Euclidean Definition and 204 Threads

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...

Hi, i would be very interested to start learning hyperbolic geometry, what would be the necessary prerequisites to begin it's study? :smile:

Hello everyone, my question is about Euclidean correlators (say a 2-pt function to be specific) in presence of non-zero chemical potential. The question in particular is: is it still true that the Minkowski time ordered 2-pt function can be simply obtained from the Euclidean one by analytic...

Reading a somewhat long and argumentative thread here inspired the following unrelated question in my mind: Where does a 2 dimensional flat Lorentzian geometry depart from Euclidean geometry as axiomatized by Euclid? I.e. Euclid's axioms (in modern language) can be taken to be: We can...

I'm working with the signature ##(+,-,-,-)## and with a Minkowski space-stime Lagrangian ## \mathcal{L}_M = \Psi^\dagger\left(i\partial_0 + \frac{\nabla^2}{2m}\right)\Psi ## The Minkowski action is ## S_M = \int dt d^3x \mathcal{L}_M ## I should obtain the Euclidean action by Wick rotation. My...

The Euclidean classical action ##S_{\text{cl}}[\phi]## for a scalar field ##\phi## is given by \begin{equation} S_{\text{cl}}[\phi]=\int d^{4}x\ \bigg(\frac{1}{2}(\partial_{\mu}\phi)^{2}+U(\phi)\bigg). \end{equation} This can be obtained from the action ##S_{\text{Mn}}[\phi]## in Minkowski...

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...

I'm seeing a presentation of Euclidean geometry that isn't hand-holdy. I've looked at some textbooks used in high schools these days, and it's hard to find the axioms and theorems in the midst of all the condescension. I just want something that states the definitions, axioms and basic...

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...

Homework Statement Let m and n be natural numbers. Suppose ## min(m, n) \geq 2^k ## for some natural number k. Show that the Euclidean Algorithm needs at most ## 2k ## iterations to calculate the gcd of m and n.The Attempt at a Solution [/B] So far I think I need to show that for all the...

Exercise 1.6 in Greenberg's Euclidean and Non-Euclidean Geometries: "Betweenness" and "Lying On" Undefined terms: point, line, lie on, between, and congruent. Postulate I. For every point $P$ and for every point $A$ not equal to $P$ there exists a unique line $\ell$ that passes through $P$ and...

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

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?!

I am reading Joseph J. Rotman's book: A First Course in Abstract Algebra with Applications (Third Edition) ... I am currently focused on Section 3.5 From Numbers to Polynomials ... I need help with Example 3.76 ... ... the example concerns Euclidean rings and their defining characteristics so...

Hi, i have some trouble with feynman rules after wick's rotation. I don't understand how the propagators transform. In particular if i take the photon's propagator in minkowskian coordinates i don't understand where the factor "-i" goes after the transformation. ## \frac{-i\eta_{\mu\nu}}{p^2}...

The line element is defined as How is dx2+dy2+dz2 be written as gijdqidqj. Is some sort of notation used??

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...

This is the Euclidean classical action ##S_{cl}[\phi]=\int d^{4}x\ (\frac{1}{2}(\partial_{\mu}\phi)^{2}+U(\phi))##. It would be nice if somebody could explain the structure of the potential. I don't understand why ##\phi## is used instead of a position vector ##\textbf{r}##. Also, how can...

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...

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...

If we choose rational numbers to represent points on a line then there will be gaps on the line and consequently the plane will be full of holes. Then we cannot say that two non-parallel line must intersect on a point (because they may meet at the gaps). So obviously we need point arranged more...

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)...

OK. A problem is simple and probably very stupid... In many (most of) maths books things like Euclidean metric, norm, scalar product etc. are defined in Cartesian coordinate system (that is one which is orthonormal). But what would happen if one were to define all these quantities in...

Hi, Just wondering when using the Euclidean Algorithm to find gcd of 4+7i and 1+3i. Where does 2 and 2+i come from in the follwoing? 4+7i = 2*(1+3i)+(2+i) 1+3i=(1+i)*(2+i) +0? I know you didvide them to get (5-i)/2 and then take closest Gaussian integer then not sure where to go.

Is there a way to map time-like curves in Minkowski space to curves in a Euclidean space such that the length of the curve in the Euclidean space is equal to the proper time of the curve in Minkowski space?

