What is Transformation: Definition and 1000 Discussions
In linear algebra, linear transformations can be represented by matrices. If
T
{\displaystyle T}
is a linear transformation mapping
R
n
{\displaystyle \mathbb {R} ^{n}}
to
R
m
{\displaystyle \mathbb {R} ^{m}}
and
x
{\displaystyle \mathbf {x} }
is a column vector with
n
{\displaystyle n}
entries, then
T
(
x
)
=
A
x
{\displaystyle T(\mathbf {x} )=A\mathbf {x} }
for some
m
×
n
{\displaystyle m\times n}
matrix
A
{\displaystyle A}
, called the transformation matrix of
T
{\displaystyle T}
. Note that
A
{\displaystyle A}
has
m
{\displaystyle m}
rows and
n
{\displaystyle n}
columns, whereas the transformation
T
{\displaystyle T}
is from
R
n
{\displaystyle \mathbb {R} ^{n}}
to
R
m
{\displaystyle \mathbb {R} ^{m}}
. There are alternative expressions of transformation matrices involving row vectors that are preferred by some authors.
What type of chemical and physical transformation occurs in the blue sphere silica gel when it is subjected to very high temperatures above the temperature supported by it for several minutes?
does it turn into a compound that will absorb and store moisture and water inside and can leak if it...
I am a physics enthusiast reading Covariant Physics by Moataz H. Emam. In his chapter about Point Particle mechanics there is a transformation equation for a displacement vector. I don't see how he arrived at the final equation 3.6. Is it a chain rule or product rule? Can't seem to figure it...
I have encountered a problem related to the Galilean Transformation. Let's consider two observers who will be referred to as ##O## and ##O^{'}##, with their corresponding coordinates ##(t,x,y,z)## and ##(t^{′},x^{′},y^{′},z^{'})## respectively. They are initially at the same location, at time...
I need to prove that under an infinitesimal coordinate transformation ##x^{'\mu}=x^\mu-\xi^\mu(x)##, the variation of a vector ##U^\mu(x)## is $$\delta U^\mu(x)=U^{'\mu}(x)-U^\mu(x)=\mathcal{L}_\xi U^\mu$$ where ##\mathcal{L}_\xi U^\mu## is the Lie derivative of ##U^\mu## wrt the vector...
Question:
Eq. 12.109:
My solution:
We’ll first use the configuration from figure 12.35 in the book Griffiths. Where the only difference is
that v_0 is in the z-direction. The electric field in the y-direction will be the same.
$$E_y = \frac{\sigma}{\epsilon _0}$$
Now we're going to derive the...
I started by expanding ##dx## and ##dt## using chain rule:
$$dt = \frac{dt}{dX}dX+\frac{dt}{dT}dT$$
$$dx = \frac{dx}{dX}dX+\frac{dx}{dT}dT$$
and then expressing ##ds^2## as such:
$$ds^2 =...
I am given an initial vector potential let's say:
\begin{equation}
\vec{A} = \begin{pmatrix}
g(t,x)\\
0\\
0\\
g(t,x)\\
\end{pmatrix}
\end{equation}
And I would like to know how it will transform under the Lorenz Gauge transformation. I know that the Lorenz Gauge satisfy...
So, I've been watching eigenchris's video series "Tensors for Beginners" on YouTube. I am currently on video 14. I am a complete beginner and just want some clarification on if I'm truly understanding the material.
Basically, is everything below this correct?
In summary of the derivation of the...
Hi,
Unfortunately I am not getting anywhere with task three, I don't know exactly what to show
Shall I now show that from ##S(T,V,N)## using Legendre I then get ##S(E,V,N)## and thus obtain the Sackur-Tetrode equation?
Hi,
unfortunately, I'm not that fit concerning the Lagrangian formalism, so I'm not sure if I solved the problem 1a correctly.
I have now proceeded as follows
the Lagrangian is
$$L=T-U$$
Since there are no constraining or other forces acting on the point mass, I assume that the...
T(α1), T(α2), T(α3) written in terms of β1, β2:
Tα1 =(1,−3)
Tα2 =(2,1)
Tα3 =(1,0).
Then there is row reduction:
Therefore, the matrix of T relative to the pair B, B' is
I don't understand why the row reduction takes place? Also, how do these steps relate to ## B = S^{-1}AS ##? Thank you.
This is a cyclic transformation. Is it safe to say thay it's irreversible because if you reverse it, it means I could extract an amount of heat from a cold reservoir and move it into a hotter reservoir with no other effect?
My Progress:
I tried to perform the coordinate transformation by considering a general function ##f(\mathbf{k},\omega,\mathbf{R},T)## and see how its derivatives with respect all variable ##(\mathbf{k},\omega,\mathbf{R},T)## change:
$$
\frac{\partial}{\partial\omega} f =...
