What is Variation: Definition and 574 Discussions

Homework Statement You probably know that intriguing problem: http://en.wikipedia.org/wiki/Ant_on_a_rubber_rope. Now, suppose 2 cars A, B are speeding at vc on two roads at 60° (in a vacuum) so that their distance is at any time rt = Tt*vc. Road A points North (edit: A 0°, B 300°). At time 10...

Hi all, This is Newton's universal law of gravitation: F = GMm/r2, where r is the distance between the centre of the two bodies. Therefore, considering two objects in mutual gravitational acceleration, with only linear motion and acceleration, they shall be moving in closer and closer. Since...

Hello everyone! I was reading the following review: http://relativity.livingreviews.org/open?pubNo=lrr-2009-4&page=articlesu23.html And I got stuck at the first equation; (10.1) So how I understand this is that there are two variations, \tilde{q}(t)=q(t)+\delta q(t)...

Homework Statement Prove that under an infinitesimal Lorentz transformation: x^\mu \to x^\mu+\omega^\mu_\nu x^\nu so: \phi\to\phi-\omega^\mu_\nu x^\nu\partial_\mu\phi the Lagrangian varies as: \delta \mathcal{L}=-\partial_\mu(\omega^\mu_\nu x^\nu \mathcal{L}) The Attempt at a...

Homework Statement i have attached the notes from 2 books below, i know that the graph of pressure of sound waves lag behind the displacement grpah by 90 degree. so it should be p=p max sin (wt-kx-(pi/2)) am i right? why the another book gives p=p max sin (wt-kx+(pi/2)) ? which is correct...

A new variational principle is presented in this paper: http://arxiv.org/ftp/arxiv/papers/1112/1112.2286.pdf When trying to derive something like the equation of motion of a Duffing oscillator, I take the following approach: Set up the functional as such: $$...

Hi all, Consider the scatter plot below (example only): The bottom subplot looks like a residual plot (y-x), but it's over (1+x). I'm hoping someone can explain what this is and the benefit of it. Thanks

Hi everyone! There is something that I would like to ask you. Suppose you have \frac{1}{\sqrt{-g}} \frac{\delta (\sqrt{-g} (g^{ab} u_a u_b + 1))}{\delta g^{cd}} The outcome of this would be ##u_{c}u_{d}## or ##-u_{c} u_{d}## ? I am really confused.

Homework Statement The following link shows the variation of vapor pressure with temperature. In which way the data should be plotted to obtain a straight line? http://www.chem.purdue.edu/gchelp/liquids/vpress.html The Attempt at a Solution For this, I need to know exactly the...

Homework Statement Suppose that we use a Michaelis-Menten growth rate in the chemostat model, and that the parameters are chosen so that a positive steady state exists. Show that N = f(V,F,C0) = (C0(F - VKm) + FKn)/(a(F - VKm)) and C = (FKn)/(F - VKm) at the positive steady state. The...

How do I set up the following problem? What geometric surface encloses the maximum volume with the minimum surface area?

Hello. A friend sent me the following problem that she wants to include in an essay: There are two epistemic peers whose mental faculties are of equal standing and who have access to all the same relevant evidence. The two go to dinner Case 1: One week later, the first man, A, states...

Homework Statement $$y'' - 2y = x + 1$$ Homework Equations $$ y_{o} = Ae^{√(2)x} + Be^{-√(2)x} $$ $$ v_{1}'e^{√(2)x} + v_{2}'e^{-√(2)x}\equiv 0 $$ $$ √(2)v_{1}'e^{√(2)x}-√(2) - v_{2}'e^{-√(2)x} = x + 1 $$ The Attempt at a Solution $$ v_{2}' = \frac{x+1}{-2√(2)e^{-√(2)x}} $$ $$...

If for some functional ##I##, ##δI=0## where ##δ## is symbol for variation functional has extremum. For ##δ^2I>0## it is minimum, and for ##\delta^2I>0## it is maximum. What if ##δI=δ^2I=0##. Then I must go with finding further variations. And if ##δ^3I>0## is then that minimum? Or what?

If for some functional ##I##, ##\delta I=0## where ##\delta## is symbol for variation functional has extremum. For ##\delta^2 I>0## it is minimum, and for ##\delta^2 I<0## it is maximum. What if ##\delta I=\delta^2 I=0##. Then I must go with finding further variations. And if ##\delta^3I>0## is...

