Physlib.SpaceAndTime.Space.Norm

For a natural number $n$ and a vector $x$ in the $d$-dimensional space $\text{Space } d$, the function returns the real value $$\sqrt{\|x\|^2 + \frac{1}{n+1}}$$ where $\|x\|$ is the Euclidean norm. This sequence of functions is differentiable everywhere and converges to $\|x\|$ as $n \to \infty$.

For any natural number $n$, the function $\text{normPowerSeries } n$ on the $d$-dimensional space $\text{Space } d$ is defined such that for any $x \in \text{Space } d$, its value is $$\text{normPowerSeries } n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$$ where $\|x\|$ denotes the Euclidean norm.

For any natural number $n$, the function $\text{normPowerSeries } n$ on the $d$-dimensional space $\text{Space } d$ is defined such that for any $x \in \text{Space } d$, its value is $$\text{normPowerSeries } n(x) = \left(\|x\|^2 + \frac{1}{n+1}\right)^{1/2}$$ where $\|x\|$ denotes the Euclidean norm.

For any dimension $d$ and any natural number $n$, the function $x \mapsto \text{normPowerSeries}_n(x)$ on $\text{Space } d$, defined by $$\text{normPowerSeries}_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}},$$ is differentiable over $\mathbb{R}$ at every point $x$, where $\|x\|$ denotes the Euclidean norm on $\text{Space } d$.

For any dimension $d$ and any non-zero vector $x \in \text{Space } d$, the sequence $\text{normPowerSeries}_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$ converges to the Euclidean norm $\|x\|$ as $n \to \infty$.

For any dimension $d$ and any non-zero vector $x \in \text{Space } d$, the sequence of the inverse of the norm power series, $(\text{normPowerSeries}_n(x))^{-1} = \frac{1}{\sqrt{\|x\|^2 + \frac{1}{n+1}}}$, converges to the inverse of the Euclidean norm $\|x\|^{-1} = \frac{1}{\|x\|}$ as $n \to \infty$.

For any dimension $d$, natural number $n$, vector $x \in \text{Space } d$, and index $i \in \{0, \dots, d-1\}$, the partial derivative of the norm power series function $\text{normPowerSeries}_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$ with respect to the $i$-th coordinate is given by: $$\frac{\partial}{\partial x_i} \left( \text{normPowerSeries}_n(x) \right) = \frac{x_i}{\sqrt{\|x\|^2 + \frac{1}{n+1}}}$$ where $x_i$ is the $i$-th component of $x$ relative to the standard orthonormal basis and $\|x\|$ is the Euclidean norm.

For any dimension $d$, natural number $n$, and vectors $x, y \in \text{Space } d$, the Fréchet derivative of the norm power series function $f_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$ at the point $x$ applied to the vector $y$ is given by: $$Df_n(x)y = \frac{\langle y, x \rangle}{\sqrt{\|x\|^2 + \frac{1}{n+1}}}$$ where $\langle y, x \rangle$ denotes the standard real inner product and $\|x\|$ is the Euclidean norm in $\text{Space } d$.

For any dimension $d$, non-zero vector $x \in \text{Space } d$, and coordinate index $i \in \{0, \dots, d-1\}$, the sequence of partial derivatives of the norm power series function $\text{normPowerSeries}_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$ with respect to the $i$-th coordinate converges to $x_i \|x\|^{-1}$ as $n \to \infty$, where $x_i$ is the $i$-th component of $x$ and $\|x\|$ is the Euclidean norm. Mathematically, $$\lim_{n \to \infty} \frac{\partial}{\partial x_i} \left( \sqrt{\|x\|^2 + \frac{1}{n+1}} \right) = \frac{x_i}{\|x\|}$$

For any dimension $d$, non-zero vector $x \in \text{Space } d$, and any vector $y \in \text{Space } d$, the Fréchet derivative of the function $f_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$ at the point $x$ applied to the vector $y$ converges to $\frac{\langle y, x \rangle}{\|x\|}$ as $n \to \infty$. Here, $\langle y, x \rangle$ denotes the standard real inner product and $\|x\|$ is the Euclidean norm in $\text{Space } d$. Mathematically, $$\lim_{n \to \infty} D \left( \sqrt{\|x\|^2 + \frac{1}{n+1}} \right) (y) = \frac{\langle y, x \rangle}{\|x\|}$$

For any natural number $n$, the function $\text{normPowerSeries}_n: \text{Space } d \to \mathbb{R}$ defined by $$\text{normPowerSeries}_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$$ is almost everywhere strongly measurable with respect to the volume measure on the $d$-dimensional space $\text{Space } d$, where $\|x\|$ denotes the Euclidean norm.

