Transformations Definition and 823 Threads

Hi, I am currently working on problem to do with conformal mappings and Mobius Transformations. My problem is: Find the Mobius Transformation which carries the points 1, i, 0 to the points ∞, 0, 1 (in precisely this order). Find the image of the domain { z : 0 < x < t} under this Mobius...

The Laplace of 1 is: $$\int_{0}^{\infty} 1 \exp(-st) dt = \left[ \frac{\exp(-st)}{-s} \right]_{0}^{\infty} = \frac{\exp(-s \infty) - \exp(-s 0)}{-s} = \frac{0 - 1}{-s} = \frac{1}{s}$$ It's result known, however, for this be true is assumed that s>0, because 0 = exp(-∞) = exp(-s∞). But we...

Homework Statement The matrix A transforms space by rotating counterclockwise around z axis by ∏/4. -What is A? -Find a unit vector s.t. Au=u (make sure the first nonzero is positive) -for what λ's does Au = λu? write answer as a set {a,b,...} Homework Equations Unit vector u =...

Helloo, I don't understand how one arrives at the conclusion that the hamiltonian is a generating function. When you have an infinitesimal canonical transformation like: Q_{i}=q_{i}+ \delta q_{i} P_{i}=p_{i}+\delta p_{i} Then the generating function is: F_{2}=q_{i}P_{i}+ \epsilon...

Hi all, I'm sort of struggling to understand the difference between active and passive transformations. I'm working with two books, and both say that in one case the vectors are transformed (active) and in the other the operators are transformed (passive). That's fine and dandy, but in terms...

I'm currently going through my courses notes for relativity. We looked at Einsteins two postulates and then said that time must therefore dilate due to constant speed of light. That I understand, however I'm still confused about the Lorentz's transformations. My notes start with a basic form of...

I got the right answer for this example problem going from reference frame A to B but when I use those fields to go back from B to A I don't get the same magnetic field I started with. Do field transformations only work one way? Surely not? I don't see how forces could be the same if this...

I understand how contravariant 4-vectors transform under a Lorentz transformation, that is: ##x'^μ= \Lambda^\mu~_\nu x^\nu## [1] and how covariant 4-vectors transform: ##x'_\mu=(\lambda^{-1})^\nu~_\mu x_\nu##. [2] Now, I have come across the following relations...

Homework Statement Find all linear transformations ##f(z)=az+b## which map half-plane ##Im(z)>0## on ##Im(z)>0##. It is a so called self-mapping transformation. Homework Equations The Attempt at a Solution I am guessing this will have something to do with Möbius transformation...

Trying to see the logic in deriving length contraction and time dilation using the Lorentz transformations and inverse Lorentz transformations. In the following treatise it leads to ambiguities. Given ##Δ\acute{t}=\gamma(Δt-\beta c^{-1}Δx)## (1) ##Δ\acute{x}=\gamma(Δx-\beta c Δt)##...

Homework Statement Hello everyone, I have a quick question about linear transformations. In my class, we were given transformation functions and asked to decide if they are linear: The transformation defined by: T(X)= X1+X2+3 The transformation defined by: T(X)=X1+X2+(X1*X2) The...

I am using a wikipedia page, Derivation of the Lorentz transformations and a lot of historical papers. To follow through I came up with my own transformations that do not contain the gamma factor: ##x^{'}=x-\beta ct## ##t^{'}=t-\beta \frac{x}{c}## When applying them to a waveform...

Homework Statement A rod of rest length 1,0m is moving longitudinally on a smooth table with a velocity 0,8c relative to the table. A circular hole of rest diameter 1,0 m lies in its path. The diameter of the hole as seen by the rod is going to be larger?Homework Equations The Attempt at a...

I know that the spacetime in special relativity is not curved and that the axis can be transformed via the lorentz transformations. I was wondering if the curved spacetime in general relativity can be transformed in such a way, and if so, how?

Hello All! A recent problem has stuck with me, and I was hoping you could help me resolve it. Consider the following premise: Let us assume that X \sim \mathcal{U}(-3,3) (U is the continuous, uniform distribution). And let the transformation Y be applied thus: Y = \left\{ \begin{align*} X+1...

Homework Statement Let V be the set of complex numbers regarded as a vector space over the real numbers R. Find a linear transformation T: V → V which is not complex linear (i.e. not a linear transformation if V is regarded as a vector space over the complex numbers). Homework Equations...

