Manuscript VI-B@1 · VI-B-v1.tex · section · 33798723

The local analytical domain

The prescribed cone and the supported presentation › The local analytical domain · lines 306–329

VI-B@1#eq:domain

Choose a regular spatial developing chart

χ : U ⟶ V \subset R^{3}

with $V$ convex and containing a closed Euclidean ball $B_{R}$ in its interior. Set

t = c (T - T_{0}), x = χ (r, ϕ, z), η = \diag (- 1, 1, 1, 1) .

Fix $0 < b < R$ , and define the shrinking Cauchy domain

D_{R, b} = {(t, x) : 0 \leq t \leq b, ∣ x ∣ < R - t}, Σ_{0} = {0} \times B_{R} .

All coefficient fields are real and smooth on neighborhoods of the portions of this domain where they are used.

Displayed formulas in this passage

VI-B@1#eq:domain · line 320
$D_{R, b} = {(t, x) : 0 \leq t \leq b, ∣ x ∣ < R - t}, Σ_{0} = {0} \times B_{R} .$
unlabeled · line 309
$χ : U ⟶ V \subset R^{3}$
unlabeled · line 315
$t = c (T - T_{0}), x = χ (r, ϕ, z), η = \diag (- 1, 1, 1, 1) .$