What is Euclidean space: Definition and 58 Discussions

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.

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1. I Homeomorphism linear subspace

Hi, consider the Euclidean space ##\mathbb R^8## and the projection map ##\pi## over the first 4 coordinates, i.e. ##\pi : \mathbb R^8 \rightarrow \mathbb R^4##. I would show that the restriction of ##\pi## to the linear subspace ##A## (endowed with the subspace topology from ##\mathbb R^8##)...

14. I Partitions of Euclidean space, cubic lattice, convex sets

If the Euclidean plane is partitioned into convex sets each of area A in such a way that each contains exactly one vertex of a unit square lattice and this vertex is in its interior, is it true that A must be at least 1/2? If not what is the greatest lower bound for A? The analogous greatest...
15. Unique smooth structure on Euclidean space

I was doing more reading in John Lee's "Introduction to smooth manifolds" and he mentioned that for every n \in \mathbb{N} such that n \neq 4 , the smooth structure that can be imposed on \mathbb{R}^n is unique up to diffeomorphism, but for \mathbb{R}^4 , there are uncountably many smooth...
16. Propagator in 2D Euclidean space

Homework Statement Consider the following scalar theory formulated in two-dimensional Euclidean space-time; S=∫d2x ½(∂μφ∂μφ + m2φ2) , a) Determine the equations of motion for the field φ. b) Compute the propagator; G(x,y) = ∫d2k/(2π)2 eik(x-y)G(k). Homework Equations Euler-Lagrange equations...
17. Finding all vectors <x,z>=<y,z>=0

Hi Guys, that's what i got <x,z>=<y,z> <x,z>-<y,z>=0 <x,z>+<-y,z>=0 <x-y,z>=0 x-y = [0,2,0] <2*[0,1,0],Z>=0 2<[0,1,0],z> = 0 <[0,1,0],z>=0 So 'im stuck at that. Any ideas?
18. Is R^n Euclidean Space a vector space too?

Dear Physics Forum personnel, I am curious if the euclidean space R^n is an example of vector space. Also can matrices with 1x2 or 2x1 dimension be a vector for the R^n? PK
19. Circle in the Euclidean space using Euler's Number

0 to 1 in Euclidean space. (1 + 1/n)^n using Euler's Number. 1 to 0 with the circle. How amazing is Euler's Number?!
20. Line element in Euclidean Space

The line element is defined as How is dx2+dy2+dz2 be written as gijdqidqj. Is some sort of notation used??
21. Lorentz transf. of a spherical wave in Euclidean space

This thread is not about the lorentz invariance of the wave equation: \frac{1}{c^2}\frac{\partial^2\Phi}{\partial t^2}-\Delta \Phi = 0 It is about an interesting feature of a standing spherical wave: A\frac{\sin(kr)}{r}\cos(wt) It still solves the wave equation above, when it is boosted in...
22. Continuing to Euclidean Space Justified in Path Integral?

It seems to me that in a path integral, since you are integrating over all field configurations, that going into Euclidean space is not valid because some field configurations will give poles in the integrand of your action, and when the integrand has poles you can't make the rotations required...
23. Euclidean space: dot product and orthonormal basis

Dear All, Here is one of my doubts I encountered after studying many linear algebra books and texts. The Euclidean space is defined by introducing the so-called "standard" dot (or inner product) product in the form: (\boldsymbol{a},\boldsymbol{b}) = \sum \limits_{i} a_i b_i With that one...
24. How to draw a 2D space in 3D Euclidean space by metric tensor

Suppose, I know the metric tensor of a 2D space. for example, the metric tensor of a sphere of radius R, gij = ##\begin{pmatrix} R^2 & 0 \\ 0 & R^2\cdot sin^2\theta \end{pmatrix}## ,and I just know the metric tensor, but don't know that it is of a sphere. Now I want to draw a 2D space(surface)...
25. I GR as a Graded Time Dilation Field in Euclidean Space?

