Transformations Definition and 823 Threads

I have obtained as such using changing of variables/ transformations: Let u= y/x and v= xy after manipulating the equation of the curves as such xy = 5, xy = 2, y/x = 1 and y/x = 4. Then u = 1, u = 4, v = 5 and v = 2 => obtain a square on the u-v plane. Change the integrand to: Outer...

Could you please help me to understand what is the difference between notions of «transformation» and «automorphism» (maybe it is more correct to talk about «inner automorphism»), if any? It looks like those two terms are used interchangeably. By «transformation» I mean mapping from some set...

I have learned that for matrix theory, for ##A \vec{x} = \vec{b}##, if there exists a particular solution ##p##, then every solution looks like ##p+k##, where ##k \in \ker A##. Can someone help me find material on this online, but only in the context of general linear transformations? For...

I wondered if anyone might have something to say about how a 'Ray Transformation' should be defined and also point out what Weinberg means by his 'ray transformations T', see Weinberg, Reference 1, pg 91. Should a ray transformation be an equivalence class of operators? I have started to try to...

Homework Statement How to prove ##max\{0, \rho(\sigma)+\rho(\tau)-m\}\leq \rho(\tau\sigma)\leq min\{\rho(\tau), \rho(\sigma)\}##? Homework Equations Let ##\sigma:U\rightarrow V## and ##\tau:V\rightarrow W## such that ##dimU=n##, ##dimV=m##. Define ##v(\tau)## to be the nullity of ##\tau##...

Homework Statement Mary and Frank are each in their own rocket ships moving along the x-axis. Mary's ship passes Frank's ship at t0=t0'=0 with a speed "v" to the right. When t=t1 in Frank's frame, Frank shoots a missile with a speed "u" where u>v in the direction of Mary. At time t=t2 in...

I was reading this article (https://www.nature.com/articles/srep40083) about designing a panoramic lens with transformation optics, and wanted to try to play around with modifying the coordinate transformations. I contacted one of the authors of the article, and she mentioned that she plotted...

Homework Statement In point form, outline the process through which electricity is generated from a wind turbine, highlighting the energy transformations that occur. Then state an advantage for this type of electricity production that have not already been discussed in the lesson. Homework...

Homework Statement This is for College Algebra. Describe what is happening to the graph of the function ##f\left(x\right)=\sqrt {1 - x}+2##. Homework Equations N/A The Attempt at a Solution This can be rewritten as ##f\left(x\right)=\sqrt {-x + 1}+2## My conundrum: Both my text, as well as...

Homework Statement Choose one form of electrical generation and in point form outline the process through which electricity is generated and the energy transformations that occur. Homework Equations NA The Attempt at a Solution Hi, I am done my work and I used two reliable sources. This...

Homework Statement Consider the metric ds2=(u2-v2)(du2 -dv2). I have to find a coordinate system (t,x), such that ds2=dt2-dx2. The same for the metric: ds2=dv2-v2du2. Homework Equations General coordinate transformation, ds2=gabdxadxb The Attempt at a Solution I started with a general...

The "Operator Norm" for Linear Transformations ... Browder, Lemma 8.4, Section 8.1, Ch. 8 ... ... I am reader Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra...

Imagine that I have a straight, statically-charged, cylinder-shaped tube with arbitrary (ideally infinite) extent. The charge is distributed evenly over the tube such that the field inside the tube is zero. For convenience, let's line up the tube centered along the x-axis such that the...

Homework Statement given that X is an n × p matrix with linearly independent columns. And $$X^∗ = XA$$ where A is an invertible p × p matrix. a) Show that: $$X^*{({X^*}^TX^*)^-}^1{X^*}^T = X{(X^TX)^-}^1X^T$$ b) Consider two alternative models $$M : Y = Xβ + ε$$ and $$M^∗ : Y = X^∗β ^∗ +...

Homework Statement Is there a more intuitive way of thinking or calculating the transformation between coordinates of a field or any given vector? The E&M book I'm using right now likes to use the vector field ## \vec F\ = \frac {\vec x} {r^3} ## where r is the magnitude of ## \vec x...

If we move towards a source of EM waves, in our reference frame the frequency appears (and it is) higher than what a stationary observer will see due to Doppler effect. The field transformations show that these two observers will se static fields differently so I would also expect that the peak...

Homework Statement "Show that every subspace of ##ℝ^n## is the set of solutions to a homogeneous system of linear equations. (Hint: If a subspace ##W## consists of only the zero vector or is all of ##ℝ^n##, ##W## is the set of solutions to ##IX=0## or ##0_vX=0##, respectively. Assume ##W## is...

