Manuscript XX-E@1 · XX-E.tex · theorem · cf7a8f40

Theorem: Complete spatial Newton image of a contact

Explicit Newton-contact images and comparison Ward images › The operation that can be constructed independently · lines 943–1004 · proof included or linked

XX-E@1#thm:contactimage, XX-E@1#eq:contactimage

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The image is the joint distribution

J^{a} (t, x; t^{'}, z) = δ (t - t^{'}) ∣ α ∣ \leq m \sum β \leq α \sum (- 1)^{∣ β ∣} (β α) \times (\partial_{y}^{β} c_{α}^{ia}) (t, z, z) \partial_{x}^{α - β} P_{i} (x - z) .

It is well defined across $x = z$ as a distribution, not as a pointwise value. For compact $z$ support, each displayed term has a spatial tail $O (r^{- 2 - ∣ α - β ∣})$ away from its singular support.

Proof. The variables

(x, y, z) ⟼ (x - y, y - z, z)

form a diffeomorphism. Thus the product of the Newton factor and the contact factor before integration is a tensor-product pullback in independent relative variables. Smooth coefficient multiplication is legitimate.

The exact distributional identity is

c (y) \partial_{y}^{α} δ^{3} (y - z) = β \leq α \sum (- 1)^{∣ β ∣} (β α) (\partial_{y}^{β} c) (z) \partial_{y}^{α - β} δ^{3} (y - z) .

Pairing a derivative of the delta with $P_{i} (x - y)$ gives $\partial_{x}^{α - β} P_{i} (x - z)$ . The two differentiation signs cancel. This proves (eq:contactimage).

The explicit formula defines the resulting joint distribution. Away from its spatial diagonal, the stated tail follows by differentiating the homogeneous dipole kernel.

Displayed formulas in this passage

XX-E@1#eq:contactimage · line 946
$J^{a} (t, x; t^{'}, z) = δ (t - t^{'}) ∣ α ∣ \leq m \sum β \leq α \sum (- 1)^{∣ β ∣} (β α) \times (\partial_{y}^{β} c_{α}^{ia}) (t, z, z) \partial_{x}^{α - β} P_{i} (x - z) .$
unlabeled · line 972
$(x, y, z) ⟼ (x - y, y - z, z)$
unlabeled · line 984
$c (y) \partial_{y}^{α} δ^{3} (y - z) = β \leq α \sum (- 1)^{∣ β ∣} (β α) (\partial_{y}^{β} c) (z) \partial_{y}^{α - β} δ^{3} (y - z) .$

Theorem: Complete spatial Newton image of a contact

Displayed formulas in this passage

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