Hello! :D I am given the following exercise: In $\mathbb{Z}_7$ apply the Euclidean division, dividing $[2]x^5+x^4+x^3+[3]x^2+[2]x+[2]$ by $[3]x^2+[2]x+[3]$. That's what I have done: $$\text{ the units of } \mathbb{Z}_7 \text{ are: } \{ 1,2,3,4,5,6\}$$ We want $[3]x^2+[2]x+[3]$ to be monic,so we...

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...

What is the difference between the normal in Minkowski spacetime and the normal in Euclidean space?

Hey! :rolleyes: I am looking at the following exercise: "Prove that $\forall n \in \mathbb{N}$ the number $17 \cdot 3^{2n+1}+41 \cdot n^2 \cdot 3^{n+1}+2$ " is not a square of an integer." We do it like that: $a(n)= 17 \cdot 3^{2n+1}+41 \cdot n^2 \cdot 3^{n+1}+2=3 \cdot (17 \cdot 3^{2n}+41...

Well I am not sure if this thread belongs here or in mathematics/groups but since it also has to do with physics, I think SR would be the correct place. An element of the Euclidean group E(n) can be written in the form (O,\vec{b}) which acts: \vec{x} \rightharpoondown O\vec{x}+\vec{b} With O...

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...

Think for example of the torus as a square with the proper edges identified. Viewed like this (i.e. using the flat metric), it clearly has zero curvature everywhere. More specifically, it seems Euclid's axioms are satisfied. But however we have non-trivial topology. So what's up? Or is...

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...

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...

Homework Statement LPN is a tangent to circle ADP. Circle BCP touches the larger circle internally at P. Chord AD cuts the smaller circle at B and C and BP and CP are joined Homework Equations The Attempt at a Solution ∠P4+5 = ∠B1 (tan chord theorem) ∠P1+2 = C1 (tan chord theorem)

Hello everybody, Based of some information that I recently learnt(which I don't know if they are right or wrong), I start asking myself this question about the euclidean geometry. Ok, this geometry is basically founded on straight lines, and what I have learned is there is no such a thing as a...

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...

Hi, I'm a Physics undergraduate, and this semester I have the option to choose between Geometry (Axiomatic Euclidean Geometry) and other disciplines. In the next year I want to be ready to study Differential Geometry, but I don't know if I need to study Euclidean Geometry first. The teacher of...

Using the Euclidean algorithm...I think... find the smallest natural number x such that 24x leaves a remainder of 2 upon division by 59 SO it seems to me that the way to approach this would be through the euclidean algorithm and a diophantine equation. Thinking about it for a moment would...

What is the difference between the Euclidean Geometry and the simple plane geometry? They seems to work with flat planes.

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...

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...

Homework Statement Suppose that u, v ∈ Z and (u,v) = 1. If u | n and v | n, show that uv | n. Show that this is false if (u,v) ≠ 1. Homework Equations a | b if b=ac [b]3. The Attempt at a Solution I understand this putting in numbers for u,v, and n but I don't know how to...

Homework Statement Let X:=ℝn with the Euclidean Topology. Is X first countable? Find a nested neighborhood basis for X at 5. Homework Equations If X is a topological space and p\inX, a collection Bp of neighborhoods of p is called a neighborhood basis for X at p if every neighborhood...

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...

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...

Homework Statement Calculate the geodesic for euclidean polar coordinates given ds^{2}=dr^{2}+r^{2}dθ^{2} Homework Equations standard euler-lagrange equation The Attempt at a Solution I was able to reduce the euler-lagrange equation to \frac{d^{2}r}{dθ^{2}}-rλ=0 where...

Is it necessary to know the proofs of all the propositions in Euclid's elements? Or one can directly jump to euclidean and precollege geometry, of course he/she will have knowledge of propositions( just the statements and all that) but not the proofs. Reply Quickly.

I am trying to show that $\mathbb{Z}[\sqrt{-3}]$ is not a euclidean domain, now I know that in every euclidean domain we have that an element is prime iff it is irreducible so I need to find an irreducible element of $\mathbb{Z}[\sqrt{-3}]$ that is not prime, I can't seem to think of one...

A book I'm reading says that the set of continuous functions is an Euclidean space with scalar product defined as <f,g> = \int\limits_a^bfg and then defines Fourier series as \sum\limits_{i\in N}c_ie_i where c_i = <f, e_i> and e_i is some base of the vector space of continuous functions. What...