Hey! :o I want to check whether the following statements are correct. At each statement I wrote my idea/question: A similitude mapping with exactly one fixed point is a scaling. A similitude mapping is a scaling, or a composition of scaling and rotation or a composition of scaling and...

Homework Statement "Determine whether the function ##T:M_{2×2}(ℝ)→ℝ## defined by ##T(A)=det(A)## is a linear transformation. Homework Equations ##det(A)=\sum_{i=1}^n a_{ij}C_{ij}## The Attempt at a Solution I'm assuming that it isn't a linear transformation because ##det(A+B)≠det(A)+det(B)##...

We use A = I.A as equation and then by transforming only A of LHS and I of RHS we come to I = P.A and we say that P is the inverse of matrix A My question is that why we only tranform A and I, why A of RHS is left as it is during the transformation, or why transformation do not take place in...

Homework Statement I have a question. I need to know the integral dxdydz/(y+z) where x>=0, y>=0, z>=0.Homework Equations It is bounded by x + y + z = 1. The transformations I need to use are x=u(1-v), y=uv(1-w), z=uvw. The Attempt at a Solution y+z = uv. J = uv(v-v^2+uv) So I get the integral...

Consider the following paragraph taken from page 15 of Thomas Hartman's lecture notes (http://www.hartmanhep.net/topics2015/) on Quantum Gravity: In an ordinary quantum field theory without gravity, in flat spacetime, there two types of physical observables that we most often talk about are...

I've been working my way through Peskin and Schroeder and am currently on the sub-section about how spinors transform under Lorentz transformation. As I understand it, under a Lorentz transformation, a spinor ##\psi## transforms as $$\psi\rightarrow S(\Lambda)\psi$$ where...

Homework Statement 2. The attempt at a solution 3. Relevant equations In the first problems of that book i was using the Galilean transformations where V1 = V2 + V But if i use that then V1 = 0.945 - 0.6 V1 = 0.345 Is not the same result, so I am confused. In this new problems we are...

Dear all, I am stuck on this question: "If f(x)=(2^x)+1, give in simplest terms the formula for h(x), which is obtained from transforming f(x) by a vertical stretch, scale factor 8 relative to y=0 a translation by vector (1,4) a horizontal stretch, scale factor 1/2 relative to x=0" This is...

Hi, I was calculating some amplitudes and I end up with an expression like this: $$(\bar{c}\gamma^\mu\gamma^\nu\gamma^\rho P_L b)(\bar{d}\gamma_\mu\gamma_\nu\gamma_\rho P_L u)$$ In the solution of the exercise they say that, from the Fierz identity: $$(\bar{c}\gamma^\mu\gamma^\nu\gamma^\rho P_L...

Say, we have two orthonormal basis sets ##\{v_i\}## and ##\{w_i\}## for a vector space A. Now, the first (old) basis, in terms of the second(new) basis, is given by, say, $$v_i=\Sigma_jS_{ij}w_j,~~~~\text{for all i.}$$ How do I explicitly (in some basis) write the relation, ##Uv_i=w_i##, for...

I was reading this article: https://arxiv.org/ftp/arxiv/papers/1005/1005.5206.pdf , regarding the mathematical description of a diamond-shaped cloaking device, and am struggling to understand how the authors found the coordinate transformations in equations (1) and (5). What is the process for...

Hi everyone. Excuse me for my poor English skills. I did an exam today and my exam result was 13 of 40. I don't understand why it was my result, because while doing the exam I though I was doing it well, then the result was a surprise for me. I will write down the questions and after show my...

Hi everyone. Excuse me for my poor English skills. I did an exam today and my exam result was 13 of 40. I don't understand why it was my result, because while doing the exam I though I was doing it well, then the result was a surprise for me. I will write down the questions and after write my...

Homework Statement Show how one can obtain the Doppler transformation for the frequency of a receding source just using the Lorentz transformations for the energy (where E=h). Homework Equations Relativistic transformations for momentum and energy: E = γ(E' + vp'x) pc/E = v/c = β The Attempt...

I was reading my textbook for my elementary modern class and the author said that a pulse of light from a light bulb would be spherical and could be expressed as x2 + y2 + z2 = c2t2 and x'2 + y'2 + z'2 = c2t'2. Then the author goes on to say that this cannot happen for both reference frames in a...

