Manuscript X-A-ST@1 · X-A-ST.tex · section · d9ccbde1

First derivatives through second order

Support, continuity, and the smooth-kernel statement › First derivatives through second order · lines 1722–1776

X-A-ST@1#eq:firstderivatives, X-A-ST@1#eq:dv, X-A-ST@1#eq:O2h

Write

B_{w}^{μ} = w^{ν} \partial_{ν} w^{μ} - v^{μ}, a_{μ} = f \partial_{μ} ϕ_{0} .

At a fixed smooth configuration, some derivatives are immediately

(O^{(1)})_{h}^{'} (O^{(1)})_{ϕ_{0}}^{'} (O^{(2)})_{ϕ_{0}}^{'} (O^{(2)})_{ϕ_{1}}^{'} = - L^{*} a, = \partial_{μ} (f w^{μ}), = - \partial_{μ} (f B_{w}^{μ}) + \frac{1}{2} \partial_{μ} \partial_{ν} (f w^{μ} w^{ν}), = \partial_{μ} (f w^{μ}), (O^{(2)})_{k}^{'} (O^{(1)})_{ϕ_{1}}^{'} (O^{(2)})_{ϕ_{2}}^{'} = - L^{*} a, = f, = f .

For completeness, if $H$ is a symmetric metric direction,

D_{h} v [H]^{μ} = - E_{0} \partial_{α} {σ [a_{1}^{α β} [H] \partial_{β} w^{μ} + a_{1}^{α β} [h] \partial_{β} (L H)^{μ} + 2 a_{2}^{α μ} [h, H]]} .

Define the compact covector

r_{λ} = - f \partial_{λ} ϕ_{1} + f (\partial_{λ} w^{μ}) \partial_{μ} ϕ_{0} - \partial_{ν} (f w^{ν} \partial_{λ} ϕ_{0}) + f w^{ν} \partial_{λ} \partial_{ν} ϕ_{0} .

Then the remaining derivative is exactly

D_{h} O^{(2)} [H] = ⟨ r, L H ⟩ - ⟨ a, D_{h} v [H] ⟩ .

Displayed formulas in this passage

X-A-ST@1#eq:firstderivatives · line 1731
$(O^{(1)})_{h}^{'} (O^{(1)})_{ϕ_{0}}^{'} (O^{(2)})_{ϕ_{0}}^{'} (O^{(2)})_{ϕ_{1}}^{'} = - L^{*} a, = \partial_{μ} (f w^{μ}), = - \partial_{μ} (f B_{w}^{μ}) + \frac{1}{2} \partial_{μ} \partial_{ν} (f w^{μ} w^{ν}), = \partial_{μ} (f w^{μ}), (O^{(2)})_{k}^{'} (O^{(1)})_{ϕ_{1}}^{'} (O^{(2)})_{ϕ_{2}}^{'} = - L^{*} a, = f, = f .$
X-A-ST@1#eq:dv · line 1752
$D_{h} v [H]^{μ} = - E_{0} \partial_{α} {σ [a_{1}^{α β} [H] \partial_{β} w^{μ} + a_{1}^{α β} [h] \partial_{β} (L H)^{μ} + 2 a_{2}^{α μ} [h, H]]} .$
X-A-ST@1#eq:O2h · line 1771
$D_{h} O^{(2)} [H] = ⟨ r, L H ⟩ - ⟨ a, D_{h} v [H] ⟩ .$
unlabeled · line 1725
$B_{w}^{μ} = w^{ν} \partial_{ν} w^{μ} - v^{μ}, a_{μ} = f \partial_{μ} ϕ_{0} .$
unlabeled · line 1763
$r_{λ} = - f \partial_{λ} ϕ_{1} + f (\partial_{λ} w^{μ}) \partial_{μ} ϕ_{0} - \partial_{ν} (f w^{ν} \partial_{λ} ϕ_{0}) + f w^{ν} \partial_{λ} \partial_{ν} ϕ_{0} .$