Variation Definition and 540 Threads

Homework Statement 1. We are considering a step junction at equilibrium(no external voltage applied). 2. The potential variation is shown as negative potential at P region(which is shown as constant) and increasing through the transient region to become positive in the n region. Homework...

Homework Statement Find the general solution of the following equation: u(t): u' = u/t + 2t Homework Equations y' + p(x)y = Q(x)....(1) yeI = ∫ dx eIQ(x) + constant.....(2) The Attempt at a Solution I rearranged the equation to give: u' - u/t = 2t Then I considered the following...

Given a ODE like this: y''(t) - (a + b) y'(t) + (a b) y(t) = x(t) The general solution is: y(t) = A exp(a t) + B exp(b t) + u(t) exp(a t) + v(t) exp(b t) So, for determine u(t) and v(t), is used the method of variation of parameters: \begin{bmatrix} u'(t)\\ v'(t)\\ \end{bmatrix} =...

Homework Statement i want to know about the variation in pressure according to varo=ious depth in different water bodies like for river, lake, stream, and all with the possibility of having fresh water or salty water. (not talking about oceans) may be shore of a sea will also work. Homework...

If we have a water filled horizontal pipe on seabed with 200 bar in it. And boat with a flexible downline is connected to the pipeline in 100m of water depth. a) If the vessel bobs up and down by 5m I think that the pressure in the pipeline read at vessel changes by 0.5 bar each way, due to...

A system is in contact with a reservoir at a specific temperature. The macrostate of the system is specified by the triple (N,V,T) viz., particle number, volume and temperature. The canonical ensemble can be used to analyze the situation. In the canonical ensemble, the system can exchange...

First time on this forum, hoping you can help clear something up for me. I am using the Van der Pauw method to characterize the sheet resistance of a metal film. In the standard setup, current is pushed through two contacts and voltage is measured across the other two contacts. My colleagues...

I was solving for a shape on Earth which has minimum potential energy. i used method of variation calculus. I assumed a function f(x) and rotated it around Y axis. sorry for uploading the problem in word.

Let's call our functional $$F[f]=\int dx\:A\left(x,f,f',f''...\right)$$ We know that the variation of F can be written as $$\delta F=\int dx\:\left[\frac{\partial A}{\partial f}\delta f+\frac{\partial A}{\partial f'}\delta f'+...\right]$$ If i wanted to get everything in terms of delta f in...

So I have an integral: ## \delta W=\int_{-\Delta}^\Delta\left[x^2\left(\frac{d\xi}{dx}\right)^2−D_S\xi^2\right]dx ## Here ##\xi## is a function of ##x## and ##D_S## is a constant. ##\Delta## is just some small ##x##. Now I need to set the variation of ##\delta W## to 0. Do do this I...

Just have a quick question about this problem in the photo... I'm not sure how they got the values 0, 1, 0, -1, 0 that they are multiplying by y=sin-x in the chart: For example. Look at the second row, pi/2. They apparently multiply 1/2 by 1 to get 1/2, but they never indicate where/how they...

Homework Statement Homework Equations F∝v The Attempt at a Solution I chose C because I thought that at time T after opening the parachute, the resisting force on the diver will increase. But it is not so since the answer is B. Is it because the diver has reached terminal velocity? Even so...

I am trying to do go over the derivations for the principle of least action, and there seems to be an implicit assumption that I can't seem to justify. For the simple case of particles it is the following equality δ(dq/dt) = d(δq)/dt Where q is some coordinate, and δf is the first variation in...

Homework Statement Determine the Lagrange-Euler equation for the functional J[y] = ∫F(x,y,y')dx = ∫F(x(y),y,1/x')*x' dx = I[x(y)] Homework Equations The Lagrange-Euler equations ∂F/∂y = d(∂F/∂y')/dx The Attempt at a Solution I'm new to the subject, so I don't really know what to do, I've...

Homework Statement Use the variation method to find a approximately value on the ground state energy at the one dimensional harmonic oscillator, H = -ħ^2/(2m) * d^2/dx^2 + 1/2mω^2*x^2 Homework Equations H = -ħ^2/(2m) * d^2/dx^2 + 1/2mω^2*x^2 u(x) = Nexp(-ax^2) <H> = <u|Hu> The Attempt at a...

Say I want to find ##\delta \bigg( \sqrt{- \eta_{\mu \nu} \frac{dx^{\mu}}{d \tau} \frac{dx^{\nu}}{d \tau}} \bigg)##. Is the following alright: ##\delta \bigg( \sqrt{- \eta_{\mu \nu}} \bigg( \frac{dx^{\mu}}{d \tau} \bigg)^{-1/2} \bigg( \frac{dx^{\nu}}{d \tau} \bigg)^{1/2} \bigg)##?

"The static pressure at all points within a free jet of a liquid is uniform throughout and is equal to the pressure surrounding the jet " I am looking for an explanation to the above statement. If we consider a vertical stream of a liquid flowing through an orifice at the bottom of a liquid...

Homework Statement I'm just wondering if I'm doing this calculation correct? eta and f are both tensors Homework EquationsThe Attempt at a Solution \frac{\delta \left ( \gamma_{3}f{_{\lambda}}^{k}f{_{k}}^{\sigma}f{_{\sigma}}^{\lambda} \right )}{\delta f^{\mu\nu}}=\frac{\delta\left (\gamma_{3}...

If load voltage vary..will Vce_Q1 changes in opposite direction in order to compensate change in Vce_Q2 in order to eliminate early effect in Q2...??

Recently we got to watch G2 as its orbit took it around the black hole at the center of the galaxy. This showed a visual change in direction along with a corresponding redshift to blue shift change on a relatively short timeline. It was really a great event. Binary stars exhibit the same...