Homework Statement Happy new year all. I was wondering if you can use matrices to translate and transform a function? So for example if I were to take the function $$f(x)=x^2+4x$$ and I want to the translate and transform the equation to $$2f(x+4)$$. Can this be done by matrices. I know how...

This isn't really a homework question, I just need help understanding the example: ===================== ==================== So transformation takes complex n-tuple as input, and it seems output is also a complex n-tuple (which is what makes it "operator"). But permutations of n entries is...

When we make a canonical transformation the volume in phase-space doesn't change. Likewise if we consider motion of a system in phase-space the volume won't change either according to Liouville's theorem. Does that mean that every physical motion of the system in phase-space is equivalent to...

$\textsf{got ? on the Greek notation. if Period = T}$ \begin{align} \displaystyle Y_{tan}&=A\tan\left[\omega\left(x-\frac{\phi}{\omega} \right) \right]+B \implies A\tan\left(\omega x-\phi \right)+B \\ T&=\left(\frac{\phi}{\omega}\right) \\ PS&=\phi \end{align} $\textsf{so on:}$ \begin{align}...

I have a question regarding these transformation formulas: ##\begin{align} & E'_x = E_x & \qquad & B'_x = B_x \\ & E'_y = \gamma \left( E_y - v B_z \right) & & B'_y = \gamma \left( B_y + \frac{v}{c^2} E_z \right) \\ & E'_z = \gamma \left( E_z + v B_y \right) & & B'_z = \gamma \left( B_z -...

I've been reading about canonical transformations in Hamiltonian mechanics and I'm a bit confused about the following: The author considers a canonical transformation $$q\quad\rightarrow\quad Q\quad ,\quad p\quad\rightarrow\quad P$$ generated by some function ##G##. He then considers the case...

We learned that the condition of a linear transformation is 1. T(v+w) = T(v)+T(w) 2. T(kv) = kT(v) I am wondering if there is any transformation which only fulfil either one and fails another condition. As obviously, 1 implies 2 for rational number k. Could anyone give an example of each...

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 Let ##C## be the space of continuous real functions on ##[0,\pi]##. With any function ##f\in C##, associate another function ##g=T(f)## defined by $$g=T(f)\equiv \int_0^\pi \cos(t-\tau) f(\tau) \, d \tau$$ a) Show ##T## is a linear transformation from ##C## to ##C##. b)What...

Hello everyone, There is something that has been bugging me for a long time about the meaning of Lorentz Transformations when looked at in the context of tensor analysis. I will try to be as clear as possible while at the same time remaining faithful to the train of thought that brought me...

Homework Statement Homework Equations none The Attempt at a Solution -amplitude is 3 -period is 180° -right 60° -down 1 Rough sketch of graph: I would like to know if the graph looks right, is there any improvements to be made? Thanks :)

It has been sometime since I have been thinking about this question and I have been quite successful in confusing myself. In Einstein's General Relativity, we say that the general coordinate transformations (or diffeomorphisms) on a manifold are the gauge transformations of the theory. The local...

Ok so... It's been a while since I first saw this problematic scenario and I want to know how to deal with it. The question arises in the context of special relativity. Suppose 2 objects moving at the same speed. The floor is the rest frame 'A' and the front object is the moving frame 'B'. The...

Homework Statement In the attached image, I'm confused on how the first circuit reduces to the second circuit. Homework Equations Is=Vs/R1 The Attempt at a Solution I know that in order to do a source transformation that I need to have a current source in parallel with a resistor or a...

I am confused about conformal transformations on Riemannian manifolds. Here's what I have so far. 1. Under a conformal transformation the metric changes by: g' -> Ω2g 2. Under a Weyl transformation the metric changes by: g' -> exp(-2f)g 3. Any 2D Riemann manifold is locally conformally...

Homework Statement Hi, I'm working on understanding how a time independent point transformation . effects the Lagrangian and to see how what values are co and contra variant.Homework Equations Would these be correct formulations, or have I overlooked something? The Attempt at a Solution and...

Homework Statement For a gauge function G(t,q) where , does or have any alternative form or can they be expressed in any other way? Homework EquationsThe Attempt at a Solution

Homework Statement Show that an infinitesimal boost by v^j along the x^j-axis is given by the Lorentz transformation \Lambda^{\mu}_{\nu} = \begin{pmatrix} 1 & v^1 & v^2 & v^3\\ v^1 & 1 & 0 & 0\\ v^2 & 0 & 1 & 0\\ v^3 & 0 & 0 & 1 \end{pmatrix} Show that an infinitesimal rotation by theta^j...