Delta Definition and 1000 Threads

Homework Statement Proof that: ##\lim_{x \to 1} \frac{1}{1+x} = \frac{1}{2}## using the epsilon-delta definition of a limit. (Problem below) Homework Equations ##\lim_{x \to 1} \frac{1}{1+x} = \frac{1}{2} \iff \forall ε>0, \existsδ>0, \forall x ∈ \mathbb{R}\backslash\{-1\}: 0 < |x-1| < δ...

Dear PF members. I have trouble understanding how zero sequence currents circulate in a delta winding. For example zero sequence currents such as 3rd harmonics, they will circulate in the delta winding and get trapped, why? Why can they not go further? I have recently started to study...

Hi PF! As outlined in my book ##\delta_{ij} \delta_{jk} = \delta_{ik}## but don't we sum over repeated indices (and the ##j## is repeated)? Can someone explain why we do not sum in this situation? Thanks!

Homework Statement hi i have to find the result of ##\int_{0}^{\pi}[\delta(cos\theta-1)+ \delta(cos\theta+1)]sin\theta d\theta## Homework Equations i know from Dirac Delta Function that ##\int \delta(x-a)dx=1## if the region includes x=a and zero otherwise The Attempt at a Solution first i...

The exercises in my imaginary textbook are giving me an ε, say .001, & are making me find a delta, such that all values of x fall within that ε range of .001. The section that I'm working on is called "proving limits." Well, that is not proving a limit. All that's doing is finding values of...

Dear all, I have a quick question, is the following statement true? $$\nabla_\textbf{x'} \delta(\textbf{x}-\textbf{x'}) = \nabla_\textbf{x} \delta(\textbf{x}-\textbf{x'})?$$ I thought I have seen this somewhere before, but I could not remember where and why. I know the identity ##d/dx...

Hey all, i've found the following expression: How do they get that? They somehow used the kronecker delta Sum_k exp(i k (m-n))=delta_mn. But in the expression above, they're summing over i and not over r_i?? Best

Hello, I'm stuck with this exercise, so I hope anyone can help me. It is to prove, that the density of states of an unknown, quantum mechanical Hamiltonian ##\mathcal{H}##, which is defined by $$\Omega(E)=\mathrm{Tr}\left[\delta(E1\!\!1-\boldsymbol{H})\right]$$ is also representable as...

Homework Statement Hey guys. I am having a little trouble answering this question. I am teaching myself calc 3 and am a little confused here (and thus can't ask a teacher). I need to find the limit as (x,y) approaches (0,1) of f(x,y) when f(x,y)=(xy-x)/(x^2+y^2-2y+1). Homework Equations...

Homework Statement Homework Equations Wye: V_Line = sqrt(3) x V_Phase Delta: V_Line = V_Phase The Attempt at a Solution My answer doesn't quite match up with the solutions in the book, and was wondering what I went wrong here? Any help is greatly appreciated!

Homework Statement δ(z*-z0*)δ(z+z0)=? δ(z*+z0*)δ(z-z0)=? where 'z' is a complex variable 'z0' is a complex number Formula is just enough, derivation is not needed.

Homework Statement What is the product of two Dirac delta functions δ(Real(z-c))δ(Img(z-c))=? 'z' and 'c' are complex numbers. This is not a problem, But I just need to use this formula in a derivation that I am currently doing. I just want the product of these two Dirac delta functions as a...

Homework Statement Differential equation: ##Ay''+By'+Cy=f(t)## with ##y_{0}=y'_{0}=0## Write the solution as a convolution (##a \neq b##). Let ##f(t)= n## for ##t_{0} < t < t_{0}+\frac{1}{n}##. Find y and then let ##n \rightarrow \infty##. Then solve the differential equation with...

Can somebody help me? I am studying Faddeev-Popov trick, following the Peskin and Schroeder's QFT book, but I can't understand one thing. After they inserted the Faddeev-Popov identity, $$I = \int {{\cal D}\alpha \left( x \right)\delta \left( {G\left( {{A^\alpha }} \right)} \right)\det \left(...

Homework Statement \text{Show that } \epsilon_{ijk} \epsilon_{mjk} = 2\delta_{im} Homework Equations \begin{equation*} \epsilon_{ijk} \epsilon_{mnp} = \left| \! \begin{array}{ccc} \delta_{im} & \delta_{in} & \delta_{ip} \\ \delta_{jm} & \delta_{jn} & \delta_{jp} \\ \delta_{km} &...

In lectures, I have learned that F(k)= \int_{-\infty}^{\infty} e^{-ikx}f(x)dx where F(k) is the Fourier transform of f(x) and the inverse Fourier transform is f(x)= \frac{1}{2\pi}\int_{-\infty}^{\infty} e^{ikx}f(k)dk . But on the same chapter in the lecture notes, there is an example solving...

