Euclidean Definition and 214 Threads

Use the Euclidean Algorithm to find the gcd of the given polynomials: (x3-ix2+4x-4i)/(x2+1) in C[x] First I got x-i R: 3x-3i, then I took the 3x-3i into x2+1 & got 1/3 x R: 1+i. Then I was going to take 1+i into 3x-3i. However that never ends it seems, unless I just confused myself...

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...

What is the famous theorem in Euclidean geometry that states that the sum of the interior angles of a triangle is 180 degrees?

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...

In this thread, I plan to try to explain (with some apropos ctensor examples) in a simple and concrete context some basic techniques and notions about Riemannian two-manifolds which also apply to general Riemannian/Lorentzian manifolds. Suppose we have a euclidean surface given by a C^2...

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.

Hi! I have a question regarding the renormalization group Beta function, i.e., \beta = \mu \frac{dg_R}{d \mu} where g_R is the renormalized coupling constant and \mu the renormalization scale. My question in a nutshell: are the Beta functions calculated for QFT and, respectively...

Homework Statement Find a positive integer solution to 1234x-4321y=1, both x and y will be positive. Homework Equations The Attempt at a Solution I created this array 4321 1234 619 615 4 3 1 3 1 1 153 1 1082 309 155 154 1 1...

Homework Statement Hey guys I need help with a lab I'm doing that is similar to the Milikan's experiment. I am given 10 bags each holding the same item (Jellybean) of various quantities. Each bag has a different mass. What I'm trying to figure out is the mass of the individual item, so mass...

Homework Statement I was just wondering how I would go about proving that the euclidean metric is always smaller than or equal to the taxicab metric for a given vector x in R^n. The result seems obvious but I am not sure how I would show this. Homework Equations The Attempt at a Solution

How do you prove that for a set of coordinates you are supposed to take \mathrm{d}s^2=\mathrm{d}x_i\mathrm{d}x^i for the distance? I mean in a very abstract fashion. All I know is that there is some coordinate mesh. Why don't I take other powers for the distance for example? Or if that...

Homework Statement This is a problem from baby rudin in Chapter 1, I've done all of them except this(problem 16). suppose k>=3,x,y belongs to [R][/k],|x-y|=d>0,and r>0.prove: (1)if 2r>d, there are infinitely many z belong to R(k) such that |z-x|=|z-y|=r. (2)if 2r=d, there is exactly one...

Prove that the number of steps of the euclidean algorithm needed to find gcd(km,kn) is exactly the same as the number of steps needed to find gcd(m,n). any help on this would be appreciated. I'm really lost.

Benjamin Crowell writes here, "The discontinuous transformations of spatial reflection and time reversal are not included in the definition of the Poincaré group, although they do preserve inner products." http://www.lightandmatter.com/html_books/genrel/ch02/ch02.html So, if I've...

Where is it proven that the unit of area in Euclidean geometry must be a square with side=1? Or is it an axiom? Why not triangles or circles to represent area?

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...

Considering that space is not curved or warped (as some pop books will falsely lead you to believe) why is that Euclidean Geometry is not true in the real world? I mean light bends in space because it falls in a gravitational field like everything else (because it has energy which is...

given sequences \left\{x_n\right\}, \left\{y_n\right\}, is it true that \sqrt{ \Sigma_{n=1}^{\infty} (x_n - y_n)^2} \leq \Sigma_{n=1}^{\infty} |x_n - y_n| this isn't a homework problem. it's just something that came up - I think it's pretty clear that it's true, but I don't know how to show...

Homework Statement The following is a worked example, I circled around the part which I couldn't follow: http://img15.imageshack.us/img15/161/untitleou.jpg Homework Equations The Attempt at a Solution To begin with, I can't understand why they wrote: x-1 =...

[b]1. Homework Statement http://img195.imageshack.us/img195/5122/200282.gif Homework Equations The Attempt at a Solution 8.2.1) Let D1 = x D4=D1 =x D4=L1 (tan chord theorem) L1=x D1=L2 L2=x angle KLM=2x KNM=2x(opp angles in //gram)...

"For example, Euclidean geometry without the parallel postulate is incomplete; it is not possible to prove or disprove the parallel postulate from the remaining axioms." http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems The parallel postulate says that, if a line segment...

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...

1)5x=1(16) is equivalent to x=5(6) is equivalent to x=1(2), x=2(3) <the equal sign here i mean congruence to> i'm a bit confused about the equivalence...how this is so? 2)3k-7n=1, k,n integers by using euclidean algorithm i got k=-2, n=-1. but the answer i got here is k=5, n=2 (the...

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!

