Manuscript Em01-D@2 · Em01-D-v2.tex · proposition · d3b49151

Proposition: Finite source Hessian

Finite source and density specifications › The unintegrated cubic source · lines 570–619 · proof included or linked

Em01-D@2#prop:sourcehessian, Em01-D@2#eq:sourcehessian

The coefficients in

(J^{'} [G] h)^{L} (x) = ∣ α ∣ + ∣ β ∣ \leq 2 \sum q_{α β}^{L, I, J} (x) \partial^{α} G_{J} (x) \partial^{β} h_{I} (x)

are given by the finite formula

q_{α β}^{L, I, J} (x) = \frac{\partial ^{2} J _{x} ^{L} ( X )}{\partial X _{J, α} \partial X _{I, β}}_{X = 0} .

They satisfy

q_{α β}^{L, I, J} = q_{β α}^{L, J, I} .

They vanish for $∣ α ∣ + ∣ β ∣ > 2$ , and transform covariantly under a simultaneous change of basis and dual basis.

Proof. The coefficient $ℓ_{3}$ is cubic in field jets, and each monomial has total derivative weight two. Euler differentiation removes one field factor. The derivatives transferred by the Euler operator restore exactly the derivative weight removed from that factor; they may also differentiate $σ$ . The resulting source is quadratic and has total derivative weight two when switch derivatives are included. In particular the field derivative weight is at most two.

For any quadratic polynomial $J$ ,

D J (G) [h] = D^{2} J (0) [G, h] .

Expanding this identity in independent ordinary jet coordinates gives (eq:sourcehessian). Commutativity of polynomial differentiation gives slot symmetry and all repeated-variable factors. The underlying bilinear tensor map is unchanged by a consistent component transformation.

Displayed formulas in this passage

Em01-D@2#eq:sourcehessian · line 581
$q_{α β}^{L, I, J} (x) = \frac{\partial ^{2} J _{x} ^{L} ( X )}{\partial X _{J, α} \partial X _{I, β}}_{X = 0} .$
unlabeled · line 573
$(J^{'} [G] h)^{L} (x) = ∣ α ∣ + ∣ β ∣ \leq 2 \sum q_{α β}^{L, I, J} (x) \partial^{α} G_{J} (x) \partial^{β} h_{I} (x)$
unlabeled · line 591
$q_{α β}^{L, I, J} = q_{β α}^{L, J, I} .$
unlabeled · line 611
$D J (G) [h] = D^{2} J (0) [G, h] .$

Proposition: Finite source Hessian

Displayed formulas in this passage

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