Just a little doubt. When we are performing a canonical transformation on a Hamiltonian and we have the equations of the new coordinates in terms of the old ones we have to find the Kamiltonian/new Hamiltonian using K = H +\frac{\partial G}{\partial t}. My question is: do we have to derive the...

So every Möbius transformation of the complex plane is holomorphic and 1-to-1 on the Riemann sphere. Is the converse also true, or are there counter-examples?

I am confused about using horizontal transformations such as f(x+a) and f(x-a) to interpret these graphs.

Hello all, I need some help to clear my doubts. Why does a horizontal translation (f(x + c)) move to the left if c is positive? Can someone graphically explain what effect a stretch and compression (vertical and horizontal) has on the original parent function? Similar to the first...

For example, I would like to visualize the transformation u=cot-1(1/α y/x) v=αx2+y2/α

I am a bit confused about something! Exactly under what kind of transformations are scalars invariant in the domain of classical mechanics? The fact which is disturbing me is, say we have a moving body of certain kinetic energy in a certain inertial frame of ref, and then we choose to.observe...

Hi everyone, :) Here is a question I encountered recently. Question: Let \(V\) be a unitary space. Give the definitions of a self adjoint and unitary linear transformations of \(V\). Prove that \(f_1 g_1=f_2 g_2\), where \(f_1,\,f_2\) are self adjoint positive and \(g_1,\,g_2\) unitary...

So a mobius transformation is defined as \frac{az+b}{cz+d}=f(z). Where ad-bc≠0. My question is just deriving this condition ad-bc≠0. I understand that the condition describes the case were the mapping leaves all points unchanged. This is described as undefined in some textbooks...(why...

Homework Statement . 1) Let ##T## be the transformation over the extended complex plane that sends the poins ##0,i,-i## to the points ##0,1,\infty##. Prove that the image of the circle centered at the origin and of radius ##1## under this transformation is the line ##\{re(z)=1\}##. 2) For...

Conformal transformations as far as I knew are defined as g_{mn}\rightarrow g'_{mn}=\Omega g_{mn}. Now I come across a new definition, such that a smooth mapping \phi:U\rightarrow V is called a conformal transformation if there exist a smooth function \Omega:U\rightarrow R_{+} such that...

Undergrad studying engineering here, and my physics class has been doing a unit about intro to special relativity. Essentially, all of our problems and studies concern themselves with velocities which are in the +x direction relative to a "home frame" (I think physicists call this standard...

hello Let me start by saying that is my 1st time posting on the forum an so I'm not sure if I should post this here or on the homework/coursework section. This is technically coursework related but doesn't seem to fit the "model" used in that section (it is not specific enough), so if this is...

I need some help with a derivation in GR. The linearized field equation in GR is: G_{ab}^{(1)} = - \frac{1}{2}{\partial ^c}{\partial _c}{{\bar \gamma }_{ab}} + {\partial ^c}{\partial _{(b}}{{\bar \gamma }_{a)c}} - \frac{1}{2}{\eta _{ab}}{\partial ^c}{\partial ^d}{{\bar \gamma }_{cd}} = 8\pi...

I've been up way too long, so pardon me if this doesn't make sense, but.. Let V and W be vector spaces. Let T and U be linear transformations from V to W. Consider the set of all x in V such that T(x) = U(x) 1.) I think that this is a subspace of V. 2.) Can I say anything about its dimension...

How do you show that a set of linear transformations from one vector space to another spans L(V,W)? This isn't a homework question, or even a question that's in the text I'm reading (Friedberg).

I was looking over my notes today, and I realized that there was a point that isn't pretty clear. If we have the image under T (being T a matrix transformation induced by a matrix A) of the unit square, then its area should be abs(det(A)). Why is this though? I was looking at the proof and I...

Homework Statement (ii) S ◦ T will be a linear transformation from P4 to R2. Write a formula for the value S(T (a4t4 + a3t3 + a2t2 + a1t + a0)) using the given formulas for T,S and use this to compute the matrix [S ◦T]B′′,B. (10p) B'' = {e1 e2} B' = {t4, t3, t2, t,1} T: P4--> M2x2 T(a4t4 +...