The title says it all, really. Are we able to describe GR in terms of a Graded Time Dilation Field in Euclidean space? From http://cpl.iphy.ac.cn/EN/Y2008/V25/I5/1571 we can see that light curvature can be analogously described via a material with a graded index refraction, so my question is...
26. MHB Euclidean Space of Polynomials

Hi everyone, :) Here's a question I encountered and I need your help to solve it. Question: Let $$V$$ be the space of real polynomials of degree $$\leq n$$. a) Check that setting $$\left(f(x),\,g(x)\right)=\int_{0}^{1}f(x)g(x)\,dx$$ turns $$V$$ to a Euclidean space. b) If $$n=1$$, find...
27. MHB Orthogonal Transformation in Euclidean Space

Hi everyone, :) Here's one of the questions that I encountered recently along with my answer. Let me know if you see any mistakes. I would really appreciate any comments, shorter methods etc. :) Problem: Let $$u,\,v$$ be two vectors in a Euclidean space $$V$$ such that $$|u|=|v|$$. Prove that...
28. Is Euclidean space an affine space?

Hi everyone, I have a question that I'm not sure about. I wanted to know if it is standard to think of Euclidean space as a linear vector space, or a (more general) affine space? In some places, I see Euclidean space referred to as an affine space, meaning that the mathematical definition of...
29. Connection forms on manifolds in Euclidean space

This question comes from trying to generalize something that it easy to see for surfaces. Start with an oriented surface smoothly embedded in Euclidean space. The embedding determines two mappings of the unit tangent circle bundle into Euclideam space. Given a unit length tangent vector,e, at...
30. Completeness of a set of basis vectors in 3D Euclidean space.

Homework Statement The problem is Exercise 2 in the picture http://postimage.org/image/3ou3x1sh7/ Homework Equations The hint says: can you express and three-dimensional vector in terms of just two linearly independent vectors? The Attempt at a Solution I have no idea where...
31. Embedding hyperbolic constant-time hypersurface in Euclidean space.

In Bernard Schutz's 'A first course in General Relativity', p325 (1st edition) he says " [the constant-time hypersurface of a FLRW spacetime with k=-1 (hyperbolic)] is not realisable as a three-dimensional hypersurface in a four- or higher-dimensional Euclidean space." On the face of it...
32. Independent fields in euclidean space

If the Lagrangian is Hermitian, then fields and their complex conjugates are not independent. That is, you can solve the EOM for one field, and if you take the complex conjugate of that field, then that's how the complex conjugate field evolves: you don't have to solve the Euler-Lagrange...
33. Wormholes in Euclidean Space?

As it would appear the universe is spatially flat, a Euclidean Plane. If this is true then how could black holes exist? Doesn't this necessitate that if black holes are embedded in flat space that the mean curvature must be zero and thus all black holes are minimal surfaces? So, if black holes...
34. Euclidean Space definition

Hi, I'm trying to fix in my head a very precise definition of what to mean for an euclidean space, as we use it in multivariable calculus. The def. I had in my mind was that an ES is (1) a real vector space (2) of finite dimension (3) with the "standard" (dot) (4) inner product I'm...
35. What is a realization of this surface in Euclidean space?

I am interested to know how to realize this abstract surface as a subset of Euclidean space. The surface as a point set is the 2 dimenional Euclidean plane minus the origin. the metric is given by declaring the following 2 vector fields to be an orthonormal frame: e_{1} = x\partialx -...
36. Relations of an affine space with R^n , and the construction of Euclidean space

(This could maybe turn out to be a little longer post, so I'll bold my questions) Hi, I was reading a little about affine geometry, and something bothered me. Namely, in some books, there were some paragraphs that were written like "blabla, let's observe an affine plane for instance, and...
37. Thought - Euclidean Space R^(-n)

While R^1, R^2, ... , R^n comes quite naturally, is it even conceivable to ponder the meaning of R^(-n)? Is this something that even can exist conceptually or is it just jibberish? This was just a random thought that rolled into my head earlier today, and it's something that I think COULD...
38. What is Euclid's Euclidean space?

