Transformation Definition and 1000 Threads

Homework Statement For an event occurring at (x,t), consider the quantity I = x^2 - (ct)^2 Find a simple expression for this in the S' frame: I' = x'^2 - (ct')^2 How are I and I' related, and why is this noteworthy? The Attempt at a Solution So the question is under "Lorentz Transformation"...

So the question looks like this: Identify the time and/or amplitude transformations that must be applied to the signal x(t) in order to obtain each of the signal specified below. Choose the transformations such that time shifting precedes time scaling and amplitude scaling precedes amplitude...

Homework Statement [/B] A spaceship is approaching Earth from the far side of the sun. The Earth and sun are 8 light minutes apart and the ship is traveling at .8c. Two events are indisputable. 1) the ship is at the sun 2) the ship is at the earth. Assume that the Earth and sun are at rest...

Homework Statement (From Di Francesco et al, Conformal Field Theory, ex .2) Derive the scale factor Λ of a special conformal transformation. Homework Equations The special conformal transformation can be written as x'μ = (xμ-bμ x^2)/(1-2 b.x + b^2 x^2) and I need to show that the metric...

I'm getting quite stuck on this problem here. Galileo said that Xb = Xa - V*Ta. (This follows from dv = dx/t --> Xa - Xb = t*dv --> the above formula) Thus, it is concluded Xa = Xb + V*Ta, but why? In my thought experiment the objects are moving relative to each other, thus if A is moving away...

Homework Statement A rocket is traveling toward a galaxy with speed v. a) If NASA says that distance from Earth to the galaxy is d, what is the distance d' from Earth to the galaxy according to the astronauts? b) The astronauts experience a travel time to the galaxy t' and NASA records the...

Homework Statement The electric and magnetic fields of a 1 Coulomb charge Q are measured by a pair of field measuring instruments. From the perspective of observers in frame O, the charge is at rest at the origin and one of the field-measuring devices is also at rest, with position (x,y,z) =...

With the change of variables-method for a many-to-one transformation function Y = t(X), what's the logic behind summing the different densities for the roots of x = t^-1(y)? Probabilities should be ok to add, but densities? Also, is there no way to extend this method for many-to-many...

Is there some geometry in which a coordinate transformation of a vector of magnitude zero transforms to a vector that does not have a zero magnitude? Since the formula for the magnitude of a vector is √(x12+x22+...xn2), I can see no way for it to have magnitude zero unless every component is...

Homework Statement Find the images of the following region in the z-plane onto the w-plane under the linear fractional transformations The first quadrant ##x > 0, y > 0## where ##T(z) = \frac { z -i } { z + i }## Homework EquationsThe Attempt at a Solution [/B] So for this, I looked at the...

Hey guys. So, as i was going through Griffith's Electrodynamics, and i came across this problem: In the solutions: How to they actually get to that expression for V = (V(bar)+vAx(bar) )Ɣ? I understand everything after that, but this just made me very confused. How do they get this from the...

Homework Statement [/B] Find the Legendre Transformation of f(x)=x^3 Homework Equations m(x) = f'(x) = 3x^2 x = {\sqrt{\frac{m(x)}{3}}} g = f(x)-xm The Attempt at a Solution I am reading a quick description of the Legendre Transformation in my required text and it has the example giving for...

Hi. First, excuse my English. In my lecture notes on classical electrodynamics, we are introduced to the Lorentz transformations: a system S' moves relative to a system S with positive veloticy v in the x-axis (meassured in S), spatial axis are parallel, origin of times t and t' coincide...

Homework Statement If the Helmholtz Free Energy remains constant, estimate the final pressure of 1.0mol of an ideal gas in the following transformation: (1.0atm, 300k) → (pfinal, 600k). Given Sgas = R. Homework Equations A = U - TS dA = -SdT - pdV The Attempt at a Solution If the Helmholtz...

In a standard problem of an electron released from the negative plate in an E field between 2 parallel plates in which the velocity must be determined why can the Lorentz transformation be used (involving v^2/c^2) when the electron is undergoing acceleration and there is nothing in the...

Let us see how the line element transforms under conformal transformations. Consider the Minkovski metric gij, a line element ds2=dxigijdxj, and a conformal transformation δk(x)=ak + λ xk + Λklxl + x2sk - 2xkx⋅s We have δ(dxk)=dδ(x)k=λ dxk + Λkldxl + 2 x⋅dx sk - 2dxkx⋅s - 2xkdx⋅s And so the...

I cant't figure out how to transform ##\dot{r}##, ##\dot{\theta}##, ##\dot{\phi}## in spherical coordinates to ##\dot{x}##, ##\dot{y}##, ##\dot{z}## in cartesian coordinates (the dot is Newton's notation for the first time-derivative which is the angular velocity and velocity). I have no...