Let F_{K}: \hat{K} \to K be defined as follows: F_{K}(\hat{x},\hat{y}) = B_{K}\left[\begin{array}{c} \hat{x}\\ \hat{y}\\ \end{array}\right] + b_{K} i.e. F_{K} maps from (\hat{x},\hat{y}) to (x,y). In a more concrete sense, for this example take the following: B_{K} =...

Few days ago, I was thinking about why we need to define V*=Hom(V,K) for a K-vector space when the dimension of V is finite because then V* and V both will have the same dimension and will be isomorphic. So, I couldn't understand why such a thing would be even called a dual vector space if it's...

Hi everyone, :) Here's a question and I'll also write down the answer for which I got zero marks. :p I would really appreciate if you can find where I went wrong. Question: Let \(\phi,\,\psi\in V^{*}\) be two linear functions on a vector space \(V\) such that \(\phi(x)\,\psi(x)=0\) for all...

I don't understand when I should use the Lorentz transformation versus time dilation or length contraction. I found this: http://www.phas.ubc.ca/~mav/p200/lttips.html but it's still unclear to me... "Length contraction applies when you are talking about a distance that is independent...

Homework Statement Show that a linear map T:R4->R4 has one of the following as its image: just the origin 0, a line through 0, a plane through 0, a copy of R3 through 0, or all of R4. Homework Equations N/a The Attempt at a Solution I'm not sure I'm even understanding the...

Please refer to the attached image (sorry for the bad cropping, they were on separate pages) I don't get what is meant by "two finite points''. Are these any two points which aren't equal to infinity? Could i have chosen f(1) and f(2) ? what am i not understanding. IF someone could explain...

Homework Statement ##{\Lambda_c}^b## is a Lorentz transformation and ##{\Lambda^c}_b## is its inverse, so ##{\Lambda_c}^b {\Lambda^c}_b## gives an identity matrix. How can I write this, assuming it's possible, in terms of ##\delta##'s ? Homework Equations The Attempt at a...

1. Give information Let T: P3 ---> P3 be the linear transformation described by: T(p(x))=p(x+1)+p(2-x). Find the matrix of T with respect to the standard basis b {1,x,x^2,x^3}. The Attempt at a Solution I found the transformations on the standard basis b: T(1) = 2 T(x) = 3 T(x^2) =...

1. Consider a four vector x^{\mu}, that is timelike (i.e x^{2}>0. show that it is always possible to find a frame where the coordinates of x are of the form (x^{0'},0). Determine the lorentz transformation relating the initial frame to this particular frame 3. I figured that assuming that the...

Hi everyone, :) Take a look at this question. Now the problem is that I feel this question is not properly worded. If the linear transformations have rank = 1 then it is obvious that \(\mbox{Im f}=\mbox{Im g}=\{0\}\). So restating that is not needed. Don't you think so? Correct me if I am...

Here is the question: Here is a link to the question: Let {e1, e2, e3} be a basis for the vector space V and T: V -> V a linear transformation.? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Homework Statement A certain practical dc voltage source can provide a current of 10.5 A when it is (momentarily) short- circuited, and can provide of 30 W to a 20 Ω load. Find the open-circuit voltage Homework Equations V=IR The Attempt at a Solution Well I'm pretty sure that...

Hi, hope this is a right place to ask this question. I work in the soil physics field and this problem has taken lots of my energy for a while! let's state it: Unsaturated horizontal water flow in 2 layer soil: we have, M(for Moisture), K (for hydraulic conductivity), h (for hydraulic...

Hello! Are the transformations such as time dilation, length contraction and relativistic mass just observed ones or real ones? Thank you.

Say I have a linear transformation T:V##\rightarrow##W. Can I necessarily say that T(V)##\subseteq##W? I feel like T being a linear transformation would make the function behave enough to force things to not be undefined but I can't be certain..

1.Hey, I am rather stuck on this question which you can see in the attached PDF. Now I began by taylor expanding the Lorentz Gamma factor (γ), up to second order and inserting this into the equation wherever I saw the gamma function, then rearranging. But I can't seem to get a function for F...

βI solved this problem but I do not know if it is correct becasue there is no way to check it: Imagine that we define the rear end of a train 120 m long to long to define the origin X'=0 in the train frame and we define a certain track signal light to define the origin X=0 in the track frame...