Manuscript Em01-C@3 · Em01-C-v3.tex · proposition · f0953702

Proposition: Finite source-coefficient rule

Finite source specification and implementation inventory › Jet-polynomial extraction · lines 2142–2181 · proof included or linked

Em01-C@3#prop:source-rule, Em01-C@3#eq:jet-Hessian

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The source coefficients are

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 vanish for $∣ α ∣ + ∣ β ∣ > 2$ . Repeated-variable factors and the factors from differentiating a quadratic source are included automatically.

The rule is covariant under a simultaneous transformation of the basis, dual basis, input components, and output conversion.

Proof. The source is homogeneous quadratic in field jets. The $J_{3}$ part contains only derivative-weight patterns $X_{0} X_{2}$ and $X_{1} X_{1}$ . The $C_{2}$ part contains $X_{0} X_{0}$ ; the $M_{2}$ part contains $X_{0} X_{1}$ . Thus total field-derivative weight is at most two.

For a quadratic polynomial,

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

Expanding the right-hand side in independent jet coordinates gives (eq:jet-Hessian) and (eq:source-polynomial). Ordinary differentiation supplies a factor two when the same quadratic variable occurs twice. A basis change transforms the Hessian and its two arguments consistently, leaving the underlying bilinear map unchanged.

Displayed formulas in this passage

Em01-C@3#eq:jet-Hessian · line 2145
$q_{α β}^{L, I, J} (x) = \frac{\partial ^{2} J _{x} ^{L} ( X )}{\partial X _{J, α} \partial X _{I, β}}_{X = 0} .$
unlabeled · line 2172
$D J_{x} (G) [h] = D^{2} J_{x} (0) [G, h] .$

Proposition: Finite source-coefficient rule

Displayed formulas in this passage

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