Transformation Definition and 1000 Threads

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 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...

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...

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...

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Within my project thesis I stumbled over the term SU(2)_V, SU(2)_A transformations. Although I know U(1)_V, U(1)_A transformations from the left and right handed quarks( U(1)_V transformations transform left and right handed quarks the same way, while U(1)_A transformations transform them with a...

Homework Statement Let L: ℝ2→ℝ2 such that L(x1, x2)T=(1, 2 ; 3, α)(x1, x2)T=Ax Determine at what values of α is L an isomorphism. Obviously L is given in matrix form. The Attempt at a Solution First of all a quick check, dim (ℝ2)=dim(ℝ2)=2 Ok. An isomorphism means linear transformation which...

my understanding of beta radiation is that an up quark in a proton changes to a down quark, forming a neutron and emitting an electron as the result of the change in charge. My questions are, 1. Why does the quark change? 2. How does it change and how does it change charge?

In the transformation of tensor components when changing the co-ordinate system, can someone explain the following: Firstly, what is the point in re-writing the indicial form (on the left) as aikTklajl? Since we're representing the components in a matrix, and the transformation matrix is also...

reading a Nature paper tonight, I read that the NV(-) separation is ~ 1.7Angstroms. There was no equation to show how they arrived at this. Any insight into how this was determined? I don't think there is a direct equation, but perhaps using a to r?

Homework Statement Let A:\mathbb R_2[x]\rightarrow \mathbb R_2[x] is a linear transformation defined as (A(p))(x)=p'(x+1) where \mathbb R_2[x] is the space of polynomials of the second order. Find all a,b,c\in\mathbb R such that the matrix \begin{bmatrix} a & 1 & 0 \\ b & 0 & 1 \\ c & 0...

I don't understand why we can write the elements of the lorentransformation in the form ## {\Lambda}^{\mu}\:_{\nu} = [exp(-\frac{i}{2}{\omega}^{\rho\sigma}M_{\rho\sigma})]^{\mu}\:_{\nu} ## I know that we can write it in the form ## {\Lambda} = exp(t\Theta) ## where ## \Theta ## are elements...

Homework Statement [/B] In Minkowski spacetime we are considering a (series of) frame(s), S', attached to a rocket with constant proper acceleration. The rocket's speed in S is v. We find with boundary conditions x = 0 at t = t' = 0 the relationships between S and S' (for x' = 0, i.e. at the...

Hey, I need help with part D2. My explanation is not right so I honestly do not know what I am suppose to write. My assignment is attached to this thread.

Homework Statement T: R^2 --> R^2 given as a projection on the line L = 5x+2y=0 decide matris T? Homework EquationsThe Attempt at a Solution L= 5,2 X=x1, x2 projL on X = (5x1+2x2)/29 *(5,2) = 1/29 [25 10 10 4] is this correct?

the question is: Rewrite the rational equation y=(-5x-18)/(x+4) to show how it is a transformation of y=1/x. describe transformations looks like it is shifted 4 to left, then stretched by factor of -5x-18. Is that accurate? would you elaborate beyond that?

1. Show that the map $\mathcal{A}$ from $\mathbb{R}^3$ to $\mathbb{R}^3$ defined by $\mathcal{A}(x,y,z) = (x+y, x-y, z)$ is a linear transformation. Find its matrix in standard basis. 2. Find the dimensions of $\text{Im}(\mathcal{A})$ and $\text{Ker}(\mathcal{A})$, and find their basis for the...

I'm confused about the notation T:R^n \implies R^m specifically about m. From my understanding if n=2 then (x1, x2). Are we transforming n=2 to another value m for example (x1, x2, x3)?

Homework Statement Hey, I posted another question yesterday, and thanks to the kindness and brilliance of hall of ivy, I was able to solve it. However when I apply the same logic to this new question I cannot seem to get it, can someone explain or show me how to do this question. Consider the...

Homework Statement Consider the linear transformation T from V = P2 to W = P2 given by T(a0 + a1t + a2t2) = (−4a0 + 2a1 + 3a2) + (2a0 + 3a1 + 3a2)t + (−2a0 + 4a1 + 3a2)t^2 Let E = (e1, e2, e3) be the ordered basis in P2 given by e1(t) = 1, e2(t) = t, e3(t) = t^2 Find the coordinate matrix...

Homework Statement Find all values a\in\mathbb{R} such that vector space V=P_2(x) is the sum of eigenvectors of linear transformation L: V\rightarrow V defined as L(u)(x)=(4+x)u(0)+(x-2)u'(x)+(1+3x+ax^2)u''(x). P_2(x) is the space of polynomials of order 2. Homework Equations -Eigenvalues and...

Hi. In the attached proof for Lorentz transformation for momentum http://www.colorado.edu/physics/phys2170/phys2170_sp07/downloads/lorentz_transformation_E_p.pdf, there is this step that I don't understand: 1/√1-u'2/c2 = γ(1-vux/c2)/√1-u2/c2 Can someone explain how they derived this? Thanks! :)

Taking a look at "http://www.space.com/30026-earth-twin-kepler-452b-exoplanet-discovery.html" I observe that planet Kepler-452b (judged to be somewhat Earth-like) is 1400 light years from Earth. If a spaceship leaves Earth at a fifth of the speed of light, traveling toward Kepler-452b, from...