Manuscript X-C@3 · X-C-v3.tex · section · 2260f5f8

Scalar and density pullbacks

C4: completed test, three-term variation, and cancellation › Scalar and density pullbacks · lines 1004–1016

For arbitrary compact $ξ, ζ$ in the open slab, put

F^{*} = 1 + κ X_{ξ} + κ^{2} (X_{ζ} + \frac{1}{2} X_{ξ}^{2}), F^{♯} = 1 + κ D_{ξ} + κ^{2} (D_{ζ} + \frac{1}{2} D_{ξ}^{2}) .

The first acts on scalars; the second is density pullback, including the Jacobian.

Displayed formulas in this passage

unlabeled · line 1007
$F^{*} = 1 + κ X_{ξ} + κ^{2} (X_{ζ} + \frac{1}{2} X_{ξ}^{2}), F^{♯} = 1 + κ D_{ξ} + κ^{2} (D_{ζ} + \frac{1}{2} D_{ξ}^{2}) .$