Manuscript XX-F@1 · XX-F.tex · theorem · de0bd24f

Theorem: Complete normal form and null test for this module

Contact normal forms before Newton composition › The displayed finite contact module · lines 1105–1175 · proof included or linked

XX-F@1#thm:contactnormal, XX-F@1#eq:contactnormal, XX-F@1#eq:Newtoncontact

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The reconstructed source has the exact unique form

b^{ia} = δ (t - t^{'}) ∣ γ ∣ \leq m \sum j_{γ}^{ia} (t, z) \partial_{y}^{γ} δ^{3} (y - z) .

It is zero if and only if every $j_{γ}^{ia}$ is zero.

The original Newton-divergence image is

(N b)^{a} (t, x; t^{'}, z) = δ (t - t^{'}) ∣ γ ∣ \leq m \sum j_{γ}^{ia} (t, z) \partial_{x}^{γ} P_{i} (x - z) .

Thus it depends on the reconstructed source, not on an arbitrary choice of coefficient presentation.

Proof. Smooth multiplication of delta derivatives gives

c_{α} (y, z) \partial_{y}^{α} δ (y - z) = γ \leq α \sum (- 1)^{∣ α - γ ∣} (γ α) (\partial_{y}^{α - γ} c_{α}) (z, z) \partial_{y}^{γ} δ (y - z) .

Collect coefficients to obtain (eq:contactnormal).

To extract a given coefficient, test in the relative variable $y - z$ with

θ (y - z) \frac{( y - z ) ^{γ}}{γ !},

where $θ = 1$ near zero, and multiply by an arbitrary compact tangential test in $(t, z)$ . Every relative derivative at zero except $γ$ vanishes. This extracts $j_{γ}$ , up to its fixed sign, proving uniqueness and the null test.

For (eq:Newtoncontact), the relative variables $x - y$ and $y - z$ are independent coordinates. Their tensor-product distribution is defined before pushforward. The contact restricts the integration to $y = z$ , making the operation proper on compact output tests. Its derivative signs give

\int P_{i} (x - y) \partial_{y}^{γ} δ (y - z) d^{3} y = \partial_{x}^{γ} P_{i} (x - z) .

This proves the image formula.

Displayed formulas in this passage

XX-F@1#eq:contactnormal · line 1108
$b^{ia} = δ (t - t^{'}) ∣ γ ∣ \leq m \sum j_{γ}^{ia} (t, z) \partial_{y}^{γ} δ^{3} (y - z) .$
XX-F@1#eq:Newtoncontact · line 1120
$(N b)^{a} (t, x; t^{'}, z) = δ (t - t^{'}) ∣ γ ∣ \leq m \sum j_{γ}^{ia} (t, z) \partial_{x}^{γ} P_{i} (x - z) .$
unlabeled · line 1135
$c_{α} (y, z) \partial_{y}^{α} δ (y - z) = γ \leq α \sum (- 1)^{∣ α - γ ∣} (γ α) (\partial_{y}^{α - γ} c_{α}) (z, z) \partial_{y}^{γ} δ (y - z) .$
unlabeled · line 1149
$θ (y - z) \frac{( y - z ) ^{γ}}{γ !},$
unlabeled · line 1167
$\int P_{i} (x - y) \partial_{y}^{γ} δ (y - z) d^{3} y = \partial_{x}^{γ} P_{i} (x - z) .$

Theorem: Complete normal form and null test for this module

Displayed formulas in this passage

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