The word Euclidean space is applied to various distinct mathematical objects. One, kind of Euclidean space is the affine space (general sense of "affine space") defined by the Euclidean group of isometries, which don't including scaling. But wouldn't Euclid's axioms apply equally well if we...
39. Hypersurfaces of Euclidean space

It seems that the tangent bundle of a hypersurface of Euclidean space is the bundle induced from the tangent bundle of the unit sphere under Gauss mapping. Is this true? The reason I think this is that tangent space at a point on the surface can be parallel translated to the tangent space on...
40. Can Complex Euclidean Space Be Defined?

Homework Statement Euclidean space is the set of n-tuples with some operations and norm. I suddenly wonder if complex euclidean space can be defined. Is it also defined? Homework Equations The Attempt at a Solution
41. Calculate Killing Vectors in 3-D Euclidean Space

Homework Statement My problem is to calculate to calculate killing vectors in 3-D euclidean space(flat space). Homework Equations The relevante equations are killing equation : d_a*V_b+d_b*V_a=0 The Attempt at a Solution I found the solution in Ray D'Inverno and that is...
42. Euclidean space, euclidean topology and coordinate transformation

Hi, I have some doubts about the precise meaning of Euclidean space. I understand Euclidean space as the metric space (\mathbb{R}^n,d) where d is the usual distance d(x,y)=\sqrt{\sum_i(x_i-y_i)^2}. Now let's supose that we have our euclidean space (in 3D for simplicity) (\mathbb{R}^3,d)...
43. Length Contraction Euclidean Space

Hi, Suppose a stationary frame S' is observing frame S moving with velocity v=0.866c in the x-direction, and let points (4,0),(10,0) define the ends of a rod in S, so its distance is 6, but as measured from S' contracts to 3 because of the Lorentz factor gamma. I'm unable to determine...
44. Topological space, Euclidean space, and metric space: what are the difference?

Hello my friends! My textbook has the following statement in one of its chapters: Chapter 8:Topology of R^n If you want a more abstract introduction to the topology of Euclidean spaces, skip the rest of this chapter and the next, and begin Chapter 10 now. Chapter 10 covers topological...
45. Our perception of Euclidean space

Ok so I was just thinking and realized that instead of a Cartesian plot, you can represent points in an n-dimensional space by drawing n parallel lines and marking a point on each line. Of course this is less appealing than the traditional plot because we perceive 3d space in a way more similar...
46. Differentiation on Euclidean Space (Calculus on Manifolds)

Homework Statement This is from Spivak's Calculus on Manifolds, problem 2-12(a). Prove that if f:Rn \times Rm \rightarrow Rp is bilinear, then lim(h, k) --> 0 \frac{|f(h, k)|}{|(h, k)|} = 0 Homework Equations The definition of bilinear function in this case: If for x, x1, x2...
47. How to determine if a set is an open subset of a Euclidean space?

I opted to not use the template because this is a pretty general question. I am not understanding how to find out if a set is an open subset of a Euclidean space. For example, {(x,y) belongs R2 | x squared + y squared < 1} The textbook is talking about open balls, greatly confusing me.
48. Properties of a volume in 3D Euclidean space

Hello, I am writing a small report and trying to be mathematically accurate in my terminology- I am trying to describe an arbitrary volume of gas, but this volume must (1) not have any holes (or bubbles where the gas cannot go) in it, and (2) must be one single volume, so a gas molecule from...
49. Canonical measure on an infinite dimensional Euclidean space R^N

I just encountered the Wikipedia page There is no infinite-dimensional Lebesgue measure, and I was left slightly confused by it. They say that a Lebesgue measure m_n on \mathbb{R}^n has the property that each point x\in\mathbb{R}^n has an open environment with non-zero finite measure, and then...
50. 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!