Homework Statement Find the Fourier spectrum of the following equation Homework Equations ##F(\omega)=\pi[\delta(\omega - \omega _0)+\delta(\omega +\omega_0)]## The Attempt at a Solution Would the Fourier spectrum just be two spikes at ##+\omega _0## and ##-\omega _0## which go up to infinity?

Given the definition: δ(x) = 0 for all x ≠ 0 ∞ for x = 0 ∫-∞∞δ(x)dx = 1 I don't understand how the integral can equal unity. The integral from -∞ to zero is zero, and the integral from 0 to ∞...

Homework Statement I am trying to determine whether $$f(x)g(x')\delta (x-x') = f(x)g(x)\delta (x-x') = f(x')g(x')\delta(x-x')$$ where \delta(x-x') is the Dirac delta function and f,g are some arbitrary (reasonably nice?) functions. Homework Equations The defining equation of a delta function...

Homework Statement [/B] A very thin plastic ring (radius R) has a constant linear charge density, and total charge Q. The ring spins at angular velocity \omega about its center (which is the origin). What is the current I, in terms of given quantities? What is the volume current density J in...

Homework Statement Consider the double well potential with two wells separated by delta function in middle. V(x) = V0 > 0 for x<-a and x>a 0 for -a<x<0 and 0<x<a αδ for x =0 1. Find Bound state energies 2. Find odd solns and their eigenvalue equation. Give solns in...

Homework Statement Consider a one-dimensional time-independent Schrodinger equation for an electron in a double quantum well separated by an additional barrier. The potential is modeled by: V (x) = -γδ(x - a) -γδ(x + a) + βδ(x) Find algebraic equations which determine the energies (or...

I want to understand why and where exactly we use dirac delta function? what is its exact use?

I need to prove that F=-constant*(delta y) if delta y < d I have no idea how to go about this. Please help!

We know that the solutions of time-independent Dirac delta potential well contain bound and scattering states: $$\psi_b(x)=\frac{\sqrt{mu}}{\hbar}e^{-\frac{mu|x|}{\hbar^2}}\text{ with energy }E_b=-\frac{mu^2}{2\hbar^2}$$ and $$ \psi_k(x)= \begin{cases} A(e^{ikx}+\frac{i\beta}{1-i\beta}e^{-ikx})...

Homework Statement For each of these sketch and provide a formula for the function (i.e. in terms of ##u(t)##, ##\delta(t)##) and its derivative and anti-derivative. Denote the ##\delta## function with a vertical arrow of length 1. (a) ##f(t)=\frac{|t|}{t}## (b) ##f(t)=u(t) exp(-t)##...

Homework Statement I need to integrate ##\frac{A}{2a\sqrt{2\pi}} \int_{-\infty}^{\infty} \frac{e^{ik(x+x')}}{(b^2+k^2)}dk## I have tried substitution and integration by parts and that hasn't worked. I can see that part of it is the delta function, but I don't really know how to use that fact...

Homework Statement I have a particle which is initially in a bound state for a given voltage in the form of a delta function at the origin, V = -αδ(x) initial state is ψα = (√αm)/h2*exp(-m*α*|x|/(h2) At t=0, voltage is changed to V = -βδ(x) both α and β are greater than zero. Right now I'm...

Homework Statement Find the solution to: $$\frac{d^2}{dt^2} x + \omega^2 x = \delta (t)$$ Given the initial condition that ##x=0## for ##t<0##. First find the general solution to ##t>0## and ##t<0##. Homework Equations The Attempt at a Solution This looks like a non-homogeneous second...

Hi, in the book 'Introduction to Quantum Mechanics' by Griffiths, on page 71 in the section 'The Delta-Function Potential' he states that the general solution to time independent Schrodinger Equation is $$\psi(x) = Ae^{-\kappa x} + B e^{\kappa x}$$ he then notes that the first term blows up as...

in calculating the delta p in vacuum chambers do we substitute p1 with 0 or the whole equation doesn't exist if air doesn't exist?

Using the Delta Netowrk from this image: http://www.belden.com/images/B23_WyevsDelta.jpg Between Y-X, there should be 120V. Between Y-Z, there should be 120V. Between X-Z, there should be 120V. But wait! If we assign values to make the above statements true... Call X = 0V, and Y = 120V, so...

Hi, friends! I have been able to understand, thanks to Hawkeye18, whom I thank again, that, if ##\mathbf{J}## is measurable according to the usual ##\mathbb{R}^3## Lebesgue measure ##\mu_{\mathbf{l}}## and bounded, a reasonable hypothesis if we consider it the density of current, if...

Hello. I'm new to this forum. I'm starting a PhD – it's going to be a big long journey through the jungle that is CFD. I would like to arm myself with some tools before entering. The machete is Cartesian Tensors. I know the rules regarding free suffix's and dummy suffixes, but I'm having...

