Sphere (mathematics)

2-sphere

In mathematics, a sphere is understood to be the surface of a sphere and the generalization of it to dimensions of any height . The unit sphere , i.e. the surface of the unit sphere in n-dimensional Euclidean space , is of considerable importance for many investigations . More generally, especially in topology and differential geometry , every topological space homeomorphic to the spherical surface is also called a sphere, see Topological Sphere .

definition

Unity sphere

The unit sphere is the set of points in -dimensional Euclidean space at a distance of one from the origin . It is defined as ${\displaystyle S^{n-1}}$ ${\displaystyle n}$ ${\displaystyle \mathbb {R} ^{n}}$

${\displaystyle S^{n-1}:=\{x\in \mathbb {R} ^{n}\colon \|x\|_{2}=1\}}$,

where is the Euclidean norm . The unit sphere can be understood as the edge of the unit sphere and is therefore also referred to as. ${\displaystyle \|\cdot \|_{2}}$ ${\displaystyle S^{n-1}}$ ${\displaystyle B^{n}}$ ${\displaystyle \partial B^{n}}$

General spheres

If there is now any point in the -dimensional space, then the -sphere with radius around this point is defined by ${\displaystyle z\in \mathbb {R} ^{n}}$ ${\displaystyle n}$ ${\displaystyle (n-1)}$ ${\displaystyle S_{r}^{n-1}(z)}$ ${\displaystyle r}$ ${\displaystyle z}$

${\displaystyle S_{r}^{n-1}(z):=\{x\in \mathbb {R} ^{n}\colon \|x-z\|_{2}=r\}}$.

Each sphere is created from the associated unit sphere by scaling with the factor and translation around the vector . ${\displaystyle S_{r}^{n-1}(z)}$ ${\displaystyle S^{n-1}}$ ${\displaystyle r}$ ${\displaystyle z}$

Examples

The closed n-dimensional unit sphere can be assigned an (n-1) -dimensional sphere as an edge manifold : ${\displaystyle \mathbb {R} ^{n}}$

• The 1-ball is the interval [−1.1]. Accordingly, the 0-sphere only consists of the two points +1 and −1. It is the only sphere that is not connected . ${\displaystyle B^{1}}$ ${\displaystyle S^{0}}$

• The 2-sphere is the circular disk with radius 1 in the plane. The 1-sphere is the unit circle line, i.e. the edge of the unit circle . The unit circle is connected, but not simply connected . It can be described by complex numbers with an amount of 1 and, when multiplied, receives a group structure , the circle group . ${\displaystyle B^{2}}$ ${\displaystyle S^{1}}$

• The 3-sphere is the full sphere in three-dimensional space. The 2-sphere is the surface of the unit sphere. It is simply connected - like all higher dimensional spheres. It is described by spherical coordinates . ${\displaystyle B^{3}}$ ${\displaystyle S^{2}}$

• The 3-sphere is no longer clearly imaginable. It is a 3-dimensional submanifold in 4-dimensional space . The 3-sphere can be understood as a set of quaternions with the amount 1 and, through their multiplication, receives a group structure that corresponds to that. ${\displaystyle S^{3}}$ ${\displaystyle \mathbb {R} ^{4}}$ ${\displaystyle SU(2)}$

Content and volume

The area or the volume of any (n − 1) sphere of radius in Euclidean space can be calculated using the formula ${\displaystyle r}$

${\displaystyle \operatorname {vol} (S_{r}^{n-1})={\frac {\mathrm {d} }{\mathrm {d} r}}r^{n}V_{n}=nr^{n-1}V_{n}={2\pi ^{\frac {n}{2}}r^{n-1} \over \Gamma ({\frac {n}{2}})}}$

Calculate, where denotes the volume of the -dimensional unit sphere and the gamma function . ${\displaystyle V_{n}}$ ${\displaystyle n}$ ${\displaystyle \Gamma }$

The sphere in topology and geometry

→ Main article : Topological sphere

In mathematics, especially in differential geometry and topology, the term sphere is usually used with a different (more general) meaning: the n-dimensional sphere is the n-dimensional topological manifold that is homeomorphic to the unit sphere . ${\displaystyle S^{n}}$ ${\displaystyle \mathbb {R} ^{n+1}}$

A sphere as defined above with that induced by the Euclidean metric of the Riemannian metric is referred to as a round sphere in differential geometry . ${\displaystyle S_{r}^{n-1}(z):=\{x\in \mathbb {R} ^{n}\colon \|x-z\|_{2}=r\}}$ ${\displaystyle \mathbb {R} ^{n}}$

Generalizations

Spheres in standardized spaces

Unit spheres with regard to the maximum norm and the sum norm in three dimensions

The concept of the sphere can be understood more generally in standardized spaces . If there is a vector space over the real or complex numbers with the corresponding norm , then the norm sphere around the vector with radius is defined as the set ${\displaystyle V}$ ${\displaystyle \|\cdot \|}$ ${\displaystyle S_{r}(v)}$ ${\displaystyle v\in V}$ ${\displaystyle r}$

${\displaystyle S_{r}(v):=\{w\in V\colon \|v-w\|=r\}}$.

The resulting spheres are point-symmetrical with respect to , but no longer necessarily round (as in the case of the Euclidean norm), but can also have corners and edges (as in the case of the maximum norm and the sum norm ). If the zero vector and the radius , one speaks again of a unit sphere. All norm spheres arise from the associated unit sphere by scaling with the factor and translation around the vector . The unit sphere is in turn the edge of the associated unit sphere. ${\displaystyle v}$ ${\displaystyle v=0}$ ${\displaystyle r=1}$ ${\displaystyle r}$ ${\displaystyle v}$

Spheres in metric spaces

Spheres can be grasped even further in metric spaces . If there is any set with a metric , then the metric sphere around the point with radius is defined as the set ${\displaystyle X}$ ${\displaystyle d}$ ${\displaystyle S_{r}(x)}$ ${\displaystyle x\in X}$ ${\displaystyle r}$

${\displaystyle S_{r}(x):=\{y\in X\colon d(x,y)=r\}}$.

In contrast to spheres in normalized spaces, metric spheres are generally not translation-invariant and accordingly the metric unit sphere no longer has any special meaning. In certain metric spaces, the unit sphere can even be empty . Furthermore, a metric sphere can generally no longer be viewed as the edge of the associated metric sphere.

literature

• IS Sharadze: Sphere . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

Web links

Commons : Spheres  - collection of images, videos and audio files

Individual evidence

1. ^ Wolfgang Walter: Analysis 2 . Springer, 2002, p. 17 .
2. ↑ Rolf Walter: Introduction to Calculus I . de Gruyter, 2007, p. 272 .