Manuscript IX-E-3@1 · IX-E-3.tex · section · 8c41417a

Derivative support and surface non-cancellation

Derivative support and surface non-cancellation · lines 436–452

IX-E-3@1#eq:metric-var, IX-E-3@1#eq:scalar-var

The two variations of the sharp insertion are

D_{h} B_{f} [h, ϕ] [h ˙] D_{ϕ} B_{f} [h, ϕ] [ϕ ˙] = - \int f (x) (L h ˙)^{μ} (x) \partial_{μ} ϕ (x) d^{4} x, = - \int f (x) (L h)^{μ} (x) \partial_{μ} ϕ ˙ (x) d^{4} x .

With $a_{μ} = - f \partial_{μ} ϕ$ , the metric derivative is represented by the full $L^{*} a$ in (eq:adjoint). The scalar variation samples $L h$ on $F$ .

Displayed formulas in this passage

IX-E-3@1#eq:metric-var · line 439
$D_{h} B_{f} [h, ϕ] [h ˙] D_{ϕ} B_{f} [h, ϕ] [ϕ ˙] = - \int f (x) (L h ˙)^{μ} (x) \partial_{μ} ϕ (x) d^{4} x, = - \int f (x) (L h)^{μ} (x) \partial_{μ} ϕ ˙ (x) d^{4} x .$
IX-E-3@1#eq:scalar-var · line 439
$D_{h} B_{f} [h, ϕ] [h ˙] D_{ϕ} B_{f} [h, ϕ] [ϕ ˙] = - \int f (x) (L h ˙)^{μ} (x) \partial_{μ} ϕ (x) d^{4} x, = - \int f (x) (L h)^{μ} (x) \partial_{μ} ϕ ˙ (x) d^{4} x .$

Derivative support and surface non-cancellation

Displayed formulas in this passage

Nearby in this section