For any natural number $n$ and any vector $x$ in the $d$-dimensional space $\text{Space } d$, the value of the function $\text{normPowerSeries}_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$ is non-negative, where $\|x\|$ denotes the Euclidean norm.

For any natural number $n$ and any vector $x$ in the $d$-dimensional space $\text{Space } d$, the function $\text{normPowerSeries}_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$ is strictly positive, where $\|x\|$ denotes the Euclidean norm.

For any natural number $n$ and any vector $x$ in the $d$-dimensional space $\text{Space } d$, the value of the function $\text{normPowerSeries}_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$ is non-zero, where $\|x\|$ denotes the Euclidean norm.

For any natural number $n$ and any vector $x$ in the $d$-dimensional space $\text{Space } d$, the value of the norm power series $\text{normPowerSeries } n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$ is less than or equal to the Euclidean norm $\|x\|$ plus 1: $$\sqrt{\|x\|^2 + \frac{1}{n+1}} \leq \|x\| + 1$$

For any natural number $n$ and any vector $x$ in the $d$-dimensional space $\text{Space } d$, the Euclidean norm $\|x\|$ is strictly less than the value of the norm power series $\text{normPowerSeries } n(x)$, defined as: $$\|x\| < \sqrt{\|x\|^2 + \frac{1}{n+1}}$$

For any natural number $n$ and any vector $x$ in the $d$-dimensional space $\text{Space } d$, the Euclidean norm $\|x\|$ is less than or equal to the value of the norm power series at $n$ and $x$, given by: $$\|x\| \leq \sqrt{\|x\|^2 + \frac{1}{n+1}}$$ where $\text{normPowerSeries } n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$.

For any dimension $d$, any natural number $n$, any integer $m$, and any non-zero vector $x \in \text{Space } d$, the $m$-th power of the norm power series $f_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$ is less than or equal to the sum of the $m$-th powers of $(\|x\| + 1)$ and $\|x\|$: $$(f_n(x))^m \leq (\|x\| + 1)^m + \|x\|^m$$ where $\|x\|$ is the Euclidean norm.

For any dimension $d$, natural number $n$, and any non-zero vector $x$ in $\text{Space } d$, the inverse of the norm power series $f_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$ is less than or equal to the inverse of its Euclidean norm $\|x\|$: $$\frac{1}{\sqrt{\|x\|^2 + \frac{1}{n+1}}} \leq \frac{1}{\|x\|}$$

For any natural number $n$ and vector $x$ in a $d$-dimensional space, the norm power series $f_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$ satisfies the inequality $$|\log(f_n(x))| \le \frac{1}{f_n(x)} + f_n(x)$$ where $\|x\|$ is the Euclidean norm and $\log$ is the natural logarithm.

For any dimension $d$, natural number $n$, and any non-zero vector $x \in \text{Space } d$, the norm power series $f_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$ satisfies the inequality $$|\log(f_n(x))| \leq \frac{1}{\|x\|} + (\|x\| + 1)$$ where $\|x\|$ is the Euclidean norm and $\log$ denotes the natural logarithm.

For any dimension $d$, natural number $n$, and integer $m$, the function $x \mapsto f_n(x)^m$ on $\text{Space } d$ is distributionally bounded (satisfies the `IsDistBounded` property), where $f_n(x)$ is the $n$-th norm power series defined as $$f_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$$ and $\|x\|$ denotes the Euclidean norm.