If you want to define a covariant derivative which transforms correctly, you need to define it as ##\nabla_i f_j = \partial_i f_j - f_k \Gamma^k_{ij}##, where ##\Gamma^k_{ij}## has the transformation property ##\bar{\Gamma}^k_{ij} = \frac{\partial \bar{x}_k}{\partial x_c}\frac{\partial...

Homework Statement Let T be a Linear Transformation defined on R4 ---> R4 Is that true that the following is always true ? KerT + ImT = R4Homework EquationsThe Attempt at a Solution Since every vector in R4 must be either in KerT or the ImT, so the addition of those subspace contains R. and ofc...

In Carroll's GR book (pg. 96), the transformation law for Christoffel symbols is derived from the requirement that the covariant derivative be tensorial. I think I understand that, and the derivation Carroll carries out, up until this step (I have a very simple question here, I believe-...

Typically an element of a vector space is called a vector, but Carroll's GR book repeatedly refers to elements of tangent spaces as "transforming as a vector" when they change coordinates as Vμ = ∂xμ/∂xν Vν. However, dual vectors are members of vector spaces (cotangent space) but obey ωμ =...

Homework Statement System of equations \frac{du_j}{dt}=u_{j+1}+u_{j-1}-2u_j-\frac{K}{2 \pi}\sin(2\pi u_j)+\bar{F}+F_{ac}\cos(2\pi \nu_o t) where ##j=1,2,3,4##. So ##\{u_j\}## is set of coordinates. If we apply symmetry transformation \sigma_{r,m,s}\{u_j(t)\}=\{u_{j+r}(t-\frac{s}{\nu_0})\} how...

Homework Statement Find the matrix that represents a rotation counterclockwise around the origin by 75 degrees followed by a reflection about the x-axis Homework Equations I know that for A rotated counter clockwise you use the 2x2 matirx [cos(theta), -sin(theta)] [sin(theta, cos(theta)] and...

hi, I always see that jacobian matrix is derived for just 2 dimension ( ıt means 2x2 jacobian matrix) in books while ensuring the coordinate transformation. After that kind of derivation, books say that you can use same principle for higher dimensions. But, I really wonder if there is a proof...

I am working in ℂ3 in this question. (If it will make it easier, we can work in a bounded subspace.) Suppose you have, in each of the three complex planes whose Cartesian product make up the space in question, a set of points. (You do not have knowledge of generators of these sets, or whether...

Homework Statement I'm asking this question because I was trying to apply the method to a Thevenin Eq problem and the answer came out wrong. Also one more related question. According to the textbook, when a current source is connected both in series with a resistor and in parallel with...

Homework Statement Show that the set of restricted canonical transformation forms a group. Verify this statement once using the invariance of Hamilton's principle under canonical transformation, and again using the symplectic condition. Homework Equations (Invariance of Hamilton's principle...

I have a problem to understand the de Broglie wavelength. We know that also particles undergo scattering and interference at a double slit. The interference pattern is calculated by the use of the de Broglie wavelength which is defined as lambda = h / p ; p is the momentum of the particle. This...

Homework Statement T:R2[x] --> R4[x] T(f(x)) = (x^3-x)f(x^2) Homework EquationsThe Attempt at a Solution Let f(x) and g(x) be two functions in R2[x]. T(f(x) + g(x)) = T(f+g(x)) = (x^3-x)(f+g)(x^2) = (x^3-x)f(x^2) + (x^3-x)g(x^2) = T(f(x)) + T(g(x)). let a be scalar in R: aT(f(x)) =...

Greetings. I'm working on a raytracer, and got stuck with trying to model a lens analytically. Given is a thin lens at position p with the axis n, radius r and a focal distance f, a ray hits it at position p1 going in the direction d. Which way would the ray be going on the other side of the...

Homework Statement Two rockets A and B are moving away from the Earth in opposite directions at 0.85c and -0.75c respectively. How fast does A measure B to be travelling? Now I have worked out v = -0.85-0.75/(1- -0.85*-0.75) = -0.997. This is correct. Now I would like to work it out backwards...

'Homework Statement Find the matrix A' for T: R2-->R2, where T(x1, x2) = (2x1 - 2x2, -x1 + 3x2), relative to the basis B' {(1, 0), (1, 1)}. Homework Equations B' = {(1, 0), (1, 0)} so B'-1 = {(1, -1), (0, 1)}. The Attempt at a Solution I'm confused at what exactly a transform matrix...

Homework Statement For the linear transformation T: R2-->R2 defined by T(x1, X2) = (x1 + x2, 2x1 - x2), use the matrix A to find T(v), where v = (2, 1). B = {(1, 2), (-1, 1)} and B' = {(1, 0), (0, 1)}.Homework Equations T(v) is given, (x1+x2, 2x1-x2) The Attempt at a Solution Okay, I see...

