Manuscript XV-B@2 · XV-B.tex · lemma · 4d3f12e8

Lemma: Exact smooth multiplication formula

Finite normal jets and complete reconstruction-null tests › Smooth multiplication and the specified module · lines 843–871 · proof included or linked

XV-B@2#lem:leibniz, XV-B@2#eq:leibniz

For $c \in C^{\infty} (U)$ and $v \in D^{'} (X)$ ,

c E_{α} v = β \leq α \sum (- 1)^{∣ β ∣} (β α) E_{α - β} ((\partial_{q}^{β} c) (x, 0) v) .

Proof. Pair the left side with $ϕ$ . The Leibniz rule gives

(- 1)^{∣ α ∣} β \leq α \sum (β α) ⟨ v, (\partial_{q}^{β} c) (x, 0) \partial_{q}^{α - β} ϕ (x, 0) ⟩ .

The insertion on the right contributes the sign $(- 1)^{∣ β ∣} (- 1)^{∣ α - β ∣} = (- 1)^{∣ α ∣}$ , giving the formula.

Displayed formulas in this passage

XV-B@2#eq:leibniz · line 846
$c E_{α} v = β \leq α \sum (- 1)^{∣ β ∣} (β α) E_{α - β} ((\partial_{q}^{β} c) (x, 0) v) .$
unlabeled · line 859
$(- 1)^{∣ α ∣} β \leq α \sum (β α) ⟨ v, (\partial_{q}^{β} c) (x, 0) \partial_{q}^{α - β} ϕ (x, 0) ⟩ .$