Transformation Definition and 1000 Threads

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...

I have a f(t) that is, e^(-t) *sin(t), now I calculate the Laplace transformation, that is: X(s) = 1 / ( 1 + ( 1 + s)^2 ) (excuse me but Latex seems not run ). Now I imagine the plane with Re(s), Im(s) and the magnitude of X(s). If i take Re(s) = -1 and Im(s) = 0, I believe I have X(s) = 1 ( s...

Is there the simplest, direct, and easy-to-understand method that only needs to apply the most basic algebra and logic to completely and strictly derive the Lorentz transformation? Thanks for your help.

It's frequently discussed Galilean transformation brings one inertial frame to another inertial frame, and such a transformation leaves Newton's second law invariant (of the same form). I wonder what happens for non-inertial frame? If we start with a non-inertial frame, and Galilean transform...

I've a transformation ##T## represented by an orthogonal matrix ##A## , so ##A^TA=I##. This transformation leaves norm unchanged. I do a basis change using a matrix ##B## which isn't orthogonal , then the form of the transformation changes to ##B^{-1}AB## in the new basis( A similarity...

REMOVED pending revision

I have a function in polar coordinates: t (rho, phi) = H^2 / (H^2 + rho^2) (1) I have moved the center to the right and want to get the new formulae. I use cartesian coordinates to simplify the transformation (L =...

I know we can prove that a Galilean transformation sends one inertial frame to another inertial frame, by proving ##\frac{d^2 f(\vec{r})}{d(f(t))^2} = \frac{d^2 \vec{r}}{dt^2}##, but can we prove the reverse? Can we prove that if the acceleration seen in two frames are the same, then the...

Let ##|z|=1## and ##1-\bar{a}z\neq 0##. Evaluate ##\frac{|z-a|}{|1-\bar{a}z|}##. It should be a real number. I read that ##f=\frac{|z-a|}{|1-\bar{a}z|}## is a mobious transformation, but I do not know what it means. @fresh_42##z=e^{i\theta_1}, a=r_2e^{i\theta_2}##...

I know that if the transformation was canonical, the form of Hamilton's equation would remain invariant. If the generating function for the transformation was time independent, then the Hamiltonian would be invariant and we could directly replace q and p with the transformation equations to get...

My textbook (from first year university physics) says that length contraction is actually real. But how can it be real when two different observers can measure two different lengths? For example, if I am in a spaceship going close to the speed of light relative to people on Earth, they will...

In the special theory of relativity, it seems impossible to derive the lorentz transformation without assuming that the lorentz factor is independent of the sign of the relative velocity. For some reason, I can't get my head around why this assumption is so easily made, as if it's trivial. Can...

While learning about Special Relativity I learned that we use the Transformation matrix to alter the space .This matrix differs for Contravariant and Covariant vectors.Why does it happen?,Why one kind of matrix (Jacobian) for basis vectors and other kind(Inverse Jacobian) for gradient...

I have the matrix above and I have to find which transformation is that. ##\begin{bmatrix} cos \theta & sin \theta \\ sin \theta & -cos \theta \end{bmatrix}## For a vector ##\vec{v}## ##v_x' = v_x cos \theta + v_y sin \theta## ##v_y' = v_x sin \theta - v_y cos \theta## If ##\phi##...

Anyone else out there convinced that MUH is on the right track? I asked the question "What would reality look like if it were all math structures", here's what I came up with: 1) Reality arises from abstract geometric objects of varying shapes and dimensionality whose transformations are being...

Hi, It's not homework but I still thought I better post it here. Please have a look on the attachment. For hi-resolution copy, please use this link: https://imagizer.imageshack.com/img922/7840/CL6Ceq.jpg I think in equations labelled "12", 'e' is electric charge and Ex is the amplitude of...

For fun, I have decided to implement a simple XOR encryption algorithm. The first step is to convert messages into bytes to perform XOR operation on each bit. The problem has started here. For instance, I want to encrypt this message. I hiked 24 miles. Now I need to turn this text into binary...

Hey! :giggle: Let $1\leq n\in \mathbb{N}$ and for $x=\begin{pmatrix}x_1\\ x_2\\ \vdots \\ x_n\end{pmatrix}, \ x=\begin{pmatrix}x_1\\ x_2\\ \vdots \\ x_n\end{pmatrix}\in \mathbb{R}^n$ and let $x\cdot y=\sum_{i=1}^nx_iy_i$ the dot product of $x$ and $y$. Let $S=\{v\in \mathbb{R}^n\mid v\cdot...

For a massless particle let\begin{align*} S[x,e] = \dfrac{1}{2} \int d\lambda e^{-1} \dot{x}^{\mu} \dot{x}^{\nu} g_{\mu \nu}(x) \end{align*}Let ##\xi## be a conformal Killing vector of ##ds^2##, then under a transformation ##x^{\mu} \rightarrow x^{\mu} + \alpha \xi^{\mu}## and ##e \rightarrow e...

hi guys I was trying to find the matrix of the following linear transformation with respect to the standard basis, which is defined as ##\phi\;M_{2}(R) \;to\;M_{2}(R)\;; \phi(A)=\mu_{2*2}*A_{2*2}## , where ##\mu = (1 -1;-2 2)## and i found the matrix that corresponds to this linear...

I want to understand bettew what this statement says. Maybe later we could try to put it mathematically, but for while i want to know if my interpretation is right. When we lie outside the light cone, the physics regarding the limit of the velocity is break, and technically we could go faster...

