PhyslibAlpha.SpaceAndTime.Space.Surfaces.SolidCylinder

The map $\text{solidCylinder} : \text{Space } 2 \times \mathbb{R} \to \text{Space } 3$ embeds a cross-sectional disk from $\text{Space } 2$ into $\text{Space } 3$ by extruding it along the third coordinate axis. For an input pair $(p, z)$ where $p \in \text{Space } 2$ and $z \in \mathbb{R}$, the function returns the point in $\text{Space } 3$ obtained by setting the first two coordinates to those of $p$ and the third coordinate to $z$. This is defined using the inverse of the coordinate slice map at index 2, $(\text{slice } 2)^{-1} : \mathbb{R} \times \text{Space } 2 \to \text{Space } 3$.

The map $\text{solidCylinder} : \text{Space } 2 \times \mathbb{R} \to \text{Space } 3$ is equal to the composition $(\text{slice } 2)^{-1} \circ \sigma$, where $\sigma : \text{Space } 2 \times \mathbb{R} \to \mathbb{R} \times \text{Space } 2$ is the swapping function $(p, z) \mapsto (z, p)$, and $(\text{slice } 2)^{-1}$ is the inverse of the continuous linear equivalence $\text{slice } 2 : \text{Space } 3 \cong \mathbb{R} \times \text{Space } 2$ that extracts the third coordinate.

The map $\text{solidCylinder} : \text{Space } 2 \times \mathbb{R} \to \text{Space } 3$ is injective.

The embedding map $\text{solidCylinder} : \text{Space } 2 \times \mathbb{R} \to \text{Space } 3$, which maps a point $p \in \text{Space } 2$ and a scalar $z \in \mathbb{R}$ to a point in $\text{Space } 3$ by extruding the 2D coordinate along the third axis, is a continuous function.

The function $\text{solidCylinder} : \text{Space } 2 \times \mathbb{R} \to \text{Space } 3$, which maps a 2D point and a real coordinate to a 3D point by extruding the point along the third axis, is a measurable embedding. This means the map is injective, measurable, and maps measurable sets in $\text{Space } 2 \times \mathbb{R}$ to measurable sets in $\text{Space } 3$ with respect to their Borel $\sigma$-algebras.

For any point $x = (p, z)$ in the product space $\text{Space } 2 \times \mathbb{R}$, where $p \in \text{Space } 2$ and $z \in \mathbb{R}$, the Euclidean norm of its image under the solid cylinder embedding satisfies: $$\|\text{solidCylinder}(x)\| = \sqrt{z^2 + \|p\|^2}$$ Here, $\|p\|$ denotes the Euclidean norm in $\text{Space } 2$, and $z^2$ is the square of the real coordinate.

The measure on $\text{Space } 3$ is defined as the pushforward of the product measure $\mu_{\bar{B}} \times \lambda$ under the embedding map $f: \text{Space } 2 \times \mathbb{R} \to \text{Space } 3$. Here, $\mu_{\bar{B}}$ is the measure of the solid unit disk in $\text{Space } 2$ (the restriction of the 2D Lebesgue volume to the closed unit ball $\bar{B}(0, 1)$), $\lambda$ is the standard Lebesgue volume measure on the real line $\mathbb{R}$, and $f$ is the function that extrudes the 2D disk along the third coordinate axis.

The measure on $\text{Space } 3$ associated with the solid cylinder, denoted as $\mu_{\text{solidCylinder}}$ (which is the extrusion of the closed unit disk in $\text{Space } 2$ along the third coordinate), has temperate growth. This property implies that the measure can be used to define a tempered distribution, as its growth at infinity is bounded by a polynomial.

The measure $\mu$ on $\text{Space } 3$ representing the solid cylinder is $s$-finite. This measure is defined as the extrusion of the 2D Lebesgue measure restricted to the closed unit disk in $\text{Space } 2$ along the third coordinate axis.

The tempered distribution on $\text{Space } 3$ (isomorphic to $\mathbb{R}^3$) that maps a test function $f$ in the Schwartz space $\mathcal{S}(\text{Space } 3, \mathbb{R})$ to its integral with respect to the solid cylinder measure $\mu_{\text{solidCylinder}}$. It is defined as: $$f \mapsto \int_{\text{Space } 3} f(x) \, d\mu_{\text{solidCylinder}}(x)$$ where $\mu_{\text{solidCylinder}}$ is the measure representing the solid cylinder formed by the extrusion of the closed unit disk in $\text{Space } 2$ along the third coordinate axis.

For any test function $f$ in the Schwartz space $\mathcal{S}(\text{Space } 3, \mathbb{R})$, the application of the solid cylinder distribution, denoted by $\text{solidCylinderDist}$, to $f$ is equal to the integral of $f$ over $\text{Space } 3$ with respect to the solid cylinder measure $\mu_{\text{solidCylinder}}$: $$\text{solidCylinderDist}(f) = \int_{\text{Space } 3} f(x) \, d\mu_{\text{solidCylinder}}(x)$$ where $\text{Space } 3$ is the 3-dimensional Euclidean space and $\mu_{\text{solidCylinder}}$ is the measure representing the solid cylinder formed by the extrusion of the closed unit disk in $\text{Space } 2$ along the third coordinate axis.

For any test function $f$ in the Schwartz space $\mathcal{S}(\text{Space } 3, \mathbb{R})$, the action of the distribution associated with the solid cylinder on $f$ is given by the integral: $$\text{solidCylinderDist}(f) = \int_{\text{Space } 2 \times \mathbb{R}} f(\text{solidCylinder}(p, z)) \, d(\mu_{B^2} \times \lambda)(p, z)$$ where $\mu_{B^2}$ is the measure of the solid unit disk in $\text{Space } 2$ (the Lebesgue measure restricted to the unit ball), $\lambda$ is the standard Lebesgue measure on $\mathbb{R}$, and $\text{solidCylinder} : \text{Space } 2 \times \mathbb{R} \to \text{Space } 3$ is the map that extrudes a 2D point $p$ along the coordinate $z$.

The measure of the universal set in $\text{Space } 3$ with respect to the solid cylinder measure $\mu_{\text{cyl}}$ is strictly positive, which is expressed as $0 < \mu_{\text{cyl}}(\mathcal{U})$, where $\mathcal{U}$ is the set of all points in $\text{Space } 3$.