For any dimension $d$ and natural number $n$, the function $x \mapsto f_n(x)$ on $\text{Space } d$ is distributionally bounded (satisfies the `IsDistBounded` property), where $f_n(x)$ is the $n$-th norm power series defined as $$f_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$$ and $\|x\|$ denotes the Euclidean norm.

For any dimension $d$ and natural number $n$, the function $x \mapsto \frac{1}{f_n(x)}$ is distributionally bounded (satisfies the `IsDistBounded` property) on the $d$-dimensional space, where $f_n(x)$ is the $n$-th norm power series defined as $$f_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$$ and $\|x\|$ denotes the Euclidean norm.

For any dimension $d$, natural number $n$, and index $i \in \{0, \dots, d-1\}$, the partial derivative of the $n$-th norm power series $f_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$ with respect to the $i$-th coordinate, mapping $x \in \text{Space } d$ to $\frac{\partial}{\partial x_i} f_n(x)$, is distributionally bounded (satisfies the `IsDistBounded` property).

For any dimension $d$, natural number $n$, and vector $y \in \text{Space } d$, the function $x \mapsto D f_n(x) \cdot y$ is distributionally bounded (satisfies the `IsDistBounded` property), where $f_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$ is the $n$-th norm power series and $D f_n(x)$ denotes its Fréchet derivative at the point $x$.

For any dimension $d$ and natural number $n$, the function $x \mapsto \log(f_n(x))$ defined on the $d$-dimensional space $\text{Space } d$ is distributionally bounded (satisfies the `IsDistBounded` property), where $f_n(x)$ is the $n$-th norm power series defined as $$f_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$$ and $\|x\|$ denotes the Euclidean norm.

For any dimension $d$, natural number $n$, and integer $m$, the function $x \mapsto (\text{normPowerSeries}_n(x))^m$ is differentiable over $\mathbb{R}$ on $\text{Space } d$, where $\text{normPowerSeries}_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$ and $\|x\|$ denotes the Euclidean norm on $\text{Space } d$.

For any dimension $d \in \mathbb{N}$ and any natural number $n \in \mathbb{N}$, the function $x \mapsto \frac{1}{\text{normPowerSeries}_n(x)}$ mapping from $\text{Space } d$ to $\mathbb{R}$ is differentiable, where $\text{normPowerSeries}_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$ and $\|x\|$ denotes the Euclidean norm.

For any dimension $d \in \mathbb{N}$ and any natural number $n \in \mathbb{N}$, the function $x \mapsto \log\left(\sqrt{\|x\|^2 + \frac{1}{n+1}}\right)$ mapping from the $d$-dimensional space $\text{Space } d$ to $\mathbb{R}$ is differentiable at every point $x$, where $\|x\|$ denotes the Euclidean norm.

For any dimension $d \in \mathbb{N}$, natural number $n$, integer $m$, and vector $x \in \text{Space } d$, the partial derivative with respect to the $i$-th coordinate of the function mapping $x$ to the $m$-th power of the norm power series $f_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$ is given by: $$\frac{\partial}{\partial x_i} \left( \sqrt{\|x\|^2 + \frac{1}{n+1}} \right)^m = m \cdot x_i \cdot \left( \sqrt{\|x\|^2 + \frac{1}{n+1}} \right)^{m-2}$$ where $x_i$ denotes the $i$-th component of $x$ and $\|x\|$ is the Euclidean norm.

For any dimension $d \in \mathbb{N}$, natural number $n$, integer $m$, and vectors $x, y \in \text{Space } d$, the Fréchet derivative of the function mapping $x$ to the $m$-th power of the norm power series $f_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$ evaluated at $x$ in the direction $y$ is given by: $$D \left( \left( \sqrt{\|x\|^2 + \frac{1}{n+1}} \right)^m \right) (y) = m \cdot \langle y, x \rangle \cdot \left( \sqrt{\|x\|^2 + \frac{1}{n+1}} \right)^{m-2}$$ where $\|x\|$ denotes the Euclidean norm and $\langle y, x \rangle$ denotes the standard inner product on $\text{Space } d$.

