PhyslibAlpha.SpaceAndTime.Space.Surfaces.Line

For a given dimension $d > 0$, the function maps a real number $r \in \mathbb{R}$ to an element in $\text{Space } d$ by scaling the first vector $e_0$ of the standard orthonormal basis by $r$. That is, the map is defined as $r \mapsto r \cdot e_0$, representing the coordinate line embedded along the first axis of the $d$-dimensional space.

For any dimension $d \in \mathbb{N}$ with $d > 0$, the function $\text{line}_d : \mathbb{R} \to \text{Space } d$ is given by $r \mapsto r \cdot e_0$, where $e_0$ denotes the first vector of the standard orthonormal basis of $\text{Space } d$.

For any dimension $d \in \mathbb{N}$ such that $d > 0$, the coordinate line map $\text{line} : \mathbb{R} \to \text{Space } d$, which maps a real number $r$ to $r \cdot e_0$ (where $e_0$ is the first vector of the standard basis), is injective.

For any positive integer $d$, the coordinate line map $\text{line}_d: \mathbb{R} \to \text{Space } d$, which maps a real number $r$ to $r \cdot e_0$ (where $e_0$ is the first basis vector of the $d$-dimensional Euclidean space), is continuous.

For any positive natural number $d$, the map $\text{line } d : \mathbb{R} \to \text{Space } d$ defined by $r \mapsto r \cdot e_0$ (where $e_0$ is the first vector of the standard orthonormal basis) is a measurable embedding. This means that the map is injective, measurable, and maps measurable sets in $\mathbb{R}$ to measurable sets in $\text{Space } d$.

For any dimension $d > 0$ and any real number $r$, the Euclidean norm of the point on the coordinate line $\text{line}(d, r)$ in $d$-dimensional space is equal to the absolute value of $r$. That is, $\|\text{line}(d, r)\| = |r|$, where $\text{line}(d, r)$ represents the vector $r \cdot e_0$ along the first basis vector $e_0$ in $\text{Space } d$.

For a dimension $d > 0$, the measure on $\text{Space}(d)$ is defined as the pushforward of the standard Lebesgue measure on $\mathbb{R}$ under the map $r \mapsto r \cdot e_0$, where $e_0$ is the first standard basis vector. This measure corresponds to integration along the coordinate line in $\text{Space}(d)$.

For any dimension $d \in \mathbb{N}$ with $d > 0$, the line measure on $\text{Space } d$ (defined as the pushforward of the standard Lebesgue measure on $\mathbb{R}$ under the map $r \mapsto r \cdot e_0$, where $e_0$ is the first standard basis vector) has temperate growth.

For a positive dimension $d$, $\text{lineDist } d$ is the tempered distribution on $\text{Space } d$ defined as the continuous linear map from the space of Schwartz functions $\mathcal{S}(\text{Space } d, \mathbb{R})$ to $\mathbb{R}$ that corresponds to integration along the first coordinate axis. For a test function $f$, it is given by the integral with respect to the line measure $\mu_L$: $$\langle \text{lineDist } d, f \rangle = \int_{\text{Space } d} f(x) \, d\mu_L(x) = \int_{-\infty}^{\infty} f(r \cdot e_0) \, dr$$ where $e_0$ is the first standard basis vector of $\text{Space } d$. This distribution can be interpreted as a mass or charge density concentrated entirely on the coordinate line.

For any dimension $d \in \mathbb{N}$ with $d > 0$ and for any Schwartz function $f: \text{Space } d \to \mathbb{R}$, the value of the tempered distribution $\text{lineDist } d$ applied to $f$ is equal to the integral of $f$ with respect to the line measure $\mu_L$ (denoted as `lineMeasure d`) on $\text{Space } d$: $$\text{lineDist } d(f) = \int_{\text{Space } d} f(x) \, d\mu_L(x)$$

For a positive dimension $d > 0$, the tempered distribution $\text{lineDist } d$ applied to a Schwartz function $f \in \mathcal{S}(\text{Space } d, \mathbb{R})$ is equal to the integral of $f$ along the first coordinate axis: $$\langle \text{lineDist } d, f \rangle = \int_{-\infty}^{\infty} f(\text{line } d(r)) \, dr$$ where $\text{line } d(r) = r \cdot e_0$ represents the coordinate line along the first standard basis vector $e_0$ of $\text{Space } d$.

For a positive natural number $d$, the `lineSubmodule` is the 1-dimensional linear subspace of $\text{Space } d$ spanned by the first element of its standard orthonormal basis $e_0$ (indexed by $0 \in \{0, \dots, d-1\}$). In other words, it is the submodule defined by $\{r \cdot e_0 \mid r \in \mathbb{R}\}$.

For a positive natural number $d$ and any real number $r$, the point $\text{line}(d, r)$ (the vector $r \cdot e_0$) in $\text{Space } d$ is an element of the 1-dimensional subspace $\text{lineSubmodule } d$ spanned by the first basis vector $e_0$.

For a positive natural number $d$, the range of the map $\text{line } d : \mathbb{R} \to \text{Space } d$ (defined by $r \mapsto r \cdot e_0$) is a subset of the 1-dimensional linear subspace $\text{lineSubmodule } d$ spanned by the first basis vector $e_0$.

For any natural number $d \ge 2$, the 1-dimensional linear subspace of $\text{Space } d$ spanned by the basis vector $e_0$ (denoted as `lineSubmodule d`) is a proper subspace of $\text{Space } d$, meaning $\text{lineSubmodule } d \neq \text{Space } d$.

For any natural number $d \ge 2$, the $d$-dimensional Lebesgue measure (volume) of the coordinate line in $\text{Space}(d)$ (the range of the map $r \mapsto r \cdot e_0$, where $e_0$ is the first vector of the standard basis) is zero. That is, $\text{volume}(\{ r \cdot e_0 \mid r \in \mathbb{R} \}) = 0$.