I'm trying to investigate the possibility of martensitic transformation in a non-iron alloy, described as a single-phase alpha-solid-solution (Nickel-Silver CuNi12Zn25Pb1, CW404J). I know that Cu-Ni-Zn alloys with higher zinc amounts show even shape memory effects. And that CuNi12Zn25Pb1 is no...
"There is a linear transformation T from R3 to R3 such that T (1, 0, 0) = (1,0,−1), T(0,1,0) = (1,0,−1) and T(0,0,1) = (1,2,2)" - why is this the case?
Thank you.
In《Introducing Einstein's Relativity Ed 2》on page 106"lowering the first index with the metric,then it is easy to establish,for example by using geodesic coordinates..."
In 《A First Course in General Relativity - 2nd Edition》on page 159 "If we lower the index a,we get(in the locally flat...
So,
##\hat{p}(\omega)=\int_{-\infty}^{\infty} p(t)e^{-i\omega t}\mathrm{d}t=A\int_{0}^{\infty}e^{-t(\gamma+i(\omega+\omega_0))}=A\left[-\frac{e^{-t(\gamma+i(\omega+\omega_0))}}{\gamma+i(\omega+\omega_0)}\right]_0^\infty,##
provided ##\gamma+ i(\omega+\omega_0)\neq 0## for the last equality. Now...
I made a tool for calculating and visualizing how the electric and magnetic fields transform under a Lorentz boost. Thought I'd share it here, in case anyone finds it interesting.
https://em-transforms.vercel.app/
Hey I have a question about the relation between Legendre transformation and Hamilton-Jacobi formalism. Is there some relation? Cause Hamilton-Jacobi is the expression of Hamiltonian with a transformation.
Hello,
I would like to reproduce the following equation, but I don't quite understand how to do the transformation:
$$ \sum_{i=1}^k \left( \frac{\langle y , x_i^* \rangle}{\sqrt{\langle x_i^*, x_i^* \rangle}} \right)^2 = \langle y, y \rangle$$
Where ##x_1^*,...,x_k^*##, are orthogonalized...
Could one derive a set of coordinate transformations that transforms events between different reference frames in the de Sitter metric using the invariant line element, similar to how the Lorentz Transformations leave the line element of the Minkowski metric invariant? Would these coordinate...
You see in the literature that the vector potentials in a gauge covariant derivative transform like:
A_\mu \rightarrow T A_\mu T^{-1} + i(\partial_\mu T) T^{-1}
Where T is not necessarily unitary. (In the case that it is ##T^{-1} = T^\dagger##)
My question is then if T is not unitary, how is...
My notes say that the Resolution of the Aperture(in the Electric field of the wave) is the Fourier transformation of the aperture.
Then gives us the equation of the aperture:
and says that for the circular aperture in particular also:
My attempt at solving this:
We know that the Fourier...
1. The 2nd line on the 3rd page of your notes, you have x=ct and
x'=ct', thus ux=dx/dt and ux'= dx'/dt' =c according to Einstein's
assumptiuon.
2. But near the end of the last page, you wrote dx'/dt' = (ux
-v)/(1-vux/c2) . Compare with 1. This equation can be valid only for ux=c
and...
The Lorentz transform for velocities is as follows:
$$u=\frac{v+w}{1+\frac{vw}{c^{2}}}$$
But which assumption exactly underlies this so that you get exactly this formula and not any other formula with approximately the same properties?
THe question is pretty simple. I was doing an exercise, in which $$p = \lambda P, Q = \lambda q$$ is a canonical transformation.
We can check it by $$\{Q,P \} = 1$$
But, if we add $$t' = \lambda ^2 t$$, the question says that the transformation is not canonical anymore.
I am a little...
I don't understand why the highlighted term is there.
This image was taken from Sean Carroll's notes available here: preposterousuniverse.com/wp-content/uploads/grnotes-three.pdf
There is no possible measurement, no matter how clever, that can measure the one way speed of light. It is a synchronization convention. In this topic I would like to apply this idea on a specific case.
I have a microwave oven with width L. In this oven I have a standing wave.
$$E(t,x)=E...
How do the energy and generalized momenta change under the following coordinate
transformation $$q= f(Q,t)$$
The generalized momenta: $$P = \partial L / \partial \dot Q = \partial L / \partial \dot q\times \partial \dot q / \partial \dot Q = p \partial \dot q / \partial \dot Q = p \partial q...
Hi,
I have to verify the sifting property of ##\frac{1}{2\pi i} \int_{-i\infty}^{i\infty} e^{-sa}e^{st} ds## which is the inverse Mellin transformation of the Dirac delta function ##f(t) = \delta(t-a) ##.
let ##s = iw## and ##ds = idw##
##\frac{1}{2\pi} \int_{-\infty}^{\infty} e^{-iwa}e^{iwt}...
