Editing N-sphere (section)

==Description==

For any [[natural number]] {{tmath|n}}, an {{tmath|n}}-sphere of radius {{tmath|r}} is defined as the set of points in {{tmath|(n+1)}}-dimensional [[Euclidean space]] that are at distance {{tmath|r}} from some fixed point {{tmath|\mathbf{c} }}, where {{tmath|r}} may be any [[Positive number|positive]] [[real number]] and where {{tmath|\mathbf{c} }} may be any point in {{tmath|(n+1)}}-dimensional space. In particular:
* a 0-sphere is a pair of points {{tmath| \{ c - r, c + r \} }}, and is the boundary of a line segment ({{tmath|1}}-ball).
* a [[1-sphere|{{math|1}}-sphere]] is a [[circle]] of radius {{tmath|r}} centered at {{tmath|\mathbf{c} }}, and is the boundary of a disk ({{tmath|2}}-ball).
* a [[2-sphere|{{math|2}}-sphere]] is an ordinary {{tmath|2}}-dimensional [[sphere]] in {{tmath|3}}-dimensional Euclidean space, and is the boundary of an ordinary ball ({{tmath|3}}-ball).
* a [[3-sphere|{{math|3}}-sphere]] is a {{tmath|3}}-dimensional sphere in {{tmath|4}}-dimensional Euclidean space.

=== Cartesian coordinates ===

The set of points in {{tmath|(n+1)}}-space, {{tmath|(x_1, x_2, \ldots, x_{n+1})}}, that define an {{tmath|n}}-sphere, {{tmath|S^n(r)}}, is represented by the equation:

:<math>r^2=\sum_{i=1}^{n+1} (x_i - c_i)^2 ,</math>

where {{tmath|1= \mathbf{c} = (c_1, c_2, \ldots, c_{n+1})}} is a center point, and {{tmath|r}} is the radius.

The above {{tmath|n}}-sphere exists in {{tmath|(n+1)}}-dimensional Euclidean space and is an example of an {{tmath|n}}-[[manifold]].  The [[volume form]] {{tmath|\omega}} of an {{tmath|n}}-sphere of radius {{tmath|r}} is given by

:<math>\omega = \frac{1}{r} \sum_{j=1}^{n+1} (-1)^{j-1} x_j \,dx_1 \wedge \cdots \wedge dx_{j-1} \wedge dx_{j+1}\wedge \cdots \wedge dx_{n+1} = {\star} dr</math>

where <math>{\star}</math> is the [[Hodge star operator]]; see {{harvtxt|Flanders|1989|loc=§6.1}} for a discussion and proof of this formula in the case {{tmath|r {{=}} 1}}.  As a result,
:<math>dr \wedge \omega = dx_1 \wedge \cdots \wedge dx_{n+1}.</math>

=== ''n''-ball ===
{{main|Ball (mathematics)}}

The space enclosed by an {{tmath|n}}-sphere is called an {{tmath|(n+1)}}-[[Ball (mathematics)|ball]]. An {{tmath|(n+1)}}-ball is [[Closed set|closed]] if it includes the {{tmath|n}}-sphere, and it is [[Open set|open]] if it does not include the {{tmath|n}}-sphere.

Specifically:
* A {{tmath|1}}-''ball'', a [[line segment]], is the interior of a 0-sphere.
* A {{tmath|2}}-''ball'', a [[Disk (mathematics)|disk]], is the interior of a [[circle]] ({{tmath|1}}-sphere).
* A {{tmath|3}}-''ball'', an ordinary [[Ball (mathematics)|ball]], is the interior of a [[sphere]] ({{tmath|2}}-sphere).
* A {{tmath|4}}-''ball'' is the interior of a [[3-sphere|{{math|3}}-sphere]], etc.

===Topological description===
[[Topology|Topologically]], an {{tmath|n}}-sphere can be constructed as a [[Alexandroff extension|one-point compactification]] of {{tmath|n}}-dimensional Euclidean space.  Briefly, the {{tmath|n}}-sphere can be described as {{tmath|1=S^n = \R^n \cup \{ \infty \} }}, which is {{tmath|n}}-dimensional Euclidean space plus a single point representing infinity in all directions.
In particular, if a single point is removed from an {{tmath|n}}-sphere, it becomes [[Homeomorphism|homeomorphic]] to <math>\R^n</math>. This forms the basis for [[stereographic projection]].<ref>James W. Vick (1994). ''Homology theory'', p. 60. Springer</ref>