Homework Statement Consider the vector space R2 with the standard inner product given by ⟨(a, b), (c, d)⟩ = ac + bd. (This is just the dot product.) PLEASE SEE THE ATTACHED PHOTO FOR DETAIlS Homework Equations T(v)=AT*v The Attempt at a Solution I was able to prove part a. I let v=(v1,v2)...

1. Show that T isn't a linear transformation and provide a suitable counterexample. ##T \begin{bmatrix}x\\y \end{bmatrix} = \begin{bmatrix}x - 1 \\ y + 1 \end{bmatrix}## 2. The attempt at a solution ##\text{let}\, \vec{v} = \begin{bmatrix}0\\0 \end{bmatrix}. \text{Then,}## ##T(\vec{v}) =...

They seem to defy the most fundamental principle of SR. The first postulate/equivalence principle. According to wikipedia, we get Lorentz boost (x direction) and slightly different formulas for the inverse Lorentz boost "This "trick" of simply reversing the direction of relative velocity...

This is where I am stuck. I studied ordered basis and coordinates vector previous to this. of course I studied vector space, basis, linear... etc too, However I can't understand just this part. (maybe this whole part) Especially this one which says [[T(b1)]]c...[[T(bn)]]c be a columns of...

Homework Statement Using the tensor transformation law applied to ##F_{\mu\nu}##, show how the electric and magnetic field ##3##-vectors ##\textbf{E}## and ##\textbf{B}## transform under (a) a rotation about the ##y##-axis, (b) a boost along the ##z##-axis. Homework Equations The Attempt at...

Hi, My Modern Physics lecturer is of the opinion that the energy dissipated during exothermic reactions is due to infinitesimally small change in mass of the reactants. Similarly, he said that an infinitesimally small part of the food we eat gets converted into the energy using which we perform...

> Admit that $V$ is a linear space about $\mathbb{R}$ and that $U$ and $W$ are subspaces of $V$. Suppose that $S: U \rightarrow Y$ and $T: W \rightarrow Y$ are two linear transformations that satisfy the property: > $(\forall x \in U \cap W)$ $S(x)=T(x)$ > Define a linear transformation $F$...

Admit that V is a linear space about \mathbb{R} and that U and W are subspaces of V. Suppose that S: U \rightarrow Y and T: W \rightarrow Y are two linear transformations that satisfy the property: (\forall x \in U \cap W) S(x)=T(x) Define a linear transformation F: U+W \rightarrow Y that...

Homework Statement [/B] A Lorentz transformation ##x^{\mu} \rightarrow x'^{\mu} = {\Lambda^{\mu}}_{\nu}x^{\nu}## is such that it preserves the Minkowski metric ##\eta_{\mu\nu}##, meaning that ##\eta_{\mu\nu}x^{\mu}x^{\nu}=\eta_{\mu\nu}x'^{\mu}x'^{\nu}## for all ##x##. Show that this implies...

Hi, I have to resample images taken from camera, whose target is a spherical object, onto a regular grid of 2 spherical coordinates: the polar and azimutal angles (θ, Φ). For best accuracy, I need to be aware of, and visualise, the "footprints" of the small angle differences onto the original...

Under the infinitesimal translation ##x^{\nu} \rightarrow x^{\nu}-\epsilon^{\nu}##, the field ##\phi(x)## transforms as ##\phi_{a}(x) \rightarrow \phi_{a}(x) + \epsilon^{\nu}\partial_{\nu}\phi_{a}(x)##. I don't understand why the field transforms as above. Let me try to do the math. The...

Hi I have to give a presentation on Bogoliubov transformations to my undergrad solid state physics class. The presentation is only supposed to be 15ish minutes long, and defining them alone will take a few minutes. Because of the time constraint, I was wondering if anyone knew of a simple...

Fourier Transform of Piecewise linear spline wavelet is defined by 1-|t|, 0<t<1; 0, otherwise, is (sinc(w/2))^2. Can anyone please show me the steps. Thanks

Homework Statement Virtually all quantum mechanical calculations involving the harmonic oscillator can be done in terms of the creation and destruction operators and by satisfying the commutation relation \left[a,a^{\dagger}\right] = 1 (A) Compute the similarity transformation...

The Galilean transforms for rotations, boosts and translations in 2D are the follows: Rotations: x' = xcosθ + ysinθ y' = -xsinθ + ycosθ Boosts: x' = x - vxt y' = y - vyt Translations: x' = x - dx y' = y - dx I wanted to combine these into a single pair of equations, so my first thought was...

Homework Statement Show that the isotropy and homogeneity of space-time and equivalence of different inertial frames (first postulate of relativity) require that the most general transformation between the space-time coordinates (x, y, z, t) and (x', y', z', t') is the linear transformation...

How do I go about finding the most general form of the canonical transformation of the form Q = f(q) + g(p) P = c[f(q) + h(p)] where f,g and h are differential functions and c is a constant not equal to zero. Where (Q,P) and (q,p) represent the generalised cordinates and conjugate momentum in...