For any dimension $d \in \mathbb{N}$, natural number $n$, vector $x \in \text{Space } d$, and index $i \in \{0, 1, \dots, d-1\}$, the partial derivative of the logarithm of the norm power series with respect to the $i$-th coordinate is given by: $$\frac{\partial}{\partial x_i} \log(\text{normPowerSeries}_n(x)) = x_i \cdot (\text{normPowerSeries}_n(x))^{-2}$$ where $\text{normPowerSeries}_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$, $\|x\|$ is the Euclidean norm, and $x_i$ denotes the $i$-th component of $x$ relative to the standard orthonormal basis.

For any dimension $d \in \mathbb{N}$ and natural number $n$, let $f_n(x) = \text{normPowerSeries}_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$ be the norm power series defined on the $d$-dimensional real inner product space $\text{Space } d$. For any vectors $x, y \in \text{Space } d$, the Fréchet derivative of the function $x \mapsto \log(f_n(x))$ at the point $x$ applied to the vector $y$ is given by: $$D(\log(f_n(x))) \cdot y = \langle y, x \rangle \cdot f_n(x)^{-2}$$ where $\langle \cdot, \cdot \rangle$ denotes the standard inner product on $\text{Space } d$.

For any dimension $d$, natural number $n$, and integer $m$, let $f_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$ be the $n$-th norm power series on the $d$-dimensional real inner product space $\text{Space } d$, where $\|x\|$ is the Euclidean norm. The distributional gradient of the function $x \mapsto f_n(x)^m$ is equal to the distribution associated with the vector-valued function $$x \mapsto m \cdot f_n(x)^{m-2} \cdot \mathbf{x}$$ where $\mathbf{x}$ denotes the coordinate representation of $x$ in the standard orthonormal basis of $\text{Space } d$.

Let $E$ be a real inner product space of dimension $d+1$. For each $n \in \mathbb{N}$, let $f_n: E \to \mathbb{R}$ be the norm power series defined by $f_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$, where $\|\cdot\|$ denotes the Euclidean norm. For any integer $m \geq -d$, any Schwartz test function $\eta \in \mathcal{S}(E, \mathbb{R})$, and any vector $y \in E$, the inner product of the distributional gradient of $f_n^m$ with $y$ converges to the inner product of the distributional gradient of $\|x\|^m$ with $y$ as $n \to \infty$: $$\lim_{n \to \infty} \langle (\nabla f_n^m)(\eta), y \rangle = \langle (\nabla \|x\|^m)(\eta), y \rangle$$ Here $\nabla$ denotes the gradient operator in the sense of distributions.

Let $E = \text{Space}(d+1)$ be a real inner product space of dimension $d+1$. For each $n \in \mathbb{N}$, let $f_n: E \to \mathbb{R}$ be the $n$-th norm power series defined by $f_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$, where $\|x\|$ is the Euclidean norm. For any integer $m$ such that $m \geq -d + 1$, any Schwartz test function $\eta \in \mathcal{S}(E, \mathbb{R})$, and any vector $y \in E$, the following limit holds: $$\lim_{n \to \infty} \langle (\nabla \mathcal{T}_{f_n^m})(\eta), y \rangle = \langle \mathcal{T}_{m \|x\|^{m-2} \mathbf{x}}(\eta), y \rangle$$ where $\mathcal{T}_g$ denotes the regular distribution associated with a function $g$, $\nabla$ denotes the gradient in the sense of distributions, and $\mathbf{x}$ denotes the coordinate representation of $x$ in the standard orthonormal basis of $E$.

For any dimension $d \in \mathbb{N}$ and natural number $n$, let $f_n: \text{Space } d \to \mathbb{R}$ be the $n$-th norm power series defined by $f_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$, where $\|x\|$ denotes the Euclidean norm. The distributional gradient of the distribution associated with the function $x \mapsto \log(f_n(x))$ is equal to the distribution associated with the vector-valued function $x \mapsto f_n(x)^{-2} \mathbf{x}$: $$\nabla \mathcal{T}_{\log(f_n)} = \mathcal{T}_{f_n^{-2} \mathbf{x}}$$ where $\mathbf{x}$ denotes the coordinate representation of the vector $x$ in the standard orthonormal basis of $\text{Space } d$, and $\mathcal{T}_g$ denotes the regular distribution induced by a function $g$.