I was playing with an online steam calculator provided by TLV, one of the bigger steam equipment vendors & the steam cost it calculates ($/ton) seems lower at high P than low Pressure. http://www.tlv.com/global/TI/calculator/steam-unit-cost.html e.g. The $/ton of 200 bar steam seems 6.09 vs...

Can someone verify that my answer is correct ? Thanks in advance. Use Variation of Parameters to find a particular solution to $y'' - y = e^t$ Solution: $y_p = \frac{1}{2}te^t - \frac{1}{4} e^t$

The functional one used for variations in calculus of variations. Had a google, can't seem to find. Thanks alot.

I should calculate the variation of the Ricci scalar to the metric ##\delta R/\delta g^{\mu\nu}##. According to ##\delta R=R_{\mu\nu}\delta g^{\mu\nu}+g^{\mu\nu}\delta R_{\mu\nu}##, ##\delta R_{\mu\nu}## should be calculated. I have referred to the wiki page...

I got the result that is consist with references in fisrt case. Is there anything wrong in 2nd way?

1. In photoelectric experiment, if anode potential w.r.t. cathode is increased, photocurrent first increases then becomes a constant, since all the photoelctorns ejected from cathode are collected at anode. If we increase the intensity of light at this point, the 'saturation current' increases...

Hi guys, I posted a thread a while ago asking for advices to create a Finite Element model of a radioisotope thermoelectric generator. I've basically finished this model mainly using conduction and radiation at the boundaries. I am wondering something though: currently my heat gradient is very...

I've been reviewing my knowledge on the technique of variation of parameters to solve differential equations and have a couple of queries that I'd like to clear up (particularly for 2nd order inhomogeneous ODEs), if possible. The first is that, given the complementary solution...

If we have two twins driving cars. They remain at the same height above sea level so gravitational time dilation is equal. Both cars travel at the same speed so time dilation due to speed is equal for both. However... Car A drives around a small circular race track and experiences centripetal...

I am not getting one line from my textbook.Can someone please explain.That line is as follows: In many crops genetic variations are available as pre-existing characters in wild relatives of the crop. My attempt for this is -i think it means if any crop shows any variations it means remaining...

I tried another approach to the problem of covariance like in Bell's theorem :from the definition ##Cov(A,B)=\langle\Psi|A\otimes B|\Psi\rangle-\langle\Psi|A\otimes 1|\Psi\rangle\langle\Psi|1\otimes B|\Psi\rangle## (##A=diag(1,-1)=B##) we can see that this 'average' is in fact a quadratic form...

[FONT=Courier New] [FONT=Arial] If we start with a Bell state 1/Sqrt(2)(|00>+|11>) [FONT=Courier New][FONT=Arial]and (after moving the second qbit a significant distance away) apply the interferometer transformation [FONT=Courier New]|0> -> 0.5(|0>+|1>) |1> -> 0.5(|0>-|1>) [FONT=Arial]to the...

Hi there, I have spent quite some time on PF as an unregistered user reading through various stuff and have learned a lot. Just registered now to seek help on something that's been bothering me a lot and to which I haven't managed to find a solution yet. Please let me know if this is not the...

So, it is defined that: Γλμυ = Γλμυ + δΓλμυ This makes obvious to see that the variation of the connection, which is defined as a difference of 2 connections, is indeed a tensor. Therefore we can express it as a sum of covariant derivatives. δΓλμυ = ½gλν(-∇λδgμν + ∇μδgλν + ∇νδgλμ) However...

Homework Statement Hey guys! So I have a Lagrangian with two coupled fields like so: \mathcal{L} = \frac{1}{2}(\partial_{\mu}\phi_{1})(\partial^{\mu}\phi_{1})...

Hello, I am in an introductory undergraduate course on ODEs, currently on the method of variation of constants to solve nonhomogenous equations. I am noticing that with many of these problems, when solving for constants after plugging in my guessed values for y I end up with enormous...

I am looking for a good and easy access reference on multi-variable calculus of variation with many examples and demonstrations. Although I have many books and references on the calculus of variation, most are focused on single-variable. Any advice will be appreciated.

if we know K^{a b}= (∇^a*ζ^b -∇^b*ζ^a)/2, ζ is a killing vector, under the variation of metric g_{a b}→g_{a b}+δ(g_{a b}) which preserves the Killing vector δ(ζ^a)=0, h_{a b} = δ(g_{a b}) = ∇^a*ζ^b +∇^b*ζ^a, how to prove δ(K^{a b})= ζ_c*∇^a*h^{b c} - h^{c a}*∇_c*ζ^b - (...

When you compute the average potential energy of a horizontal spring mass system from the mean position to the positive amplitude A, the value comes to be (1/6)kA^2. For the average kinetic energy over the same range and direction, it is (1/3)kA^2, which is double the average potential energy...

I'm having a bit of difficulty conceptually understanding a problem. Hoping someone here can clear it up for me. In a simple circular condenser tube (which is cooled by water or air) what happens the tube wall temperature as the steam inside is condensed? Intuitively I would have thought that...

The variation of the Einstein Hilbert action is usually done in coordinate basis where there is a crucial divergence term one can neglect which arise in the variation of the Ricci tensor, and is given by ##g^{ab}\delta R_{ab} = \nabla_c w^c## where $$w^c = g^{ab}(g^{db} \delta \Gamma^{c}_{db} -...

Suppose I have such an equation: A= f(B) so A is a function of B. Can I really use the fact that a variation of A is like taking the differential of it? \delta A= dA so that: \delta A = \frac{d f(B)}{dB} \delta B ?

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&amp;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.