Transformation Definition and 1000 Threads

I have a problem on which I am stuck and would like help on how to proceed. The problem and my work is fairly lengthy, so please bear with me. **Problem:** A model for transport of a solute (moles of salt) and solvent (volume of water) across a permeable membrane has the form...

Homework Statement A polynomial of degree two or less can be written on the form p(x) = a0 + a1x + a2x2. In standard basis {1, x, x2} the coordinates becomes p(x) = a0 + a1x + a2x2 equivalent to ##[p(x)]_s=\begin{pmatrix}a0\\ a1\\ a2 \end{pmatrix}##. Part a) If we replace x with...

Hello everyone, i am now working on a problem with three fingers robot hand to grab a cube to undergo some motion however i face some difficulties on deriving the grasp transformation matrix which help to switching the local coordinate frame at first i was given three point vectors [0 1...

The two properties every linear transformation T: V -> W has to satisfy is T(u + v) = T(u) + T(v), for u,v in V (i) T(cu) = cT(u) for u in V and scalar c (ii) I'm trying to find a transformation which satisfies (i) but doesn't satisfy (ii) [I've been able to find the opposite for what it's...

Homework Statement ##D_{ijk}## is an array with ##3^3## elements, which is not known to represent a tensor. If for every symmetric tensor represented by ##a_{jk}## $$b_i = D_{ijk}a_{jk},$$ represents a vector, what can be said about the transformation properties under rotations of the...

[I've tried asking this question on math.stackexchange.com, but haven't got any responses, so I thought I'd try here] I’m building a model in a 3D simulation program (MSC Adams) and part of that model is a triangular platform which can translate and rotate in the virtual world, as shown in...

I am starting to learn Special and General realitivity by reading through Bernard F. Schutz's book "A First Course in General Realitivity". However I can't seem to grasp the relationship between two reference frames as compared with a Space-Time diagram. I understand the geometry of the diagrams...

Homework Statement Use resistance reduction and source transformation to find Vx in the circuit below. All resistance values are in ohms. Homework Equations The Attempt at a Solution For this problem, I know I can combine the 16 Ω resistors, but where I'm having a little trouble...

For two body decay, in CM frame, we know that the magnitude of the final particle momentum is a constant, which can be described by a delta function, ##\delta(|\vec{p^*}|-|\vec{p_0^*}|)##, ##|\vec{p_0^*}|## is a constant. When we go to lab frame (boost in z direction), what's the Lorentz...

Homework Statement A particle's movement is described by \vec{r} in the inertial system IS. Find the velocity of the particle \vec{\dot{r'}} in the system IS', which is moving with arbitrary velocity v from IS. Both inertial systems are arbitrary.Homework Equations For the position vector the...

Hi, I was wondering when you need to write the decomposition of a substance, how do you know if the number is going to be a coefficient or subscript ? Example: 2H2O ===>2H2 + 02 would be the answer But why not 2H2O ==> 2H2 + 2O Knowing that we have 2 moles of O in the beginning...

Homework Statement x(t) = 5cos(2*pi*1000*t) and g(t) = ∑ from n=-infinity to infinity delta(t-n/10000) find Fourier transform of x(t) and g(t) and the product of the two The Attempt at a Solution x(w) = 5*sqrt(pi/2) [delta(w-2000pi)+delta(w+2000pi)] g(w) = 1 so would the...

Homework Statement Starting with the Lorentz transformation for the components of the velocity, derive the transformation for the components of acceleration.Homework Equations Lorentz Transformation for position and time : ##x'={\gamma}(x - vt)## ##t'={\gamma}(t - {\frac{vx}{c^2}})## Resulting...

I've been working on a problem that I can't seem to get started on. Here is how it is posted: Metric of a space is: ds^2 = (1+2\phi^2)dt^2 - (1-2\phi)(dx^2+dy^2+dz^2), where |\phi | << 1 everywhere. Given a point (t_0 , x_0 , y_0, z_0) find a coordinate transformation to a locally...

Homework Statement let f be the linear transformation represented by the matrix M = ( -3, 2) ( 0, -2) state what effect f has on areas, and whether f changes orientation. Find the matrix that represents the inverse of f. Homework Equations N/A The Attempt at a...

For two-body decay ##A\rightarrow B+C##, if A is polarized, it is clear that we have: ##\frac{dN}{d\Omega}\propto 1+\alpha \cos\theta^*##, for final particle distribution. where, ##\theta^*## is the angle between the final particle's momentum ##p^*## and the polarization vector of ##A## in the...

http://i.imgur.com/MDigPh5.png if i have my original coordinate (white) and i am transforming this into the red coord. , could someone explain to me why y=y'cos\phi is incorrect and why y'=ycos\phi is correct?

