Form Definition and 1000 Threads

I grok, am NOT asking about, the answers below. Rather, how can I calculate the final answer DIRECTLY, without division? I don't know why my Latex isn't rendering here? Please see: [web link redacted by the Mentors] Orange underline 1. Unquestionably, $\color{#FFA500}{4 \times 3/2} = 3!$ But...

On page 180 they write the following passage: My question is how do I get ##F^M_{(0,0)}## from ##F^M_{(0,0)}(q^2)## for large ##q^2##. I tried using Mathematica's free input to get the series ##1/(1+x^2)^2=1-2x^2+3x^4+O(x^6)##, in our case ##x=\gamma/q##. But it doesn't seem to fit the result...

Imagine a ball floating in space, and there is an observer at rest next to it. Then the observer goes for a trip at high speeds and returns to be at rest with the ball once again. Would the world line of the ball be curved because of the frame of reference of the observer, or would the world...

I am sure you are all familiar with the cross product in 3D space. i cross into j gives k. Cyclic Negative, if reversed, etc. I am sure you are all familiar with the definition as: norm of the first vector, norm of the second, sine of the angle, perpendicular (but direction using right hand...

Energy and mass are interdependent and electrons can manifest as particles and fields as do all other particles, but is it generally true that physical(classical) matter is a peculiar type of energy that can(for some reason? What?) manifest as physical objects?

##S## is a set of all vectors of form ##(x,y,z)## such that ##x=y## or ##x=z##. Can ##S## have a basis? S contains either ##(x,x,z)## type of elements or ##(x,y,x)## type of elements. Case 1: ## (x,x,z)= x(1,1,0)+z(0,0,1)## Hencr, the basis for case 1 is ##A = \{(1,1,0), (0,0,1)##\} And...

I put it in echelon form but don't know where to go from there.

Hi, I would like to ask for a clarification about the difference between a differential k-form and a generic (0,k) tensor field. Take for instance a (non simple) differential 2-form defined on a 2D differential manifold with coordinates ##\{x^{\mu}\}##. It can be assigned as linear combination...

Based on the photo electric effect. Maybe its perception that is the problem? Charge doesn't travel through a vaccuum. Electro magntic waves alway carry photons. Photons can make charge. Charge creates magnetism? Would explain why Coulumbs equation is the same as the one for gravity but on a...

Would you be able to help me figure out the equation to be used at both lenses to get the final light sheet?

Is anybody aware of how to install FORM(hep) for symbolic calculation? Is that support on windows? Can you share any pdf about installation

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I'm working on a model which produces a form of concavity which I'm not familiar with. Does anyone know what this form is called and if it has been studied before? The definition in its differentiable version reads: Let ##X\subset \mathbb{R}^{n}##. A differentiable function ##f##, defined on...

#17 Find values of $r$ of the form $y = e^{rt}$ $y''+y'-6y=0$ $r^2+r-6=(r-2)(r+3)\quad \therefore r=2 \quad r=-3$ well so far it that all there is to do? However I didn't see clearly what the purpose of this was or how it is eventually used.

I don't know how to solve for u(x,y) from where I left of after 5.

I have the following integration - $$\int \frac{2}{x - b \frac{x^{m - n + 1}}{(-x + 1)^m}} \, dx $$ To solve this I did the following - $$\int \frac{1 - b \frac{x^{m - n}}{(-x + 1)^m}+1 + b \frac{x^{m - n}}{(-x + 1)^m}}{x(1 - b \frac{x^{m - n}}{(-x + 1)^m})} \, dx $$ Which gives me -...

The covariant form for the Levi-Civita is defined as ##\varepsilon_{i,j,k}:=\sqrt{g}\epsilon_{i,j,k}##. I want to show from this definition that it's contravariant form is given by ##\varepsilon^{i,j,k}=\frac{1}{\sqrt{g}}\epsilon^{i,j,k}##.My attemptWhat I have tried is to express this tensor...

Phlogiston lead to the discovery of oxygen but was later on proven to be imaginary. Can the same be said of the quite strange Higgs field and Higgs mechanism following from the strange Mexican hat potential? Is that potential just imaginary? It lead to the discovery of the eponimous particle...

Here I'm asking solely about the circle pictograms. Please eschew referring to, or using, numbers as much as possible. Please explain using solely the circle pictograms. Undeniably, I'm NOT asking about how to divide numbers. I don't understand 1. How do I "visually" divide Circle 1...

