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ID: INPUT{id.yaml} Title: > Volume of the $d$-dimensional unit sphere Definition: > $S_d = \text{vol}(\{x\in\mathbb{R}^{d+1}: |x| = 1\})$ is the $d$-dimensional volume of the unit sphere in $(d+1)$-dimensional Euclidean space. Parameters: d: display: $d$ type: Z constraints: $d \geq -1$ Comments: Formulas: formula-gamma: > $S_d = \frac{2\pi^{(d+1)/2}}{\Gamma((d+1)/2)}$. formula-ball: > $S_{d-1} = dB_d$ for $d\geq 0$, where $B_d$ denotes the $d$-dimensional volume of the unit ball in $\mathbb{R}^d$. Programs: References: Links: Wiki: title: "Wikipedia: N-sphere" url: https://en.wikipedia.org/wiki/N-sphere Similar tables: Keywords: Tags: - elementary geometry - integral - period Data properties: type: R complete: no Display properties: number-header: $S_d$ Numbers: INPUT{numbers.yaml}
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Volume of the $d$-dimensional unit sphere
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Numbers
$d$
$S_d$
INPUT{numbers.yaml} (not shown in preview)
Definition
$S_d = \text{vol}(\{x\in\mathbb{R}^{d+1}: |x| = 1\})$ is the $d$-dimensional volume of the unit sphere in $(d+1)$-dimensional Euclidean space.
Parameters
$d$
— integer ($d \geq -1$)
Formulas
(1)
$S_d = \frac{2\pi^{(d+1)/2}}{\Gamma((d+1)/2)}$.
(2)
$S_{d-1} = dB_d$ for $d\geq 0$, where $B_d$ denotes the $d$-dimensional volume of the unit ball in $\mathbb{R}^d$.
Links
[1]
Wikipedia: N-sphere
Data properties
Entries are of type: real number
Table is complete: no