For any natural number $d$, let $V = \text{Space}(d+2)$ be a real inner product space of dimension $d+2$. Let $f_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$ be the $n$-th norm power series. For any Schwartz test function $\eta \in \mathcal{S}(V, \mathbb{R})$ and any vector $y \in V$, the sequence of real values obtained by taking the inner product of the distributional gradient of $x \mapsto \ln f_n(x)$ (evaluated at $\eta$) with $y$ converges to the inner product of the distributional gradient of $x \mapsto \ln \|x\|$ (evaluated at $\eta$) with $y$ as $n \to \infty$. Mathematically, this is expressed as: $$\lim_{n \to \infty} \langle (\nabla T_{\ln f_n})(\eta), y \rangle = \langle (\nabla T_{\ln \|x\|})(\eta), y \rangle$$ where $T_g$ denotes the distribution associated with a function $g$ and $\nabla$ denotes the distributional Fréchet derivative.

For any natural number $d$, let $V = \text{Space}(d+2)$ be a real inner product space of dimension $d+2$. Let $f_n(x) = \sqrt{\|x\|^2 + \frac{1}{n+1}}$ be the $n$-th norm power series, where $\|x\|$ denotes the Euclidean norm. For any Schwartz test function $\eta \in \mathcal{S}(V, \mathbb{R})$ and any coordinate vector $y \in \mathbb{R}^{d+2}$, the limit as $n \to \infty$ of the inner product of the distributional gradient of $x \mapsto \ln(f_n(x))$ (evaluated at $\eta$) with $y$ is: $$\lim_{n \to \infty} \langle (\nabla \mathcal{T}_{\ln f_n})(\eta), y \rangle = \langle \mathcal{T}_{\|x\|^{-2} \mathbf{x}}(\eta), y \rangle$$ where $\mathcal{T}_g$ denotes the regular distribution induced by a function $g$, $\nabla$ denotes the distributional gradient, and $\mathbf{x}$ denotes the coordinate representation of the vector $x$ in the standard orthonormal basis of $V$.

Let $E = \text{Space}(d+1)$ be a real inner product space of dimension $d+1$. For any integer $m$ such that $m \geq -d + 1$, the distributional gradient of the function $x \mapsto \|x\|^m$ is equal to the distribution associated with the vector-valued function $x \mapsto m \|x\|^{m-2} \mathbf{x}$. Mathematically, $$\nabla \mathcal{T}_{\|x\|^m} = \mathcal{T}_{m \|x\|^{m-2} \mathbf{x}}$$ where $\|x\|$ denotes the Euclidean norm, $\mathbf{x}$ is the position vector (represented in the standard orthonormal basis of $E$), and $\mathcal{T}_f$ is the regular distribution induced by a function $f$.

For any natural number $d$, let $V = \text{Space}(d+2)$ be a real inner product space of dimension $d+2$. The distributional gradient of the regular distribution induced by the function $x \mapsto \ln \|x\|$ is equal to the regular distribution induced by the vector-valued function $x \mapsto \frac{\mathbf{x}}{\|x\|^2}$, where $\|\cdot\|$ denotes the Euclidean norm and $\mathbf{x}$ denotes the coordinate representation of $x$ in the standard orthonormal basis of $V$. Mathematically, this is expressed as: $$\nabla T_{\ln \|x\|} = T_{\|x\|^{-2} \mathbf{x}}$$ where $T_f$ denotes the distribution associated with a function $f$.

For any natural number $d$, let $n = d + 1$. In the $n$-dimensional real inner product space $\text{Space } n$, the distributional divergence of the vector-valued function $x \mapsto \|x\|^{-n} x$ (where $x$ is identified with its coordinate representation in the standard basis) is equal to $n \cdot \text{Vol}(B_1(0)) \delta_0$, where $\|x\|$ is the Euclidean norm, $\text{Vol}(B_1(0))$ is the volume of the unit ball in the space, and $\delta_0$ is the Dirac delta distribution at the origin.