I saw this question at https://math.codidact.com/posts/283253.

##\vec{L} = \vec{P} \times\vec{r}## ##L = mvr sin \phi##, where P = mv Since ##\vec{r}## and ##\vec{v}## are always perpendicular, ##\phi## = 90. Then, ##L = mvr## At this point, I don't see how to get ##L = mvr = mr^2\omega##, using ##\omega = \dot{\phi}## I know that ##\omega =...

My book claims that the diff. form of Gauss' law is $$\nabla\cdot\mathbf E=4\pi\rho$$ Can someone tell me why it isn't ##\nabla\cdot\mathbf E=\rho/\epsilon_0##?

Attempting this question without any guidance from my professors unfortunately as they did not teach this bit. Searched online and also there aren't many questions like this. From what I know, (I) Having n-1 means you should shift right by 1, which means x[0] is now equals to 0? So x[n-1] = [0 5...

I've been trying for a very long time to show that the following integral: $$ I_D=2{\displaystyle \int} \, {\displaystyle \prod_{i=1}^3} d \Pi_i \, (2\pi )^4\delta^4(p_H-p_L-p_R) |{\cal M}({e_L}^c e_R \leftrightarrow h^*)|^2 f_{L}^0f_{R}^0(1+f_{H}^0). $$ can be reduced to one dimension: $$ I_D...

Ref https://arxiv.org/abs/quant-ph/9903047. They say the form of their no-which-path-info interference curves (figs 3,4) is "standard". But a standard interference curve has a zero base line. Their base line is a humped curve of the form of their Fig.5, but with about 1/3 of its height. They...

A school district purchases a high-volume printer, copier, and scanner for $24,000. After 10 years, the equipment will have to be replaced. Its value at that time is expected to be $2000. Write a linear equation giving the value V of the equipment during the 10 years it will be in use. Let t =...

Chapter 1, Section 1.2 Write the standard form of the equation of the circle with the given characteristics. 72. Center: (−2, −6); Solution point: (1, −10) Solution: given: Center: (−2, −6); => h=-2, k=-6 => then (x - (-2))^2 + (y - (-6))^2 = r^2 (x +2)^2 + (y +6)^2 = r^2... I then use...

Chapter 1, Section 1.2 Write the standard form of the equation of the circle with the given characteristics. 74. Endpoints of a diameter: (11, −5), (3, 15) I want to know if the following steps are correct for me to answer the above question. Steps: 1. Find the distance between the points...

Hello: I'm not sure if there's an accepted canonical form for a quadratic equation in two (or more) variables: $$ax^2+by^2+cxy+dx+ey+f=0$$ Is it the following form? (using the orthogonal matrix Q that diagonalizes the quadratic part): $$ w^TDw+[d \ \ e]w+f=0$$ $$w^TDw+Lw+f=0$$ where $$...

Problem statement : Given the equation ##\sin\theta+2\cos\theta=1##, find the general solution for the angle ##\theta##. Attempt : For the general case where we have ##a\sin\theta+b\cos\theta=c##, the line of approach is to take ##a=r\cos\alpha## and ##b=r\sin\alpha## wherein we will have...

Hey guys! Sorry if this is a stupid question but I'm having some trouble to express this charge distribution as dirac delta functions. I know that the charge distribution of a circular disc in the ##x-y##-plane with radius ##a## and charge ##q## is given by $$\rho(r,\theta)=qC_a...

I understand how waves undergo superposition. However, for a standing wave, the reflected wave is a mirror opposite of the incoming wave. By the superposition principle, won’t the 2 waves add up to 0, at all points?

Hi, I just have a quick question regarding image processing. What is the correct form of the Laplacian for image processing? I have seen different versions online and don't understand which one is meant to be the conventional one. I know that: \nabla^2 f(x, y) = \frac{\partial^2 f}{\partial...

Since the question asks for Cartesian coordinates, I wrote dV as 2pi(x^2+y^2+z^2)dxdydz and did the integral over the left hand side of the equation with x, y, z from 0 to R. My integral returned (0, 2*pi*R^5, 5/3*pi*R^6) which doesn't seem right. I also tried to compute the right-hand side of...

