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

Internal lines and tensor formulas

The retained target and the available inherited formulas › Internal lines and tensor formulas · lines 1756–1821

XV-B@2#eq:L, XV-B@2#eq:K5a, XV-B@2#eq:K5z, XV-B@2#eq:K47, XV-B@2#eq:K6, XV-B@2#eq:Didentity

The supplied coordinate-response relation is

L^{μ, I} (s, a) = σ_{ε} (a) A^{λ μ, I} \partial_{a^{λ}} R (s - a),

with the source's fixed smooth switch and tensor map CheungXIVH,CheungXIVI. This recovers a definition omitted from the original XV-B excerpt. It does not evaluate the tensor map by identifying it with an unrelated candidate formula.

The two $K_{5}$ placements are

K_{5}^{[a], I, J} (a, b, z) K_{5}^{[z], I, J} (a, b, z) = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)] .

The grouped kernel and the local supported term are

K_{47}^{I, J} (a, b, z) = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)],

K_{6}^{I, J} (a, b, z) = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ^{(4)} (a - z)] .

The tensor $D$ is fixed by the source identity

a_{2} ˙^{α μ} [G; h] = D^{α μ, I, J} h_{I} G_{J} .

These formulas are inherited internal formulas CheungXIVH,CheungXIVI. Equation (eq:K47) retains the grouped coefficient; no canceled explicit scalar-Hessian term is reintroduced.

Displayed formulas in this passage

XV-B@2#eq:L · line 1759
$L^{μ, I} (s, a) = σ_{ε} (a) A^{λ μ, I} \partial_{a^{λ}} R (s - a),$
XV-B@2#eq:K5a · line 1771
$K_{5}^{[a], I, J} (a, b, z) K_{5}^{[z], I, J} (a, b, z) = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)] .$
XV-B@2#eq:K5z · line 1771
$K_{5}^{[a], I, J} (a, b, z) K_{5}^{[z], I, J} (a, b, z) = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)] .$
XV-B@2#eq:K47 · line 1790
$K_{47}^{I, J} (a, b, z) = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)],$
XV-B@2#eq:K6 · line 1803
$K_{6}^{I, J} (a, b, z) = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ^{(4)} (a - z)] .$
XV-B@2#eq:Didentity · line 1812
$a_{2} ˙^{α μ} [G; h] = D^{α μ, I, J} h_{I} G_{J} .$

Internal lines and tensor formulas

Displayed formulas in this passage

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