While deriving Lorentz transformation equations, my professor assumes the following:
As ##\beta \rightarrow 1,##
$$-c^2t^2 + x^2 = k$$
approaches 0. That is, ##-c^2t^2 + x^2 = 0.## But the equation of the hyperbola is preserved in all inertial frames of reference. Why would ##-c^2t^2 + x^2##...
Consider a set of ##n## position operators and ##n## momentum operator such that
$$\left[q_{i},p_{j}\right]=i\delta_{ij}.$$
Lets now perform a linear symplectic transformation
$$q'_{i} =A_{ij}q_{j}+B_{ij}p_{j},$$
$$p'_{i} =C_{ij}q_{j}+D_{ij}p_{j}.$$
such that the canonical commutation...
I don't know where to start. I understand that the constraint ##ad-bc=1## gives us one less parameter since ##d=1+bc/a##. So we can rewrite our original function. I know how to compute the generators of matrix groups but in this case the generators will be functions. I also know there should be...
I'm trying to understand the so-called polaron transformation as frequently encountered in quantum optics. Take the following paper as example: "Quantum dot cavity-QED in the presence of strong electron-phonon interactions" by I. Wilson-Rae and A. Imamoğlu. We have the spin-phonon model with...
Hello!
Im having some trouble with solving ODE's using Laplace transformation,specifically ODE's that require partial fraction decomposition.Now I know how to do partial fraction decomposition,and have done it many times on standard polynoms but here some things just are not clear to me.For...
I have read that canonical transformation is basically a symplectomorphism which leaves the symplectic form invariant. My understanding is that the canonical transformation is a passive picture where we keep the point on the phase space fixed and change the coordinate chart, where...
I found that the
a) invariant points are all points on y-axis
b) invariant lines are y-axis and ##y=c## where ##c## is real
I am confused what the final answer should be. How to state the answer as "subset of domain"? Is it:
$$\{x,y \in \mathbb R^2 | (0, y) , x = 0, y=c\}$$
Thanks
Standard matrix for T is:
$$P=\begin{bmatrix}
1 & 0 & 0\\
0 & 1 & -1
\end{bmatrix}$$
(i) Since matrix P is already in reduced row echelon form and each row has a pivot point, ##T## is onto mapping of ##\mathbb R^3 \rightarrow \mathbb R^2##
(ii) Since there is free variable in matrix P, T is...
Edit: Ugh accidentally posted instead of previewing, this is a lot of latex to write to give my attempted solution, but I'll keep doing that. I am using the chain rule (or dividing the differential of ##\vec v'## by that of ##t'##). I get
$$d \vec v' = \frac{d \vec v \cdot \vec u}{\gamma c^2...
Two spaceships are heading towards each other on a collision course. The following facts are all as measured by an observer on Earth: spaceship 1 has speed 0.74c, spaceship 2 has speed 0.62c, spaceship 1 is 60 m in length. Event 1 is a measurement of the position of spaceship 1 and Event 2 is a...
Summary:: Special relativity and Lorentz Transformations - I got this problem from a first-semester course at university. I have been struggling for a few days and decided to get some help.
A rocket sets out from x = x' = 0 at t = t' = 0 and moves with speed u in the negative x'-direction, as...
Hello!
Consider this transferfunction H(s);
$$ H(s) =\frac{s-1}{1-2(s^2-s)-As-\frac{A}{2}} $$
Now I need to determine A (note that A is coming from R) so that the impulse response h(t) (so in time domain) so that it contains components with $$te^{at} \sigma(t) $$.
Now I honestly really have...
We got two vectors ##\mathbf{v_1}## and ##\mathbf{v_2}##, their sum is, geometrically, :
Now, let us rotate the triangle by angle ##\phi## (is this type of things allowed in mathematics?)
OC got rotated by angle ##\phi##, therefore ##OC' = T ( \mathbf{v_1} + \mathbf{v_2})##, and similarly...
I want to understand how the domain and range change upon applying transformations like (left/right shifts, up/down shifts, and vertical/horizontal stretching/compression) on functions.
Let f(x)=2-x if 0 ≤x ≤2 and 0 otherwise.
I want to describe the following functions 1) f(-x) 2) -f(x) 3)...
We have a transformation ##T : V_2 \to V_2## such that:
$$
T (x,y)= (x,x)
$$
Prove that the transformation is linear and find its range.
We can prove that the transformation is Linear quite easily. But the range ##T(V_2)## is the the line ##y=x## in a two dimensional (geometrically) space...
I believe this does not belong to the homework category. I hope I won't be mistaken.
I am reading a book to self-study special relativity, the following is an example mentioned in the book.
When clock C' and clock C1 meet at times t'=t1=0, both clocks read zero. The Observer in reference frame...