Homework Statement For a problem I'm doing, I am considering a universe in which k=0, and I'm told that I can consider most of the expansion to have happened during a phase when only one of the density parameters was dominant (I know which one, as well), but I don't know the scale factor or...

Homework Statement The problem is attached as TheProblemAndSolution.png, and everything is typewritten, so it should be easily legible (but you will likely need to zoom into read the text since the image's height is significantly larger than its width). Homework Equations Differential...

For a set with n points of data, why is the "degree of freedom" of the standard variance n-1? Hell, what does "degree of freedom" actually mean? Heck, my book "proves" this by saying that since ##\sum_1^n (x_i - \bar{x}) = 0## (obviously), then ##\sum_1^n (x_i - \bar{x})^2## must have n-1...

Homework Statement Hi, I have a problem calculating the variation of the action using tensor algebra because two derivative indices are causing some problem. Homework Equations Generally you have the action S = \int L(A^{\mu}, A^{\mu}_{\;,\nu}, x^{\mu})d^4x where: A ^{\mu}=...

Homework Statement solve 4y''-4y'+y=16et/2 Homework Equations v1= -∫ y2g/w v2= ∫ y1g/w The Attempt at a Solution http://imgur.com/gxXlfdH the correct answer is 2t^2 e^(t/2) instead of what i have though, i am not sure what i am doing wrong?

I am reading Apostol's section on Riemann-Stieltjes integral and I have doubts on one statement: Let ##α## be a function of bounded variation on ##[a,b]## and suppose ##f \in R(α)## on ##[a,b]##. We define ##F## as ##F(x)=\int_a^x f(x)dα## if ##x \in [a,b]##, then ##F## is a function of...

Homework Statement . Prove or disprove that the function ##f(x)= x^2sin^2(\dfrac{\pi}{x})## if ##0<x\leq 1## and ##f(x)=0## if ##x=0## is of bounded variation. The attempt at a solution. I've seen the graph of this function on wolfram and for me it's clearly not of bounded variation since it...

Let (X,R) be a measure space. v=u_{1}+iu_{2} be a complex charge. Find the variation of v when u_{1}, u_{2} are positive disjunctive charges. Does disjunctive charges mean that there is a partition A, B of X such that u_{1}(A)= u_{2}(B)=0?

Refer to "2.jpg", it said that the shortest path on the surface of a sphere is Ay-Bx=z , which is a plane passing through the center of the sphere. I cannot really understand about this. Does it mean that the shortest path is a ring that connects two points with its center at the center of the...

So the expression for Gibb's free energy is: dG = -SdT + VdP + μdN, Here, we see that the Gibb's free energy changes with temperature (dT), change in pressure (dP) and change in chemical potential (as a result of change in particle number). My question is: we know chemical potential...

Hi, I've worked out that for the particle-in-a-box model (square well) the energy E E = 3h^2/8mL^2 where m is the effective mass of the electron. The next question asks Hence derive an expression for the variation of the effective energy gap in the quantum dot as a function of its...

y'' + y = f(x) -- Variation of Parameters? Homework Statement Use variation of parameters to solve ##y'' + y = f(x), y(0) = y'(0) = 0.##Homework Equations A description of the method is here: http://en.wikipedia.org/wiki/Variation_of_parametersThe Attempt at a Solution The complementary...

The following inequality can easily be proved on ##ℝ## : ## ||x|-|y|| \leq |x-y| ## I was wondering if it extends to arbitrary normed linear spaces, since I can't seem to prove it using the axioms for linear spaces. (I can however, prove it using the definition of the norm on ##ℝ## by using...

Consider a charged wire with constant linear charge density λ. The wire has length 2πa and is attached to the edge of a disc with radius a. In the central region of the disc (a circular region of radius b<a) a constant magnetic field B is applied (perpendicular to the disc). The magnetic...

Theoretically , a change in either electric or magnetic field will cause a current to flow , i am already familiar to Faraday's law of electromagnetic induction , so i tried to search about producing a current using a varying electric field, didn't find anything , i found an interseting...

Hi Guys, I am not a physicist, just an average blue collar Joe who finds QM fascinating. I was wondering why they always use parallel slits. Has anybody ever tried varying the angle between the slits? And what would the interference pattern look like at different angles, starting at 1...

Hello everyone, I have a concern regarding the conservation of momentum for an incompressible Newtonian fluid with constant viscosity. Say you have a volume of fluid sliding down an inclined plane with a velocity Vx with the perpendicular axis facing upward in the y-direction. When you try...

