PhyslibAlpha.SpaceAndTime.Space.Surfaces.SphericalShell

For any dimension $d \in \mathbb{N}$, the map $\text{sphericalShell} : S^{d-1} \to \text{Space } d$ is the canonical inclusion of the unit sphere into the $d \text{-dimensional}$ Euclidean space $\text{Space } d$. The domain $S^{d-1}$ is defined as the set of points $x \in \text{Space } d$ such that $\text{dist}(x, 0) = 1$. The function maps each element $x$ of the sphere to itself as an element of the ambient space.

For any dimension $d \in \mathbb{N}$, the canonical inclusion map $\text{sphericalShell}_d: S^{d-1} \to \text{Space } d$ from the unit sphere $S^{d-1} = \{ x \in \text{Space } d \mid \text{dist}(x, 0) = 1 \}$ to the $d$-dimensional Euclidean space $\text{Space } d$ is injective.

For any dimension $d \in \mathbb{N}$, the canonical inclusion map $\text{sphericalShell}_d : S^{d-1} \to \text{Space } d$, which maps each point of the unit sphere $S^{d-1} = \{x \in \text{Space } d \mid \text{dist}(x, 0) = 1\}$ to itself in the $d$-dimensional Euclidean space $\text{Space } d$, is continuous.

For any dimension $d \in \mathbb{N}$, the canonical inclusion map $i: S^{d-1} \hookrightarrow \text{Space } d$, which maps each point on the unit sphere $S^{d-1} = \{x \in \text{Space } d \mid \text{dist}(x, 0) = 1\}$ to itself in the ambient $d$-dimensional Euclidean space $\text{Space } d$, is a measurable embedding.

For any dimension $d \in \mathbb{N}$, and for any point $x$ on the unit sphere $S^{d-1}$ (the sphere centered at the origin $0$ with radius $1$ in the $d$-dimensional Euclidean space $\text{Space } d$), the Euclidean norm of $x$ as an element of the ambient space is $1$. That is, $$\|x\| = 1$$ where $x \in \{p \in \text{Space } d \mid \text{dist}(p, 0) = 1\}$.

For any dimension $d \in \mathbb{N}$, the spherical shell measure is the measure on $\text{Space } d$ defined as the pushforward of the standard surface measure on the unit sphere $S^{d-1} = \{x \in \text{Space } d \mid \|x\| = 1\}$ via the inclusion map $i : S^{d-1} \hookrightarrow \text{Space } d$. This measure corresponds to integration over the spherical shell of radius 1 in the $d$-dimensional Euclidean space.

For any dimension $d \in \mathbb{N}$, the spherical shell measure on $\text{Space } d$ has temperate growth. The spherical shell measure is the measure on $\text{Space } d$ defined as the pushforward of the standard surface measure on the unit sphere $S^{d-1} = \{x \in \text{Space } d \mid \|x\| = 1\}$ via the inclusion map $i : S^{d-1} \hookrightarrow \text{Space } d$. A measure has temperate growth if it defines a tempered distribution, typically meaning it is bounded by a polynomial.

For any dimension $d \in \mathbb{N}$, the spherical shell distribution is the tempered distribution on $\text{Space } d$ (isomorphic to $\mathbb{R}^d$) defined by integration against the spherical shell measure. This distribution maps a Schwartz test function $f \in \mathcal{S}(\mathbb{R}^d, \mathbb{R})$ to its integral over the unit sphere $S^{d-1} = \{x \in \mathbb{R}^d \mid \|x\| = 1\}$: $$\langle \text{sphericalShellDist}(d), f \rangle = \int_{\text{Space } d} f(x) \, d\mu_{S^{d-1}}(x)$$ where $d\mu_{S^{d-1}}$ is the pushforward of the standard surface measure on the unit sphere to the ambient space.

For any dimension $d \in \mathbb{N}$ and any Schwartz function $f \in \mathcal{S}(\text{Space } d, \mathbb{R})$, the evaluation of the spherical shell distribution on $f$ is equal to the integral of $f$ with respect to the spherical shell measure on $\text{Space } d$: $$\langle \text{sphericalShellDist}(d), f \rangle = \int_{\text{Space } d} f(x) \, d\mu_{S^{d-1}}(x)$$ where $\mu_{S^{d-1}}$ denotes the `sphericalShellMeasure` in dimension $d$.

For any dimension $d \in \mathbb{N}$ and any Schwartz test function $f \in \mathcal{S}(\text{Space } d, \mathbb{R})$, the spherical shell distribution $\text{sphericalShellDist}(d)$ applied to $f$ is equal to the integral of $f$ over the unit sphere $S^{d-1}$ with respect to the spherical surface measure induced by the Lebesgue volume: $$\langle \text{sphericalShellDist}(d), f \rangle = \int_{S^{d-1}} f(x) \, d\sigma(x)$$ where $S^{d-1} = \{x \in \text{Space } d \mid \|x\| = 1\}$ and $\sigma$ denotes the measure on the sphere obtained from the ambient volume measure.