Homework Statement Find the scalar product of diracs delta function ##\delta(\bar{x})## and the bessel function ##J_0## in polar coordinates. I need to do this since I want the orthogonal projection of some function onto the Bessel function and this is a key step towards that solution. I only...

The following integral arises in the calculation of the new density of a non-uniform elastic medium under stress: ∫dx ρ(r,θ)δ(x+u(x)-x') where ρ is a known mass density and u = ru_r+θu_θ a known vector function of spherical coordinates (r,θ) (no azimuthal dependence). How should the Dirac...

Lim as x approaches 4 of 1/x = 1/4 Given epsilon > 0, come up with a delta, d?Limits have been introduced. So far my instructor has had us make tables to see what value x was approaching. Although I don't understand exactly how limits are EVALUATED (different from looking at a chart & saying...

Homework Statement a) ∫δ( x² cos x) dx with x from -∞ to +∞ b) Integral over C, with C = unit circle ∫C δ(α-α0) a ds a =(1,1) Homework Equations a) δ(g(x)) =∑δ(x-xi)/|g'(xi)| b)? The Attempt at a Solution a) g(x) = x² cosx g'(x) = 2x cos x - x² sin x xi = 0 and (2n+1)π/2...

Homework Statement Use the epsilon delta definition to show that lim(x,y) -> (0,0) (x*y^3)/(x^2 + 2y^2) = 0 Homework Equations sqrt(x^2) = |x| <= sqrt(x^2+y^2) ==> |x|/sqrt(x^2+y^2) <= 1 ==> |x|/(x^2+2y^2)? The Attempt at a Solution This limit is true IFF for all values of epsilon > 0, there...

Will epsilon delta test fail if curve changed direction within the +/- delta of the limit point? Is there a scenario where no matter how small we pick delta to be, the frequency of the graph changing directions is always going to be a higher than delta's distance? In that scenario the limit...

Homework Statement What can be special cases, in the study of electrical circuits, in which one can not apply wye delta transformation? Can someone graph them in some way? Homework EquationsThe Attempt at a Solution

I am currently reading Modern Electrodynamics by Andrew Zangwill and came across a section listing some delta function identities (Section 1.5.5 page 15 equation 1.122 for those interested), and there is one identity that really confused me. He states: \begin{align*} \frac{\partial}{\partial...

Hello there ! I found this discussion http://physics.stackexchange.com/questions/155304/how-do-we-normalize-a-delta-function-position-space-wave-function about dirac notation and delta function . The one that answers to the problem says that ##<a|a>=1## and ##<a|-a>=0## . As far as i know: 1)...

Homework Statement Compute the average value of the function: f(x) = δ(x-1)*16x2sin(πx/2)*eiπx/(1+x)(2-x) over the interval x ∈ [0, 8]. Note that δ(x) is the Dirac δ-function, and exp(iπ) = −1. Homework Equations ∫ dx δ(x-y) f(x) = f(y) The Attempt at a Solution Average of f(x) = 1/8 ∫from...

I understand that there are vortices formed on the delta wing for moderate angles and low speeds. I observed recently 2 asymmetric vortices forming on a small delta wing model for a moderate angle and a low speed. But, why would there be asymmetriic flow over a delta wing? If the flow is...

Homework Statement https://z-1-scontent-sin1-1.xx.fbcdn.net/hphotos-xtf1/v/t34.0-12/12435980_942112312550457_1708451532_n.jpg?oh=b29a5e9b2ff96bbf0e11932d081eb25e&oe=568A9B79 [/B] We need to find the equivalent resistance along A to B ie AB Homework Equations R=(1/R1 + 1/R2+...+1/Rn)^-1 ...

Hello, I feel like I am fudging these integrals a bit and would like some concrete guidance about what's going on. 1. Homework Statement Evaluate ##I = \int_{-1}^{1} dx \delta'(x)e^3{x} ## Homework EquationsThe Attempt at a Solution [/B] I use integration by parts as follows, ##u =...

Hi - firstly should I be concerned that the dirac function is NOT periodic? Either way the problem says expand $\delta(x-t)$ as a Fourier series... I tried $\delta(x-t) = 1, x=t; \delta(x-t) =0, x \ne t , -\pi \le t \le \pi$ ... ('1' still delivers the value of a multiplied function at t)...

Homework Statement Two isotopes N1, N2. You are given the value of the proportions N1(t1)/N2(t1) and N1(t2)/N2(t2). From this calculate Δt. Homework Equations dN/dt = -λN The Attempt at a Solution Solve dN1/N = -λdt and dN2/N = -λdt obtain: N1(t2) = N1(t1)exp(-λ1(t2-t1)) (1) and N2(t2) =...

This is not a homework question per se, but it's a simple enough computation question from some notes I'm reading that I think it belongs in this section. I will do my best to keep to you guys' formatThis is not a homework question per se, but it's a simple enough computation question from some...