Question, within the conformal group of say standard euclidean space can the inversion be obtained by exponentiating the standard generators? Presumably it would be with some combination of translation and special conformal transformation in parallel directions but I'm not seeing how it can work...

So I've been reading Einstein's theory of relativity, and at one point when discussing the Lorentz equations' proof that light remains constant, he just states it without mathematically doing it. Probably because it wasn't the super scientific version (?) but I wanted to see how he did it, so I...

Hello, this is something basic I have hard to understand and would like to have help!:) this is a exemple from My book and I Dont understand the input! "Let T: P_2->P_2 be the linear transformation defines by T(P(x))=p(2x-1) I Dont understand how this work T(1)=1, T(x)=2x-1, T(x^2)=(2x-1)^2...

Homework Statement Question as stated: In special relativity consider the following coordinate transformation between inertial frames: first make a velocity boost v_x in the x-direction, then make a velocity boost v_y in the y-direction. 1) Is this a Lorentz transformation? 2) Find the matrix...

Hi everyone, :) Here's a question I encountered recently and did partway. I need your advice on how to proceed. Question: What can be said about the Jordan normal form of a linear transformation \(f:V\rightarrow V\) where \(V\) is a vector space over \(\mathbb{C}\), if we know that...

Homework Statement Hi, I am not sure if this is the right place for my question but here goes! The stress tensor in the Si coordinate system is given below: σ’ij = {{-500, 0, 30}, {0, -400, 0}, {30, 0, 200}} MPa Calculate the stress tensor in the L coordinate system if: cos-1a33=45°, and...

Hey, I consider a diagonalized Hamiltonian: H=\sum\limits_{k} \underbrace{ (\epsilon_{k} u_{k}^2 -\epsilon_{k} v_{k}^2 -2\Delta u_{k} v_{k} )}_{E_{k}}(d_{k \uparrow}^{\dagger}d_{k \uparrow} + d_{k \downarrow}^{\dagger}d_{k \downarrow}) +const with fermionic creation and annihilation...

I'm working on a problem that requires me to take the cartesian metric in 2D [1 0;0 1] and convert (using the transformation equations b/w polar and cartesian coords) it to the polar metric. I have done this without issue using the partial derivatives of the transformation equations and have...

Why ##\rho,\rho^2,\rho^3,\rho^4## are even transformation and ##\rho\sigma,\rho^2\sigma,\rho^3\sigma## are odd transformation. I'm talking about case of ##D_4## group, where ##\rho## is rotation and ##\sigma## is reflection.

Hello, I am trying to numerically evaluate a convolution integral of two functions (f*g) using Fourier transform (FT) i.e using FT(f*g) = FT(f) multiplied by FT(g) (1) I am testing for a known case first. I have taken the gaussian functions (eq. 5, 6 and 7) as given in...

Homework Statement I'm given two circles in the complex plane. |z|=1 and |z-1|=\frac{5}{2}. The goal is to find a "Linear Fractional Transformation" or Mobius Transformation that makes these two circles concentric about the origin. Homework Equations w=f(z)=\frac{az+b}{cz+d} The...

Why parity is discrete transformation? ##Px=-x## ##P\psi(x)=\psi(-x)## when ##x## is continual variable. Could you explain me difference between discrete and continual transformation?

I'm given the following transformation X=x \cos \alpha - \frac{p_y}{\beta} \sin \alpha Y=y \cos \alpha - \frac{p_x}{\beta} \sin \alpha P_X=\beta y \sin \alpha + p_x \cos \alpha P_Y=\beta x \sin \alpha + p_y \cos \alpha and I'm asked to find what type(s) of transformation it is. I'm not...

Homework Statement Let ##S = \{1, e^x, e^{-x}, e^{2x}, e^{-2x}\}## and ##B = \{1, sinh(x),cosh(x), sinh(2x), cosh(2x)\}##. S spans the vector space V, and a linear transformation T: V -> V is defined by T(y) = y'' - 3y' - 4y. (a) Find the representation matrix of T with respect to the bases S...

Homework Statement Let Q^1 = (q^1)^2, Q^2 = q^1+q^2, P_{\alpha} = P_{\alpha}\left(q,p \right), \alpha = 1,2 be a CT in two freedoms. (a) Complete the transformation by finding the most general expression for the P_{\alpha}. (b) Find a particular choice for the P_{\alpha} that will reduce the...

Hello all, I am developing a model of multiple gaps in a square lattice. I simplified the associated Hamiltonian to make it quadratic. In this approximation it is given by, H = \begin{pmatrix} \xi_\mathbf{k} & -\sigma U_1 & -U_2 & -U_2\\ -\sigma U_1 & \xi_{\mathbf{k}+(\pi,\pi)} & 0 &...

