Euclidean area
Euclidean area is in that Area, for that the laws that apply. Euclidean areas exist in arbitrary n. A two-dimensional Euclidean area is called also Euclidean Even one.
Algebraic description
a Euclidean area in arbitrary dimensions leaves itself n (n > 0) by that nfold that real number set R describe. Since during this description no information is lost, the term is narrowed frequently on this special area, then as <math>R^n</math> or also <math>E^n</math> one designates.
By koordinatenweise Addition and also Scalar one becomes it a real to be defined can, by those Coordinates in pairs multiplies and the developing Products are added. In three arises in such a way for example:
- <math>
x\cdot y=x_{1} y_{1}+x_{2} y_{2} +x_{3} y_{3} </math>
That the algebraic definition of distances makes possible and Angles. In addition first one becomes for each point x Standard length mentioned fixed, by those from the dot product of the vector with itself is defined. Again in three dimensions arises for example:
- <math>
||x||=|x|=\sqrt{(x_{1})^2+(x_{2})^2+(x_{3})^2} </math>
The distance of two points x and y arises now as a result of the Euclidean d(x, y) (Euclidean distance), itself as Standard that XY calculates. As example in three dimensions is considered then:
- <math>
d(x, y)=\sqrt{(x_{1}-y_{1})^2+(x_{2}-y_{2})^2+(x_{3}-y_{3})^2}. </math>
between two vectors x and y become by those Cosine- fixed the cosine of the angle defines itself as from that of x,y and that Product their standards:
- <math>
\cos \alpha(x, y)=\frac{x\cdot y}{|x||y|} </math>
Euclidean areas in higher mathematics
By its Metrik D is each Euclidean area R n and thus also . As it is besides the classical example of one topological vector space. In particular is it in Praehilbertraum and, because in the finite-dimensional also completely, Banachraum and thus also Hilbertraum. After a proof of Brouwer are not Euclidean areas of different dimension homoeomorph one on the other imagable.
A Euclidean area is at the same time the prototype of a topological and differentiable . For all dimensions except four one is to R n homoeomorphe also one to R n diffeomorphe. The exceptions existing in four dimensions are called exotic 4-Raeume.
Euclidean vector spaces
A Euclidean area must not necessarily by the special area R n are described: Each finite-dimensional vector space, on which a dot product is defined, ever two vectors one real Number assigns, is at the same time a model of a Euclidean area to same dimension; one calls such a vector space therefore also Euclidean vector space. However each Euclidean vector space is isomorphic to the special area R by selection of a basis n, i.e., there are at least regarding geometry no differences between both.
Euclidean ones vector spaces are for their part special examples of Praehilbertraeume or dot product areas; their characteristics and exact definition are more near discussed therefore in this article.
See also: Hierarchy of mathematical structures
