PhyslibAlpha.SpaceAndTime.Space.Surfaces.Ring

The function `Space.ring` represents the embedding of the unit circle $S^1$ (the unit sphere in $\mathbb{R}^2$) into the 3-dimensional Euclidean space $\mathbb{R}^3$. Specifically, for a point $x \in S^1 \subset \mathbb{R}^2$, the map identifies $x$ as an element of $\mathbb{R}^2$ and then embeds it into $\mathbb{R}^3$ by setting the third coordinate (at index 2) to $0$ and assigning the remaining coordinates to those of $x$.

The map $\text{ring} : S^1 \to \text{Space } 3$ is equal to the composition $(\text{slice}_2)^{-1} \circ f$, where $f : S^1 \to \mathbb{R} \times \text{Space } 2$ is the map $x \mapsto (0, \text{sphericalShell}_2(x))$. Here, $\text{sphericalShell}_2$ is the canonical inclusion of the unit circle into $\text{Space } 2$, and $\text{slice}_2 : \text{Space } 3 \cong \mathbb{R} \times \text{Space } 2$ is the continuous linear equivalence that extracts the third coordinate (at index 2).

The embedding $\text{ring}: S^1 \to \mathbb{R}^3$, which maps the unit circle $S^1$ (the unit sphere in $\mathbb{R}^2$) into the 3-dimensional Euclidean space $\mathbb{R}^3$, is injective.

The embedding map $\text{ring} : S^1 \to \mathbb{R}^3$, which maps the unit circle $S^1$ (the unit sphere in $\mathbb{R}^2$) into 3-dimensional Euclidean space $\mathbb{R}^3$, is continuous.

Let $S^1 \subset \mathbb{R}^2$ be the unit circle, defined as the sphere of radius $1$ centered at the origin in $2$-dimensional Euclidean space. The map $\text{ring} : S^1 \to \mathbb{R}^3$, which embeds this circle into $3$-dimensional Euclidean space by identifying $S^1$ with its image in the $xy$-plane (setting the third coordinate to $0$), is a measurable embedding.

The measure `Space.ringMeasure` is defined on the 3-dimensional Euclidean space $\mathbb{R}^3$ as the pushforward of the uniform surface measure (arc length measure) on the unit circle $S^1 \subset \mathbb{R}^2$ via the embedding map $f: S^1 \to \mathbb{R}^3$. The embedding $f$ maps a point $(x, y)$ on the unit circle to the point $(x, y, 0)$ in $\mathbb{R}^3$. This measure corresponds to integration along the unit ring situated in the $xy$-plane of 3D space.

The measure $\mu_{\text{ring}}$ on $\mathbb{R}^3$ (3-dimensional Euclidean space), defined as the pushforward of the uniform arc length measure on the unit circle $S^1 \subset \mathbb{R}^2$ into the $xy$-plane, has temperate growth.

Let $\mu_{\text{ring}}$ be the measure on the 3-dimensional Euclidean space $\mathbb{R}^3$ defined as the pushforward of the uniform arc length measure on the unit circle $S^1 \subset \mathbb{R}^2$ via the embedding $(x, y) \mapsto (x, y, 0)$. Let $\lambda$ be the Lebesgue volume measure on a finite-dimensional real inner product space. Then the product measure $\mu_{\text{ring}} \times \lambda$ has temperate growth.

The measure on the unit ring in $\mathbb{R}^3$ (defined as the pushforward of the uniform surface measure on the unit circle $S^1 \subset \mathbb{R}^2$ via the embedding $(x, y) \mapsto (x, y, 0)$) is $s$-finite. A measure is $s$-finite if it can be expressed as a countable sum of finite measures.

The measure $\mu_{\text{ring}}$ on $\mathbb{R}^3$ (defined as the pushforward of the uniform surface measure on the unit circle $S^1$ in the $xy$-plane) is a finite measure. That is, the total measure of the space under $\mu_{\text{ring}}$ is finite.

Let $f : \text{Space } 3 \to \mathbb{R}$ be a real-valued function. If the restriction of $f$ to the unit ring (the composition $f \circ \text{ring}$, where $\text{ring} : S^1 \to \text{Space } 3$ embeds the unit circle into the $xy$-plane) is continuous, then $f$ is integrable with respect to the ring measure $\mu_{\text{ring}}$.

Let $\mu_{\text{ring}}$ be the ring measure on the 3-dimensional Euclidean space $\mathbb{R}^3$, which is defined as the pushforward of the uniform arc length measure on the unit circle $S^1 \subset \mathbb{R}^2$ via the embedding $\text{ring}: S^1 \to \mathbb{R}^3$ (where $(x, y) \mapsto (x, y, 0)$). For any function $f: \mathbb{R}^3 \to \mathbb{R}^n$, if the composition $f \circ \text{ring}$ is continuous, then $f$ is integrable with respect to the measure $\mu_{\text{ring}}$.

Let $\mu_{\text{ring}}$ be the measure on $\mathbb{R}^3$ corresponding to the unit ring in the $xy$-plane (defined as the pushforward of the uniform measure on the unit circle $S^1$ via the map $(x, y) \mapsto (x, y, 0)$), and let $\lambda$ be the standard Lebesgue volume measure on $\mathbb{R}^3$. For the transformation $\Phi: \mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}^3 \times \mathbb{R}^3$ defined by $\Phi(x, y) = (x, y + x)$, the pushforward of the product measure $\mu_{\text{ring}} \otimes \lambda$ under $\Phi$ is equal to $\mu_{\text{ring}} \otimes \lambda$.

The total measure of the 3-dimensional Euclidean space $\mathbb{R}^3$ under the ring measure (denoted as `ringMeasure`) is equal to $2\pi$. This corresponds to the total length (circumference) of the unit circle embedded in the $xy$-plane of $\mathbb{R}^3$.

The tempered distribution $\text{ringDist}$ on $\mathbb{R}^3$ is defined as the continuous linear map that sends a Schwartz function $f \in \mathcal{S}(\mathbb{R}^3, \mathbb{R})$ to its integral with respect to the ring measure $\mu_{\text{ring}}$, i.e., $$\langle \text{ringDist}, f \rangle = \int_{\mathbb{R}^3} f(x) \, d\mu_{\text{ring}}(x).$$ The measure $\mu_{\text{ring}}$ corresponds to the arc length measure on the unit circle embedded in the $xy$-plane of $\mathbb{R}^3$.

For any Schwartz function $f \in \mathcal{S}(\mathbb{R}^3, \mathbb{R})$, the action of the tempered distribution $\text{ringDist}$ on $f$ is equal to the integral of $f$ with respect to the ring measure $\mu_{\text{ring}}$ on $\mathbb{R}^3$: $$\langle \text{ringDist}, f \rangle = \int_{\mathbb{R}^3} f(x) \, d\mu_{\text{ring}}(x)$$ where $\mu_{\text{ring}}$ is the arc length measure on the unit circle in the $xy$-plane.

For any Schwartz function $f \in \mathcal{S}(\mathbb{R}^3, \mathbb{R})$, the action of the ring tempered distribution $\text{ringDist}$ on $f$ is equal to the integral of the Dirac delta distribution at $z$ (denoted $\delta_z$) evaluated at $f$, with respect to the ring measure $\mu_{\text{ring}}$: $$\langle \text{ringDist}, f \rangle = \int_{\mathbb{R}^3} \delta_z(f) \, d\mu_{\text{ring}}(z).$$ Here, $\mu_{\text{ring}}$ is the arc length measure on the unit circle situated in the $xy$-plane of $\mathbb{R}^3$.