Euclidean space Definition and 59 Threads

Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension, including the three-dimensional space and the Euclidean plane (dimension two). It was introduced by the Ancient Greek mathematician Euclid of Alexandria, and the qualifier Euclidean is used to distinguish it from other spaces that were later discovered in physics and modern mathematics.
Ancient Greek geometers introduced Euclidean space for modeling the physical universe. Their great innovation was to prove all properties of the space as theorems by starting from a few fundamental properties, called postulates, which either were considered as evident (for example, there is exactly one straight line passing through two points), or seemed impossible to prove (parallel postulate).
After the introduction at the end of 19th century of non-Euclidean geometries, the old postulates were re-formalized to define Euclidean spaces through axiomatic theory. Another definition of Euclidean spaces by means of vector spaces and linear algebra has been shown to be equivalent to the axiomatic definition. It is this definition that is more commonly used in modern mathematics, and detailed in this article.In all definitions, Euclidean spaces consist of points, which are defined only by the properties that they must have for forming a Euclidean space.
There is essentially only one Euclidean space of each dimension; that is, all Euclidean spaces of a given dimension are isomorphic. Therefore, in many cases, it is possible to work with a specific Euclidean space, which is generally the real n-space





R


n


,


{\displaystyle \mathbb {R} ^{n},}
equipped with the dot product. An isomorphism from a Euclidean space to





R


n




{\displaystyle \mathbb {R} ^{n}}
associates with each point an n-tuple of real numbers which locate that point in the Euclidean space and are called the Cartesian coordinates of that point.

View More On Wikipedia.org
  1. J

    What is an Example of a Closed Set with an Empty Interior in Euclidean Space?

    Give an example of a closed set S in R^2 such that the closure of the interior of S does not equal to S (in set notation). I have no idea where to start...any help would be nice! Thanks!
  2. P

    Euclidean Space - Maximum Value

    Homework Statement Find the maximum of \frac{x+2y+3z}{\sqrt{x^2+y^2+z^2}} as (x,y,z) varies among nonzero points in R^{3} Homework Equations I'm not sure. The section in which this problem lies in talks about scalar products, norms, distances of vectors, and orthognality. However, I...
  3. G

    What is the Matrix of Reflection in Euclidean Space?

    Homework Statement V is a three-dimensional euclidean space and v1,v2,v3 is a orthonormal base of that space. Calculate the Matrix of the reflection over the subspace spanned by v1+v2 and v1+2*v2+3*v3 . Homework Equations The Attempt at a Solution To determine the matrix I...
  4. K

    Metric space and subsets of Euclidean space

    I am having some troubles understanding the following, any help to me will be greatly appreciated. 1) Let S1 = {x E R^n | f(x)>0 or =0} Let S2 = {x E R^n | f(x)=0} Both sets S1 and S2 are "closed" >>>>>I understand why S1 is closed, but I don't get why S2 is closed, can anyone...
  5. D

    Are There General Rules for Limits in Multivariable Real-Valued Functions?

    When dealing with real valued functions (one output for now) of more than one real variable, can the usual rules from R --> R be generalised in the natural way? Specifically the sum, product, quotient and composite rules. Any pathological cases? Also I was also wondering if there are any...
  6. M

    Killing vectors on a 3D Euclidean space

    I have read that the Killing vectors in a 3D euclidean space are the 3 components of the ordinary divergence plus the 3 components of the ordinary rotational. I have being trying to find a derivation of this but it isn´t being easy. I really apreciates any clues. Thanks
  7. S

    Euclidean and Non Euclidean Space?

    Hi Can someone explain the difference between Euclidean and Non Euclidean Space and how does one classify a space as Euclidean or Non Euclidean?? I heard about Gauss coming up with Non Euclidean Spaces when he was doing surveying of a piece of land. I am wondering what the word 'FLAT' really...
  8. H

    Flat torus embedding in euclidean space?

    hi, for most of you this might be a simple question: Is it possible to embed the flat torus in Euclidean space? If we, for example, take a rectangle and identify the left and the right sides we get a cylinder shell, that can be embedded easily in R^3. If we construct the...
  9. Loren Booda

    Curvature of reciprocal Euclidean space

    A triangle in Euclidean space can be described as having a hypotenuse of one, and legs of Lorentz parameters \beta and \gamma. What spatial curvature underlies a triangle with hypotenuse one, and legs 1/ \beta and 1/ \gamma?
Back
Top