Transformation Definition and 1000 Threads

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...

Transformations always give me trouble, but this one does in particular. Assume X_1, X_2 independent with binomial distributions of parameters n_1, n_2, and p=1/2 for each. Show Y = X_1 - X_2 + n_2 has a binomial distribution with parameters n= n_1 + n_2, p = 1/2. My first instinct was...

Homework Statement A spaceship is approaching a planet at a speed v. Suddenly, the spaceship explodes and releases a sphere of photons traveling outward as seen in the spaceship frame. The explosion occurs in the planet frame when the spaceship is a distance L away from the planet. In the...

Hi all, The context of this problem is as follows: I'm trying to implement a discontinuous finite element method and the formulation calls for the computation of line integrals over the edges of the mesh. Anyway, more generally, I need to evaluate \int_{e}f(x,y)ds, where e is a line segment...

I know that every linear transformation from Rn to Rm can be represented in a matrix form. What about a transformation from a 1. Infinite dimension to infinite dimension 2.finite to infinite dimension 3.infinite to finite dimension Can they represented by matrix form...? Before...

Dear All: For a line structure(say a very long atomic chain) and a closed loop structure( connect the head and tail of this atomic chain). Does there exists a transformation between these two structures? For example if we want to study the vibration mode of these two cases. If we already know...

Does there exist a transformation between a line and a closed loop ? Dear All: For a line structure(say a very long atomic chain) and a closed loop structure( connect the head and tail of this atomic chain). Does there exists a transformation between these two structures? For example if we...

Homework Statement Use Y to Δ transformation to find i0 and i/x Homework Equations The Attempt at a Solution Here's my transformation. Calculated i0, which is equal to 3A. I have no clue how to find ix.

Homework Statement Show that Q_{1}=\frac{1}{\sqrt{2}}(q_{1}+\frac{p_{2}}{mω}) Q_{2}=\frac{1}{\sqrt{2}}(q_{1}-\frac{p_{2}}{mω}) P_{1}=\frac{1}{\sqrt{2}}(p_{1}-mωq_{2}) P_{2}=\frac{1}{\sqrt{2}}(p_{1}+mωq_{2}) (where mω is a constant) is a canonical transformation by Poisson bracket test. This...

Homework Statement If \phi \in \mathcal{M} (group of all linear fractional transformations or Mobius Transformations has three fixed points, then it must be the identity. (The proof should exploit the fact that \mathcal{M} is a group. The Attempt at a Solution Hi all, So...

Given the following ODE \[ \left(\frac{du}{dx}\right)^2 = \mu u^2 - \frac{2\alpha}{\sigma + 2}u^{\sigma + 2} - \frac{\gamma}{\sigma + 1}u^{2(\sigma + 1)} \] How do I obtain \[ u(x) = \left(\frac{A}{B + \cosh(Dx)}\right)^{1/\sigma} \] where \(A = \frac{(2 + \sigma)B\mu}{\alpha}\), \(B =...

When performing transformation, after adding SOC media to the newly transformed cells, we place them at 37 degrees Celsius for an hour to allow growth. I understand the need for an incubator but I'm confused regarding how shaking helps in microbial growth? I did some searching on the internet...

Hi everyone, :) Here's another question I encountered recently. I am writing the question and my full solution. Many thanks if you can go through it and find a mistake, or confirm whether it's correct, or can contribute with any other useful comments. :) Problem: Find the matrix of a linear...

I am using an excel sheet to generate some URLs that need a input number encoded. I figured out the pattern -- it is a simple digit manipulation Input --> Encoded Output -------- ---------------------- 271678 --> 01303032373136373875 261268 -->...

Homework Statement Two RVs X1 and X2 are continuous and have joint pdf f_{X_1,X_2}(x_1, x_2) = \begin{cases} x_1+x_2 &\mbox{for } 0 < x_1 < 1; 0 < x_2 < 1 \\ 0 & \mbox{ } \text{otherwise}. \end{cases} Find the pdf of Y = X_1X_2.Homework Equations I'm using the transformation "shortcut' that...

Homework Statement Find the Möbius transformation that maps 0 -> -1 1 -> infinity infinity -> 1 Homework Equations w = f(z) = \frac{az + b}{cz+d} The Attempt at a Solution My first idea was to attempt to solve it as a normal system of eq's but that quickly falls apart due to...

Hi everyone, :) Here's a question I was stuck on. Hope you people can help me out. :) The definition of root vectors is given >>here<<. Now a \(n\times n\) matrix can be diagonalized if it has \(n\) distinct eigenvalues. So I don't see how the given condition (all root vectors are...

Hi everyone, :) Here's a question I got stuck. Hope you can shed some light on it. :) Of course if we write the matrix of the linear transformation we get, \[A^{t}.A=\begin{pmatrix}a_1^2 & a_{1}a_2 & \cdots & a_{1}a_{n}\\a_2 a_1 & a_2^2 &\cdots & a_{2}a_{n}\\.&.&\cdots&.\\.&.&\cdots&.\\a_n...

Hi everyone, :) This is one of those questions I encountered when trying to do a problem. I know that a eigenvector of a linear transformation should be non-zero by definition. So does that mean every linear transformation has eigenvectors? What if there's some linear transformation where no...