Let's same I have an observer A and B that initially occupy the same point at ##t=0## but they have a relative velocity to each other. Now let's assume there is an object C that moves in a circular motion around some point from A's frame. The initial condition/position is given (in A's frame)...

I am taking a course on General Relativity. Recently, I was given the following homework assignment, which reads > Derive the following transformation rules for vielbein and spin connection: $$\delta e_a^\mu=(\lambda^\nu\partial_\nu e_a^\mu-e_a^\nu\partial_\nu\lambda^\mu)+\lambda_a^b e_b^\mu$$...

∂/∂x ((ρh^3)/12μ ∂p/∂x) + ∂/∂z ((ρh^3)/12μ ∂p/∂z) = ∂/∂x (ρh (U_1+U_2)/2) + ∂/∂z (ρh (W_1+W_2)/2) + (∂(ρh))/∂t (1) 1/r ∂/∂r (r (ρh^3)/12μ ∂p/∂r) + 1/r ∂/∂θ ((ρh^3)/12μ ∂p/r∂θ) = rω/2 ∂(ρh)/r∂θ + (∂(ρh))/∂t (2)

Hello! I need to check if this transformation (not sure if it is the right word in English) from ## R^3 to R^3 ## is linear f(x1,x2,x3) = f(sin(x1),x2+x3,0). Now we are given that the transformation is linear if this you can prove this statement. $$f(\lambda * u + \mu * v) = \lambda * f(u) +...

I posted a thread yesterday and I think that I did not formulated it properly. So I have a metric ##{ds}^{2}=-{dt}^{2}+{dx}^{2}+2{a}^2(t)dxdy+{dz}^{2}## I was asked to find the the coordinate transformation so that I can get a diagonalized metric. so what I've done is I assumed a coordinate...

hey there :) So I had a homework, and I was asked to diagonalize the metric ##{ds}^2=-{dt}^2+{dx}^2+2a^2(t)dxdy+{dz}^2## and to find the coordinate transformation for the coordinates of the new metric. so I found the coordinate transformation but the lecturer said that what I found is a...

The metric tensor in an inertial frame is ## \eta = diag(-1, 1)##. Where I amb dealing with only 1-D space. The metric tranformation rule after a crtain coordinate chane is the following: $$ g_{\mu \nu} = \frac{\partial x^\alpha}{\partial x'^{\mu }} \frac{\partial x^\beta}{\partial x'\nu }...

Let us suppose we have a covariant derivative of a contravariant vector such as $$\nabla_{\mu}V^{\nu}=\partial_{\mu}V^{\nu} + \Gamma^{\nu}_{\mu \lambda}V^{\lambda}$$ If ##\Delta_{\mu}V^{\nu}## is a (1,1) Tensor, it must be transformed as $$\nabla_{\bar{\mu}}V^{\bar{\nu}} = \frac{ \partial...

Hello, Let's consider a vector ##X## in 2D with its two components ##(x_1 , x_2)_A## expressed in the basis ##A##. A basis is a set of two independent (unit or not) vectors. Any vector in the 2D space can be expressed as a linear combination of the two basis vectors in the chosen basis. There...

Hi, Context: I was looking through some circuits material in the topic of filter synthesis for two port networks. For simplicity, these networks have been synthesized for normalized conditions: 1 rad/s cut-off frequency and a termination resistor of 1 ## \Omega ##. An example filter is shown...

if $Q(\theta)$ is $\left[\begin{array}{rr} \cos{\theta}&- \sin{\theta}\\ \sin{\theta}&\cos{\theta} \end{array}\right]$ how is $Q(\theta)$ is a linear transformation from R^2 to itself. ok I really didn't know a proper answer to this question but presume we would need to look at the unit...

a. I believe that y=ln(2x) is a horizontal stretch of y=ln(x) of scale factor 1/2. In the transformation y=ln(2x), each x-value is multiplied by 2 before the corresponding y-value is calculated. b. I think that y=ln(4-x) is a reflection in the y-axis followed by a translation by the vector...

Hi, starting from this thread I'm a bit confused about the content of the principle of relativity from a mathematical point of view. Basically the "Galilean principle of Relativity" puts requirements on the transformation laws between Inertial Frame of Reference (IFR); thus they have to...

I tried hard to understand what this author proposed, but I feel like I failed miserably. My attempt of solution is here: Item (a) is verified in the case where ##n = 2##, since ##F## being a linear transformation, by the Corollary of the Nucleus and Image Theorem, ##F## takes a basis of...

Dear Everybody, I am in the process of relearning high school geometry through Khan Academy. I am current an graduated undergraduate student in mathematics. I am doing this because geometry is one of my weakest subject in mathematics. Second reason is that I want to reason out a problem...

Hi, I have a question about probability transformations when the transformation function is a many-to-one function over the defined domain. Question: How do we transform the variables when the transformation function is not a one-to-one function over the domain defined? If we have ## p(x) =...

Attempt at a solution To show φ satisfies our PDE, we first solve the substitution for φ ##\mathrm{ln(\phi) = -\frac {1} {2} \int u dx}## which gives ##\mathrm{\phi = e^{-\frac {1} {2} \int u dx} }## and plug it into our PDE, which simplifies to ##\mathrm{\frac {\partial } {\partial t} -...

So while reading T. Frankel's "The Geometry of Physics", I was going through the part on cotangent bundles which ended with the definition of Poincare 1-form. The author argued that cotangent bundles are better suited than tangent bundles for some problems in physics and that there is no natural...