I know the basics of GR, but I'm far from an expert. I'm interested in the general formula for the angle shift experienced by a light ray in presence of a gravitational field. Light bending is topic covered in many books, of course, however the usual treatments would just not do for my purposes...

Bell's inequality in it's original form is: |cor(a,b) - cor(a,c)| \le 1 - cor(b,c) where ##a,b## and ##c## are random variables with values ##\pm 1##, and the correlation is then simply the expectation value of their products, ##cor(a,b)=E[ab]## or as usually expressed ##\langle ab\rangle##...

Hi guys, I'm new here. I am doing my final degree project and it's hard for me to understand what this paragraph means in one of the papers that I'm reading, it's about primordial black hole formation. [Talking about a spherically density perturbation] The rarefaction wave starts at the surface...

Hi, I found the above observable canonical form using this source: https://www.mathworks.com/help/control/ug/canonical-state-space-realizations.html#mw_a76b9bac-e8fd-4d0e-8c86-e31e657471cc I'm almost certain that I did do it correctly. But the code below gives me different values for B, C...

Hey! :giggle: Let \begin{equation*}A=\begin{pmatrix}0 & -2 & 2 & 0 & 0 & -6 \\ 0 & 0 & 0 & 1 & 1 & 3\\ 0 & 0 & 0 & 1 & -1 & -1 \\ 0 & 1 & -1 & 0 & 2 & 7\\ 0 & 3 & -3 & 1 & 2 & 14\end{pmatrix}, \ b_1=\begin{pmatrix}0 \\ 0 \\ 0 \\ 0 \\ 0\end{pmatrix} , \ b_2=\begin{pmatrix}-2 \\ 1 \\ -1 \\ 3 \\...

Hi, I was recently being taught a control theory course and was going through a 'derivation' of the controllable canonical form. I have a question about a certain step in the process. Question: Why does the coefficient ## b_0 ## in front of the ## u(t) ## mean that the output ## y(t) = b_0 y_1...

2 separate big conductors initially charged Q1 and Q2. Then connect them in a circuit with a battery of emf V. The charges Q1 and Q2 will go to the 4 surfaces (marked red). All the 4 surfaces have an area A. Suppose the 2 conductors form an ideal parallel plate capacitor and the wires in the...

Hi guys, I am having difficulties in solving this limit. Below, I'll attach my procedure which ends up in the indeterminate form ##0\cdot \infty##... How could I solve it? $$\lim_{x \to +\infty}(\sqrt[3]{x^3-4x^2}-x) \rightarrow \lim_{x \to +\infty}(x\sqrt[3]{1-\frac{4}{x}}-x) \rightarrow...

Greetings. I'm having a bit of difficulty with getting from the first to the second equation. I know some basic identities, but it all just feels like a mess. My approach was just going to be to write whatever I could, but some of the terms are confusing me...

Kepler's equation is M = E - e*sinE According to wiki - Kepler's equation - "Kepler's equation is a transcendental equation because sine is a transcendental function, meaning it cannot be solved for E algebraically. Numerical analysis and series expansions are generally required to evaluate...

The first two parts I think were fine, I expressed the tensors in coordinate basis and wrote for the first part$$ \begin{align*} \mathcal{L}_X \omega = \mathcal{L}_X(\omega_{\nu} dx^{\nu} ) &= (\mathcal{L}_X \omega_{\nu}) dx^{\nu} + \omega_{\nu} (\mathcal{L}_X dx^{\nu}) \\ &= X^{\sigma}...

A self test question from Chemical Principles(Atkins/Jones). Other than rote memorization I don’t see how to do this. The review section references the Periodic Table and, for elements on the right side of the table, one can use 18 minus the group number which would give Al(superscript 5+) since...

As we know that matter is Anything which has mass and occupies space then are the Subatomic particles like electron,proton and neutron a matter. Is electron a matter ? Is proton a matter ? Is neutron a matter ? As these particles have mass and they occupy space so they are matter, But the...

In the book "The Variational Principles of Mechanics" by Cornelius Lanczos, the following statement is made about a lagrangian ##L_1## where time is given as an dependent parameter, and a new parameter ##\tau## is introduced as the independent variable, see (610.3) and (610.4) pg. 186,187 Dover...

My idea was to consider first the structure of the matrix element and to see if there are any possible constraints that we could use for parametrization. If I am not mistaken, we are dealing with the hadronic decay governed by QCD which conserves parity. Since we have a derivative operator...