Homework Statement Let X_1, X_2 have the joint pdf h(x_1, x_2) = 8x_1x_2, 0<x_1<x_2<1 , zero elsewhere. Find the joint pdf of Y_1=X_1/X_2 and Y_2=X_2. Homework Equations p_Y(y_1,y_2)=p_X[w_1(y_1,y_2),w_2(y_1,y_2)] where w_i is the inverse of y_1=u_1(x_1,x_2) The Attempt at a Solution We can...

This is from a chapter on distributions of two random variables. Let X and Y have the pdf f(x,y) = 1, 0<x<1 and 0<y<1, zero elsewhere. Find the cdf and pdf of the product Z=XY. My current approach has been to plug in X=Z/Y in the cdf P(X<=x) , thus P(Z/Y<=x), and integrate over all values of...

T is a surjective linear transformation $$T: \mathbb{R^4}-> \mathbb{R^2}$$. Decide dim ker T. How many free variables do I get if I solve equation system $$T(x)=y$$ for a vector $$y \in \mathbb{R^2}$$? Construct a transformation matrix belonging to a surjective linear transformation...

I have checked many textbooks and papers on SUSY and it seems that none of them mentions anything about the infinitesimal susy transformation on component fields in the case N\neq 1. So I am wondering what does it looks like, say for N=2 vector multiplet? Another related question is, do we need...

In West's book "Introduction to Strings and Branes", page 2, I don't understand why the auxiliary field e transformed as e'(\tau')d\tau'=e(\tau)d\tau.

I tried to verify that the SYM lagrangian is invariant under SUSY transformation, but it turned out there is a term that doesn't vanish. The SYM lagrangian is: \mathscr{L}_{SYM}=-\frac{1}{4}F^{a\mu\nu}F^a_{\mu\nu}+i\lambda^{\dagger a}\bar{\sigma}^\mu D_\mu \lambda^a+\frac{1}{2}D^a D^a the...

Homework Statement The question asks to find the current I going into the 2k resistor path using Y-delta or delta-Y transformations. Homework Equations Resistance in parallel 1 / R = 1 / R1 + 1 / R2 .. Converting Delta to Y, R1 = RaRb / (Ra + Rb + Rc) Current divider formula Ix = (Rt...

Homework Statement Calculate the result of the transformation of the vector operator \hat{V_{y}} by rotation \hat{R_{x}} around an angle \alpha . Homework Equations I believe that \hat{R_{x}} = \begin{pmatrix} 1& 0& 0\\ 0& cos\alpha & -sin\alpha \\ 0 & sin\alpha & cos\alpha...

Homework Statement Consider the transformation T from ℝ2 to ℝ3 given by, $$T\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = x_1\begin{bmatrix} 1 \\ 2 \\ 3\end{bmatrix} + x_2\begin{bmatrix} 4 \\ 5 \\ 6\end{bmatrix}$$ Is this transformation linear? If so, find its matrix Homework Equations A...

Hi all, I have an ODE of the form \frac{d^{3}\psi}{d\xi^{3}}-A\left(\psi+\xi\frac{d\psi}{d\xi}\right)=0, where \psi=C_{1}U(\xi)+C_{2}V(\xi). Is there any transformation or inventive manipulation I can use here to obtain an ODE for \sigma=U(\xi)+V(\xi)? As this is the quantity I would...

proof onto Prove: A linear Map T:Rn->Rm is an onto function : The only way I have thought about doing this problem is by proving the contrapositive:

Appendix 1 - simple Lorentz transformation derivation found at - http://www.bartleby.com/173/a1.html Given in equation (3) (x'-ct') = Y(x-ct) [Y = const.] by rearrangement, it yields, (x'-ct')/(x-ct) = Y. But it is stated that both (x-ct) and (x'-ct') are zero, so...

Homework Statement I need to find the laplace transformation of the following function (and it's ok to leave it expressed as an integral). After doing the initial steps and algebra I got Y(s)= g(t)/(s+2)^2 + 7(1/(s+2)^2)+ 2(1/(s+2)^2) the answer is y(t)=2e^-2t +te^-2t...

Problem: A linear transformation T: Rm->Rm is invertible if and only if, for any basis {v1, ...vm} of Rm, {T(v1),...,T(vm)} is also a basis for Rm.Ideas: Since the inverse exists, we can say that some vector u in the inverse of T can be represented as linear combinations of basis vectors...

Hey This is from Numerical analysis course (Legendre polynom) - they gave us the polynomial transformation from [-1,1] to [a,b] as: x = 2/(b-a) * z - (b+a)/(b-a) what is the proof of this tranformation? where did it come from? thanks

Hello, I have few question for deriving the Lorentz transformation (LT): While deriving the LT, we draw a graph as x,y,z in one frame of reference and x',y',z' in the other frame of reference as S and S' as two frames of reference. Now the factor ct comes in, which is the flash of...

Suppose I am in a stationary frame of reference S and there is a lamp post at a distance X from my origin in the positive X direction. Say you move at a velocity V along that axis and the distance of the lamp post in your frame of reference S' is X'. Then by Lorentz transformation equation X'...