Source of Question J.N. Reddy states in Finite Element Method: "Variational forms of the laws of continuum physics may be the only natural and rigorously correct way to think of them. While all sufficiently smooth fields lead to meaningful variational forms, the converse is not true: There...

I was given the problem, "Find the general solution of the given differential equation." \(y^{\prime\prime}+9y=9\sec^2(3t)\) My work as follows, please let me know if this is correct and where to go from here. I have hit a roadblock of sorts. \(y^{\prime\prime}+9y=9\sec^2(3t)\)...

Helo, I've been working out this exercise, but my solution and the text's aren't the same. Homework Statement We have a spotlight on the floor located at a distance of 30.5 m from a wall of a building. There is a person 1.83 m tall between the spotlight and the wall moving away from the...

In calculus of variation, we use Euler's equation to minimize the integral. e.g. ∫f{y,y' ;x}dx why we treat y and y' independent ?

This #94 from the 2008 GRE: An observer O at rest midway between two sources of light at x = 0 and x = 10 m observes the two sources to flash simultaneously. According to a second observer O′, moving at a constant speed parallel to the x-axis, one source of light flashes 13 ns before the...

Hi, I derived the equation: 1+(y')^2-y y''-2y\left(1+(y')^2\right)^{3/2}=0 Letting y'=p and y''=p\frac{dp}{dy}, I obtain: \frac{dp}{dy}=\frac{1+p^2-2y(1+p^2)^{3/2}}{yp} I believe it's tractable in p because Mathematica gives a relatively simple answer: p=\begin{cases}\frac{i...

Suppose we are taking the variation of a multiple integral and the integrand contains some terms with \frac{\partial g}{\partial x}. When is it ok to put \delta\frac{\partial g}{\partial x}=\frac{\partial}{\partial x}(\delta g) ?

Hello! Im currently reading Ryder's QFT book and am confused with the variation of a scalarfield. He writes that the variation can be done in two ways, \phi(x) \rightarrow \phi'(x) = \phi(x) + \delta \phi(x) and x^\mu \rightarrow x'^\mu = x^\mu + \delta x^\mu. This seems...

Homework Statement Prove that if ##\left |x-x_{0} \right | < \frac{\varepsilon }{2}## and ##\left |y-y_{0} \right | < \frac{\varepsilon }{2}## then ##|(x+y)-(x_0+y_0)| < \varepsilon ## and ##|(x-y)-(x_0-y_0)| < \varepsilon ##Homework Equations Postulate and proof with real numbers as well...

Hi! I do have this problem - Consider that for a set of values, I do have a Poisson distribution with mean value <m> - Now, I need to gather another set of dataset, which I should vary the mean value by 5% - My question is, how can I translate each one of these new values to sigma deviations...

So we know that there is variation of EM radiation in different latitudes we receive from Sun. My question is, it same through all EM radiation like uv rays, visible spectrum , IR spectrum or specific to only to one type of radiation like only uv or IR rays? i.e. higher latitude, there is less...

Hi all, Doing some calculations on an air-cooled heat exchanger at the moment and could use some help understanding the variation of air-side heat transfer coefficient (htc) with flow rate. It's more of an intuitive problem really as I'm okay with the math, etc. So I know that as you...

I have a composite function f(g(x,y)). When is it true that ∂f/∂g = (∂f/∂x)(∂x/∂g) + (∂f/∂y)(∂y/∂g)? Does g have to be invertible with respect to x and y for this to be true?

Homework Statement Okay, I have no idea about the method they want me to solve it with. What in this case is the indicator that a function is appropriate? A particle mass m affects a potential of the form ##V(x)=V_0 \frac{|x|}{a}## where ##V_0## and ##a## are positive constants. a) Draw a...

I am working on the following problem: Can someone please show or explain the steps to invert the phi matrix? I've given it a few tries, but I can't reach what the book has for the answer. Please help! Thanks

I'm reading Classical Mechanics (Taylor), and the 6th chapter is a basic introduction to calculus of variations. I'm super confused :confused: I've tried to go to other sources for an explanation, but they just make it even worse! So, let me see if I can get some help here...

If we take the variation of a functional of some function \phi(x_1,...,x_n) with \partial_{j}\phi being the partial deriviative of phi with respect x_j, when is it ok to set \delta \partial_j \phi equal to \partial_j (\delta\phi)?

The coefficient of variation tells us about the consistency in the data. I know that the lower the coefficient of variation is, the higher will be the consistency in the data. What I don't understand is what is being meant by 'consistency' here. Could someone please explain that?