What is Approximation: Definition and 761 Discussions

Homework Statement Given the pendulum equation \sin\frac{\theta}{2}=\sin\frac{\kappa}{2}\sin\phi=a\sin\phi where \theta oscillates between the limits \pm\kappa, \phi is a new variable which runs from 0 to 2\pi for one cycle of \theta, and...

For k_D<<k_F |\frac{\hbar^2k^2_F}{2m}-\frac{\hbar^2k^2}{2m}|\approx \frac{\hbar^2k_F}{m}|k_F-k| Where k goes from k-k_D to k+k_D k_F - Fermi wave vector k_D - Debay wave vector

I know how to derive the binomial approximation for (1+\alpha x)^{\gamma} using a Mellin transform, but for (1-\alpha x)^{\gamma} the method appears to fail because I can't take x to infinity. Here is the basics of the method. Take the Mellin transform of (1+\alpha x)^{\gamma}: M(p) =...

To those who have Cohen-Tannoudji vol 2, QM textbook. On page 920, he gives there the differential cross section, in equation B-48, which he writes it as: \sigma^{(B)}_k(\theta,\phi)=\frac{\mu^2}{4\hbar^4 \pi^2}|\int d^3 r e^{-iK.r} V(r)| Now shouldn't it be \frac{1}{\pi} instead of factor...

Hi... I have a quick question. I'm given V = ([0,1], <f,g> = interval from 0 to 1 of f(x)g(x)dx, S = {1, 2x-1}, W = lin(s), and P exists in W. I was determining the best approximation (P) to a function (F). F was some polynomial (3x + 5) and when I did the work I got P to be the same...

Can somebody demonstrate : \[ \frac{n}{{\sqrt[n]{{n!}}}} < \left( {1 + \frac{1}{n}} \right)^n \] ITS not A HOMEWORK

Hi People. I was looking at the derivation(s) of Fermi-Dirac Statistics by means of the "most probable distribution" (I know the correct way is to use ensembles, but my point is related to this derivation) and it usually employs Lagrange multipliers and Stirling's approximation on the...

The historic roots of string theory are in an explanation of the strong force. Nowadays QCD is the accepted theory of strong force. But having heard several lectures on the large N limit (SU(N)) of gauge theories it seems these theories start to looklike string theories in this limit. I believe...

Hi all, if you have a small slope approcimation, what can you say about the curvature? and higher derivatives of the slope?

If someone says something about a "fourth-order approximation," does that mean...? ...that, say, if something is being approximated by a Taylor series expansion in which only the first few terms are retained, and the expansion is in a small parameter \kappa, we only keep the terms up to order...

When you are not given an acceptable level of error in a problem, is there any rule of thumb I should use for how large Theta can be before I stop using the small angle approximation(Sin Theta=Theta) ?

Hi there! In some papers related to statistical field theory and condensed matter I've encountered the ladder approximation. It apparently corresponds to the summation of a certain class of Feynman diagrams. I've tried Google and some field theory books to learn more about this but I've found...

Homework Statement This comes from a book on relativity but it basically comes down to a math problem. The problem is to prove that if T^2 \ll\frac{c^2}{\alpha^2} then {t}\approx{T}(1-\frac{\alpha^2{T^2}}{6c^2}) given \frac{\alpha{T}}{c}=sinh(\frac{\alpha{t}}{c}) Homework Equations...

Hello, In few days, I have an examination, and I still have some dark zone in my head! If somebody could help me by giving me some advices/answers/way of reflexion/books to consult, it could be very great! Here is my questions: How to determine energy levels and wavefunction of the...

Hi, I don't undestand this limits to infinity. [ (2n)! / (n!)^2 ]^1/n and [ ( n^n + 2^n ) / (n! + 3^n) ]^1/n I've absolutely no idea how the first one can be "4", and the seconda one "e". Assuming (1 + 1/n)^n = e I don't see how can I go from that form to this one. Any...

The scattering amplitude in the Born approximation is as f(θ) = (-2m / (h/2π )2 K ) * integral 0 to ∞ (r sin Kr V(r) dr) Substituting V (r) = -V0 exp(-r2/2a2) We get f(θ) = (-mVoa3√π / 2(h/2π) ) * exp (-k2a2sin2θ/2 ) Differential crossection dσ / dΩ = / f(θ) /^2...

Homework Statement Find the perturbation approximation of the following in terms of powers of θ0. T=\sqrt{\frac{8L}{g}}\int^{\theta_0}_{0} \frac{d\theta}{\sqrt{cos\theta - cos\theta_0}} It is helpful to first perform the change of variable u = θ/θ0 in the integral Homework Equations...

In a text, it introduces an rotating frame and applies it on evolution of density matrix of two-level system. In the original frame, the first diagonal element of the time-derivitative of density matrix gives \frac{d\rho_{11}}{dt} = i e^{i\omega_r t} K \rho_{21} - i e^{-i\omega_r t} K^*...

