Manuscript VIII-B@3 · VIII-B-v3.tex · lemma · 3aa6baa1

Lemma: Descent and constant physical norms

The retuned polyhedral slicing family › Descent and the physical gradient identities · lines 695–732 · proof included or linked

VIII-B@3#lem:gradients, VIII-B@3#eq:oblique, VIII-B@3#eq:gradient-identities

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The functions in (eq:family) agree across sector rays and the seam. The oblique spatial metric is

G_{α} = (1 z_{α} z_{α} 1), G_{α}^{- 1} = \frac{1}{S _{α} ^{2}} (1 - z_{α} - z_{α} 1) .

For a covector $p$ , its physical squared norm is $p^{T} G_{α}^{- 1} p$ . In particular,

∥\nabla h_{μ} ∥^{2} ∥\nabla (b ξ) ∥^{2} = \frac{2 a ^{2}}{1 + z _{α}} = \frac{a ^{2}}{cos ^{2} ( α /2 )} = \frac{8}{9}, = ∥\nabla (b ζ) ∥^{2} = \frac{b ^{2}}{S _{α} ^{2}} = \frac{9}{25} .

Proof. On a sector ray, $ρ$ is its ray parameter, $h_{μ} = a ρ$ , and $k_{μ} = 0$ . Adjacent sectors and the seam-identified boundaries therefore agree.

The Gram matrix of the unit boundary rays is $G_{α}$ . Substitute $(a, a)^{T}$ , $(b, 0)^{T}$ , and $(0, b)^{T}$ into its inverse. Equation (eq:coefficients) gives the constant values.

Displayed formulas in this passage

VIII-B@3#eq:oblique · line 699
$G_{α} = (1 z_{α} z_{α} 1), G_{α}^{- 1} = \frac{1}{S _{α} ^{2}} (1 - z_{α} - z_{α} 1) .$
VIII-B@3#eq:gradient-identities · line 710
$∥\nabla h_{μ} ∥^{2} ∥\nabla (b ξ) ∥^{2} = \frac{2 a ^{2}}{1 + z _{α}} = \frac{a ^{2}}{cos ^{2} ( α /2 )} = \frac{8}{9}, = ∥\nabla (b ζ) ∥^{2} = \frac{b ^{2}}{S _{α} ^{2}} = \frac{9}{25} .$

Nearby in this section

section: Descent and the physical gradient identities