Σ) - Q (Y; Σ)) .$
unlabeled · line 708
$U (υ) = u_{0} + υ u_{1} + υ^{2} u_{2} .$