Manuscript XV-C@1 · XV-C.tex · section · 5afd124d

Available internal expressions

Frozen branch and placement ledger › Available internal expressions · lines 312–373

XV-C@1#eq:K5a, XV-C@1#eq:K5z, XV-C@1#eq:K47, XV-C@1#eq:K6

The two supplied $K_{5}$ 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 supplied XIV-H proposition expresses them as finite differential operations on the tensor-product pullbacks associated with

R (b - a) R (a - z), R (b - z) R (z - a)

Hkernel. XV-B independently checks the existence mechanism for these internal products XVB.

The other displayed source expressions 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)]

and

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)] .

Their complete inherited ingredients are not defined by these formulas alone Iamplitude.

In particular, the existence of internal products in independent variables does not establish their product with additional singular cross-lines in the attachment variables.

Displayed formulas in this passage

XV-C@1#eq:K5a · line 315
$K_{5}^{[a], I, J} (a, b, z) K_{5}^{[z], I, J} (a, b, z) = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)] .$
XV-C@1#eq:K5z · line 315
$K_{5}^{[a], I, J} (a, b, z) K_{5}^{[z], I, J} (a, b, z) = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)] .$
XV-C@1#eq:K47 · line 344
$K_{47}^{I, J} (a, b, z) = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)]$
XV-C@1#eq:K6 · line 358
$K_{6}^{I, J} (a, b, z) = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ^{(4)} (a - z)] .$
unlabeled · line 337
$R (b - a) R (a - z), R (b - z) R (z - a)$