I was going through my professor's notes about Canonical transformations. He states that a canonical transformation from (q, p) to (Q, P) is one that if which the original coordinates obey Hamilton's canonical equations than so do the transformed coordinates, albeit for a different Hamiltonian...

Hi everyone, :) Recently I encountered the following problem. Hope you can confirm whether my method is correct. My answer seems so trivial and I have doubts whether it is correct. Problem: Find the Jordan normal form of a unitary linear transformation. My Solution: Now if we take the...

Homework Statement DIAGRAM ATTACHED AT BOTTOM Q. The following statements are true for an element in plane stress state. (this is 2D) (1) one of the principle stresses is 40Mpa; (2) σx= -2τxy; (the algebraic values) (3) in x'oy' with θ=30°, the two normal stresses σx'=σy' Determine...

I am not following the template for the reason that this is a generic question. Consider that the change in kinetic energy is 1J. Suppose further you have two particles, both of equal mass that are gravitationally attracted to each other (and the change in energy comes from the fact that they...

I've managed to confuse myself and don't understand the difference between the formula for Lorentz time transformation (t'=γ(t-vx/c^2) and the time dilation equation t'=γ(t_proper) As I understand, proper time is difference between two events that happen in same place in a given reference...

Given a substance, for example water, does the heat of vapourization vary with pressure or any other variables? Also, at a specific pressure, water (like all other substances, but at its own respective pressure) changes phases from solid to gas without any intermediate phase. Would the heat...

Hi All, I'm working through the theory of the strong interaction and I roughly follow it. However I have some questions about the meaning of the terms. The book I use gives the gauge transformation as: \psi \rightarrow e^{i \lambda . a(x)} \psi First question ... What are the a(x)...

Hi everyone, :) I don't understand how to use the given linear transformation so as to calculate the exterior power of \(V\); \(\wedge^2(f)\). I hope you can help me with this. :) Problem: Find the trace of the linear transformation \(\wedge^2(f)\), if \(f\) is given by the matrix...

Homework Statement Given a square and the respective distension tensor, ε, find the position on his vortices after the transformation. ε = 0.1...0.25 ...0.25...0.1 Homework Equations The Attempt at a Solution I got kind of lost in this question. I started thinking that maybe...

Homework Statement The question is quite basic; what is the Lorentz transformation of the follows 4-vectors from S to S' frame: A photon (P) in S frame with 4-momentum P = (E/c,p,0,0) and frequency f where hf = pc = E. h is the planks constant, p is the magnitude of 3-momentum...

Homework Statement Compute the inverse, eigenvalues and eigenvectors of the following matrix, M. Are the eigenvectors orthogonal? Determine a unitary similarity transformation matrix U such that U-1MU is diagonal.With M being {2, 0, 2i, 0, 1} {0, -1, 0,-2i,0} {-2i, 0, 1, 1, 1} {...

Hello everyone I hope someone can check the solution for me. Here is the problem: Let $V=V_1\oplus V_2$, $f$ is the projection of $V$ onto $V_1$ along $V_2$( i.e. if $v=v_1+v_2, v_i\in V_i$ then $f(v)=v_1$). Prove that $f$ is self-adjoint iff $<V_1,V_2>=0$ my solution is this...

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...

Homework Statement (a) Consider a system with one degree of freedom and Hamiltonian H = H (q,p) and a new pair of coordinates Q and P defined so that q = \sqrt{2P} \sin Q and p = \sqrt{2P} \cos Q. Prove that if \frac{\partial H}{\partial q} = - \dot{p} and \frac{\partial H}{\partial p} =...

Hi all, I got a 3 part Qs: γ=1/√1-v^2-c^2 Part A Homework Statement Consider the Lorentz transformation tensor Matrix Row 1: [ γ 0 0 -vγ/c] Row 2: [ 0 1 0 0 ] Row 3: [ 0 0 1 0 ] Row 4:-[vγ/c 0 0 γ ] for transforming 4-vectors from frame S...

Hi all, I got a 3 part Qs: γ=1/√1-v^2-c^2 Part A Homework Statement Consider the Lorentz transformation tensor Matrix Row 1: [ γ 0 0 -vγ/c] Row 2: [ 0 1 0 0 ] Row 3: [ 0 0 1 0 ] Row 4:-[vγ/c 0 0 γ ] for transforming 4-vectors from frame S...

Let V and W be two finite-dimensional vector spaces over the field F. Let B be a basis of V, and let C be a basis of W. For any v 2 V write [v]B for the coordinate vector of v with respect to B, and similarly [w]C for w in W. Let T : V -> W be a linear map, and write [T]C B for the matrix...