PhyslibAlpha.SpaceAndTime.Space.Surfaces.SolidSphere

For a given dimension $d$, let $\text{Space } d$ be the $d$-dimensional Euclidean space with origin $0$. The closed unit ball $\bar{B}(0, 1)$ is the set of points $x \in \text{Space } d$ such that $\text{dist}(x, 0) \leq 1$. The function `Space.solidSphere` is the canonical inclusion map $\iota: \bar{B}(0, 1) \hookrightarrow \text{Space } d$, which maps each point $x$ in the unit ball to itself in the ambient space.

For any natural number $d$, the canonical inclusion map $\iota: \bar{B}(0, 1) \hookrightarrow \text{Space } d$, which maps each point in the closed unit ball of the $d$-dimensional Euclidean space to itself in the ambient space, is injective.

For any dimension $d \in \mathbb{N}$, let $\text{Space } d$ be the $d$-dimensional Euclidean space and $\bar{B}(0, 1) \subset \text{Space } d$ be the closed unit ball centered at the origin. The inclusion map $\iota: \bar{B}(0, 1) \to \text{Space } d$ (denoted as `solidSphere d`) is continuous.

For any dimension $d \in \mathbb{N}$, the inclusion map $\iota: \bar{B}(0, 1) \hookrightarrow \text{Space } d$, which maps the closed unit ball centered at the origin to the $d$-dimensional Euclidean space $\text{Space } d$, is a measurable embedding. Both the ball and the ambient space are equipped with their respective Borel $\sigma$-algebras.

For any dimension $d \in \mathbb{N}$, let $\text{Space } d$ be the $d$-dimensional Euclidean space. For any point $x$ in the closed unit ball $\bar{B}(0, 1)$ centered at the origin, the Euclidean norm of $x$ satisfies: $$\|x\| \leq 1$$

For a $d$-dimensional Euclidean space $\text{Space } d$, the measure $\mu$ is defined as the restriction of the ambient Lebesgue volume measure $\lambda$ to the closed unit ball $\bar{B}(0, 1)$, which is the set of points $x \in \text{Space } d$ satisfying $\|x\| \leq 1$.

For any dimension $d \in \mathbb{N}$, the measure on the $d$-dimensional Euclidean space $\text{Space } d$ defined as the restriction of the ambient Lebesgue volume measure to the closed unit ball $\bar{B}(0, 1) = \{x \in \text{Space } d \mid \|x\| \leq 1\}$ is a finite measure. That is, the total measure of the space under this measure is finite.

For any dimension $d \in \mathbb{N}$, let $\mu$ be the measure on the $d$-dimensional Euclidean space $\text{Space } d$ defined as the restriction of the Lebesgue volume measure to the closed unit ball $\bar{B}(0, 1) = \{x \in \text{Space } d \mid \|x\| \leq 1\}$. Then $\mu$ has temperate growth.

For any dimension $d \in \mathbb{N}$, the tempered distribution $\text{solidSphereDist}(d)$ is the continuous linear map from the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{R})$ to $\mathbb{R}$ that maps a test function $f$ to its integral over the closed unit ball $\bar{B}(0, 1) = \{x \in \text{Space } d \mid \|x\| \leq 1\}$ with respect to the ambient Lebesgue volume measure.

For any dimension $d \in \mathbb{N}$ and any test function $f$ in the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{R})$, the value of the tempered distribution $\text{solidSphereDist}(d)$ applied to $f$ is equal to the integral of $f$ over the space with respect to the measure $\text{solidSphereMeasure}(d)$: $$\text{solidSphereDist}(d)(f) = \int_{\text{Space } d} f(x) \, d(\text{solidSphereMeasure } d)(x)$$ where $\text{solidSphereMeasure}(d)$ is the Lebesgue volume measure restricted to the closed unit ball $\bar{B}(0, 1) \subset \text{Space } d$.

For any dimension $d \in \mathbb{N}$ and any test function $f$ in the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{R})$, the tempered distribution $\text{solidSphereDist}(d)$ applied to $f$ is equal to the integral of $f$ over the closed unit ball $\bar{B}(0, 1) \subset \text{Space } d$ with respect to the ambient volume: $$\text{solidSphereDist}(d)(f) = \int_{\|x\| \leq 1} f(x) \, dx$$ where $\bar{B}(0, 1) = \{x \in \text{Space } d \mid \|x\| \leq 1\}$.

For any natural number $d$, the volume (Lebesgue measure) of the closed unit ball centered at the origin with radius $1$ in $\text{Space } d$ is strictly positive.

For any natural number $d$, the total measure of the $d$-dimensional space $\text{Space } d$ under the solid sphere measure is strictly positive. Here, the solid sphere measure is defined as the restriction of the ambient Lebesgue volume measure to the closed unit ball $\bar{B}(0, 1) = \{x \in \text{Space } d \mid \|x\| \leq 1\}$.