PhyslibAlpha.SpaceAndTime.Space.Surfaces.HalfPlane

The half-plane domain is the set of points in the 2-dimensional space $\mathbb{R}^2$ whose second coordinate is non-negative, defined as $\{x \in \mathbb{R}^2 \mid 0 \le x_1\}$, where $x_1$ denotes the second component of the vector $x$.

The function $\text{halfPlane} : \text{Space } 2 \to \text{Space } 3$ maps a 2-dimensional vector $x$ to a 3-dimensional vector by setting the third coordinate (at index 2) to $0$ and using the components of $x$ for the first two coordinates. Specifically, for $x = (x_0, x_1)$, $\text{halfPlane}(x) = (x_0, x_1, 0)$. This map provides the coordinate plane embedding used for the half-plane surface in $\text{Space } 3$.

The embedding $\text{halfPlane} : \text{Space } 2 \to \text{Space } 3$ is equal to the composition $(\text{slice}(2))^{-1} \circ g$, where $g : \text{Space } 2 \to \mathbb{R} \times \text{Space } 2$ is the map $x \mapsto (0, x)$, and $\text{slice}(2)$ is the continuous linear equivalence between $\text{Space } 3$ and $\mathbb{R} \times \text{Space } 2$ that extracts the third coordinate (index 2).

The function $\text{halfPlane} : \text{Space } 2 \to \text{Space } 3$, which embeds a 2-dimensional vector $x = (x_0, x_1)$ into 3-dimensional space as $(x_0, x_1, 0)$, is injective.

The map $\text{halfPlane} : \text{Space } 2 \to \text{Space } 3$, which embeds the 2-dimensional Euclidean space into the 3-dimensional Euclidean space by mapping $(x_0, x_1)$ to $(x_0, x_1, 0)$, is continuous.

The map $\text{halfPlane} : \mathbb{R}^2 \to \mathbb{R}^3$ defined by $\text{halfPlane}(x_0, x_1) = (x_0, x_1, 0)$, which embeds the 2-dimensional space into the 3-dimensional space as the plane $x_2 = 0$, is a measurable embedding. This means that the map is injective and a set $S \subseteq \mathbb{R}^2$ is measurable if and only if its image $\text{halfPlane}(S)$ is measurable in $\mathbb{R}^3$.

For any vector $x \in \text{Space } 2$, the Euclidean norm of its image under the embedding $\text{halfPlane} : \text{Space } 2 \to \text{Space } 3$ is equal to the Euclidean norm of the original vector $x$. That is, $\|\text{halfPlane}(x)\| = \|x\|$.

The measure on $\mathbb{R}^3$ (identified as `Space 3`) is defined as the pushforward of the Lebesgue measure on $\mathbb{R}^2$ (identified as `Space 2`), restricted to the half-plane domain $D = \{x \in \mathbb{R}^2 \mid x_1 \ge 0\}$, under the embedding $f: \mathbb{R}^2 \to \mathbb{R}^3$ given by $f(x_0, x_1) = (x_0, x_1, 0)$. This measure represents the surface measure associated with integration over the half-plane in three-dimensional space.

The surface measure $\mu_H$ on the half-plane in $\mathbb{R}^3$, defined as the pushforward of the Lebesgue measure on the region $\{(x_0, x_1) \in \mathbb{R}^2 \mid x_1 \ge 0\}$ under the embedding $f(x_0, x_1) = (x_0, x_1, 0)$, has temperate growth. This means that there exists some power $n$ such that the integral $\int (1 + \|x\|)^{-n} d\mu_H(x)$ is finite, allowing the measure to define a tempered distribution.

The measure on the half-plane in $\mathbb{R}^3$ (denoted by `halfPlaneMeasure`) is $\sigma$-finite.

This definition characterizes the tempered distribution on $\mathbb{R}^3$ (identified as `Space 3`) that corresponds to integration over a half-plane. It is a continuous linear map from the Schwartz space $\mathcal{S}(\mathbb{R}^3, \mathbb{R})$ to $\mathbb{R}$, defined by: $$f \mapsto \int_{\mathbb{R}^3} f(x) \, d\mu_H(x)$$ where $\mu_H$ is the surface measure on the half-plane $\{(x_0, x_1, 0) \in \mathbb{R}^3 \mid x_1 \ge 0\}$.

For any test function $f$ in the Schwartz space $\mathcal{S}(\mathbb{R}^3, \mathbb{R})$, the value of the tempered distribution associated with the half-plane (denoted by `halfPlaneDist`) applied to $f$ is equal to the integral of $f$ over $\mathbb{R}^3$ with respect to the half-plane surface measure $\mu_H$ (denoted by `halfPlaneMeasure`): $$\text{halfPlaneDist}(f) = \int_{\mathbb{R}^3} f(x) \, d\mu_H(x)$$ Here, the half-plane is defined as the set $\{(x_0, x_1, 0) \in \mathbb{R}^3 \mid x_1 \ge 0\}$.

For any Schwartz function $f \in \mathcal{S}(\mathbb{R}^3, \mathbb{R})$, the tempered distribution associated with the half-plane, denoted by $T_H$ (`halfPlaneDist`), applied to $f$ is equal to the integral of $f$ over the half-plane domain in $\mathbb{R}^2$. Specifically, $$T_H(f) = \int_{H} f(x_0, x_1, 0) \, dx$$ where $H = \{(x_0, x_1) \in \mathbb{R}^2 \mid x_1 \ge 0\}$ is the half-plane domain in $\mathbb{R}^2$, and the integral is taken with respect to the 2-dimensional Lebesgue measure.

The $\mathbb{R}$-submodule of $\text{Space } 3$ (isomorphic to $\mathbb{R}^3$) consisting of all vectors $x$ whose third coordinate $x_2$ (indexed by $2 \in \text{Fin } 3$) is zero. This subspace defines the coordinate plane $\{x \in \text{Space } 3 \mid x_2 = 0\}$ that contains the half-plane surface.

For any vector $x \in \text{Space } 2$, its image under the mapping $\text{halfPlane} : \text{Space } 2 \to \text{Space } 3$ is an element of the $\mathbb{R}$-submodule $\text{halfPlaneSubmodule}$. Here, $\text{halfPlane}$ is the embedding that maps $(x_0, x_1)$ to $(x_0, x_1, 0)$, and $\text{halfPlaneSubmodule}$ is the subspace of $\text{Space } 3$ consisting of vectors whose third coordinate is zero.

The image of the domain $\text{halfPlaneDomain} = \{x \in \mathbb{R}^2 \mid x_1 \ge 0\}$ under the embedding $\text{halfPlane} : \mathbb{R}^2 \to \mathbb{R}^3$ (defined as $x \mapsto (x_0, x_1, 0)$) is a subset of the subspace $\text{halfPlaneSubmodule} = \{x \in \mathbb{R}^3 \mid x_2 = 0\}$.

The $\mathbb{R}$-submodule of $\text{Space } 3$ consisting of all vectors $x$ whose coordinate $x_2$ is zero is a proper subspace; that is, it is not equal to the entire space $\text{Space } 3$.

Let $D = \{x \in \mathbb{R}^2 \mid x_1 \ge 0\}$ be the half-plane domain in 2-dimensional space. Let $f: \mathbb{R}^2 \to \mathbb{R}^3$ be the embedding map defined by $f(x_0, x_1) = (x_0, x_1, 0)$. The Lebesgue measure (volume) of the image $f(D)$ in $\mathbb{R}^3$ is equal to $0$.