I have this question: Inside a square ABDE, take a point C so that CDE is an isosceles triangle with angles 15 degrees at D and E. What kind of triangle is ABC? I put C close to the bottom to get my isosceles triangle. According to the answer in back, the triangle ABC is equilateral...

see the attachment please answer

I have this question which is rather simple, basically reiterating a general theorem. Show that S={z in H||z-i|=3/5} is a hyperbolic circle S={w in H| p(w,w0)=r} for r>0 and find sinh(r/2) and w0. Now to show that it's hyperbolic is the easy task, I just want to see if I got my...

hey guys im trying to prove a fact. it is supposed to be really easy but I am having some trouble. this is it: http://en.wikipedia.org/wiki/Topological_manifold "It follows from invariance of domain that Euclidean neighborhoods are always open sets." Invariance of Domain -...

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...

A general question. A unit element is one that has it's multiplicative inverse in the ring. An element p is prime if whenever p=ab then either a or b is a unit element. Can a prime be a unit element? The answer is, i think, no but thus far I've been unable to find a contradiction.

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...

I would say by now, I'm an expert in manipulating equations and playing with algebra. However, I've also realized I have no idea why some of the operations I do are valid. For example... why is (x+2)(x-2) = x^2 - 4? Why does this expansion work? I'm guessing it preserves some kind of field...

Hi I am currently studying Information Theory. Could I please have anyone's ideas on the following question: Using the Euclidean algorithm, show that coprime numbers always have multiplicative inverses modulo each other. I tried the following proof, using Fermat's little theorem, let me...

I have been wondering now for quite some time about the meaning of Euclidean Quantum Field Theory. The Wick rotation t\to it allows us to transform a QFT in Minkowski space to a QFT in Euclidean space (positive definite metric). After that the expectation values of observables can be...

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...

I am teaching euclidean geometry this fall and realized i don't know it that well. there are some famous modern versions of the axioms which do not completely satisfy me, such as hilberts, gasp. i said it. i especially like the new book by hartshorne, geometry euclid and beyond, because he...

Homework Statement If F is a field than does it imply it must also be a Euclidean domain?The Attempt at a Solution Yes since for any a,b in F. a=bq for some q in R. In fact let q=(b^-1)a. So the remainder which occurs in a ED is always 0. So the rule for being a ED is satisfied in any field.

Well I am doing a minor project on dimensions and probablity.Please friends try this out:----------- A coin of diameter d is tossed randomly onto the rectangular cartesian plane . What is the probablity that the coin does not intersect any line whose equation is of the forms :------- (a)...

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...

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

While I'm on the topic, here is another ring I need to show Euclidean. I'll show more of the work this time too. The ring is Z[{\sqrt 2 \over 2}(1+i)] So, using the standard norm difference approach, we pick any element alpha in the field and try and show we can always find an element beta in...

Let \displaystyle{\zeta = e^{{2\pi i} \over 5}} I need to show that Z[\zeta] is a Euclidean ring. The only useful technique I know about is showing that given an element \epsilon \in Q(\zeta) we can always find \beta \in Z[\zeta] such that N(\epsilon - \beta) < 1 (using the standard norm for...

i am learning path integral for quantum field theory, and my professor used euclidean time (imaginary time) and most textbooks use minkowski time. does actually changing the time from real (minkowski) into euclidean (imaginary) CHANGE the physics in some way?

Hi folks, I have to find the generalized solution for the following Ax=y : [1 2 3 4;0 -1 -2 2;0 0 0 1]x=[3;2;1] The rank of A is 3 so there is one nullity so the generalized solution is: X= x+alpha.n (where alpha is a constant , and n represents the nullity) I found the...

http://img82.imageshack.us/img82/4458/divisonfx9.jpg

Speculation for helical mechanics in space-time. These illustrations may demonstrate [not prove] how Euclidean space [anthropic perspective] may coexist with hyperbolic space of Gaussian curvature and elliptic space of Riemannian curvature. [The hyperbola is reciprocal to the ellipse in...

I wanted to know if euclidean geometry is to do with the real world ? generalisations of vector space to anything that satisfies the axioms for a vector space can be made, but how can geometry be studied without reference to the real world ? roger

Hi to everynoe! I have a bit of trouble in understanding the following thing : Suppose we have a phase space, in which a dynamical system evolves: for example a two dimensional vector space: temperature and time. Now, does it make a sense to define the euclidean norm of a vector in such...

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...

Abstract A Euclidean interpretation of special relativity is given wherein proper time \tau acts as the fourth Euclidean coordinate, and time t becomes a fifth Euclidean dimension. Velocity components in both space and time are formalized while their vector sum in four dimensions has invariant...