Homework Statement Here is a picture of the problem: http://i3.photobucket.com/albums/y62/Phio/34.jpg" Homework Equations y - f(c) = f '(c) (x - c) The Attempt at a Solution 1/(5+x)^1/2 = (5 + x) ^ -1/2 = (1/5)^(1/2) * (1 + x/5)^(-1/2) I have this. But I don't know if I'm on...

In physics texts, its customary to write (and even to define the gradient as) the following: dT = (\nabla T) \cdot dl Working in Cartesian coordinates, we can expand this as follows: dT = \frac{\partial T}{\partial x} dx + \frac{\partial T}{\partial y} dy + \frac{\partial T}{\partial...

you are given a contour map of a hill from which you are to approximate a cone and hence find volume. each contour is an ellipse my question: does centreing the ellipses effect the volume. I am pretty sure it doesn't but i want to verify also. if i created 4 linear lines from the...

I am working on a ballistics model, and have been trying to find some material on impact forces. Does anyone have any reference material for creating an approximation of the brittle behaviour of materials over short time scales? At sufficient impact velocity a material will appear to be more...

I`m studying Series/Sequences/Approximation by polynomials.. -We approximate a function f(x) by getting a polynomial (I don`t know how we get it, and I don`t know what characteristics it should have, and I`d like to know please) -when we need more accuracy we add a higher derivatives, but why...

Homework Statement N weakly interacting distinguishable particles are in a box of volume V. A particle can lie on one of the M possible locations on the surface of the box and the number of states available to each particle not on the surface (in the gas phase) is aV, for some constant a. 1...

hello all I am new here so i don't know if this is the correct place for this topic so sorry for any inconvenience caused. this is my question: Use the small-signal model of the diode to predict the output voltages for the input voltages given. Plot a graph of | vout(t) | based on the small...

Hello friends, I've been reading Schiff's book on QM (3rd Edition), esp the section on the WKB approximation. (This isn't homework.) I have a few questions: What is the physical significance of the arrow on the connection formulas, like \frac{1}{2}\frac{1}{\sqrt{\kappa}} e^{-\zeta_{2}}...

Homework Statement Using the sum of the first 10 terms , Estimate the sum of the series (1/n^2) n from 1 to infinity ? How good the estimate is ? c) Find a value for n that will ensure that the error in the approximation s= sn is less than .001. Homework Equations I think Rn = s -...

Homework Statement Q1) Use the Taylor series of f (x), centered at x0 to show that F1 =[ f (x + h) - f (x)]/h F2 =[ f (x) - f (x - h) ]/h F3 =[ f (x + h) - f (x - h) ]/2h F4 =[ f (x - 2h) - 8 f (x - h) + 8 f (x + h) - f (x + 2h) ]/12h are all estimates of f '(x). What is the error...

Homework Statement \sum n^n/ n! This example is in the book, it concludes that the above series is : (1 +1/n)^n, which converges to e and n->infinity how is this so? If i take the root test then, is not answer 1 as n->infinity Can you explain?

Homework Statement Use linear approximation to approximate sqrt((3.2)^2 + 2(2.1) + (4.3)^2) with a rational number (a ratio of integers). Homework Equations f(x,y) = sqrt(x^2 + 2y + z^2) f(x,y) = (x^2 + 2y + z^2)^1/2 The Attempt at a Solution x = 3 ∆x = 2/10 y = 2 ∆y =...

Homework Statement Show that the multiplicity of an Einstein solid with large N and q is \frac{\left(\frac{q+N}{q}\right)^q\left(\frac{q+N}{N}\right)^N}{\sqrt{2\pi q\left(q+N\right)/N}} Homework Equations N! \approx N^N e^{-N} \sqrt{2 \pi N} The Attempt at a Solution Well...

Hello all, I am taking a Numeric Methods course this semester and my professor asked us to investigate Harold Hotelling's method( I suppose this would be and approximation) of finding the inverse of a matrix. I have searched for day and have found many cool things linked to Hotteling but...

Hi everyone, I'm an 18-year-old from Germany and I'm making use of MIT's OpenCourseWare programme. Currently, I'm watching the Calculus II course, and am having some trouble understanding how to find the equation: z=z0 + a(x-x0) + b(y-y0) by using parametric equations/vectors. a and b being...

Homework Statement Light of wavelength 587.5 nm illuminates a single 0.75 mm wide slit. (a) At what distance from the slit should a screen be placed if the first minimum in the diffraction pattern is to be 0.85 mm from the central maximum? (b) Calculate the width of the central maximum...

While studying the Einstein Equation, I noticed something curious, at least to me with little experience in General Relativity. Start with the usual formulation of the equation: R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R + g_{\mu\nu}\Lambda = \frac{8{\pi}G}{c^2}T_{\mu\nu} Then, apply the...

Homework Statement the value of f(x) = (sqrrt e^x +3) at x=0.08 obtained from the tangent to the graph at x=0 is...? Homework Equations The Attempt at a Solution i used linear approximation. (sqrrt e^o +3) + (1/2(sqrrt3+e^0)(0.08) i got an answer but i know its wrong. i...

Homework Statement I am a first-year physics student learning calculus. my question is about the approximation of the area of a region bounded by y = 0. Homework Equations Use rectangles (four of them) to approximate the area of the region bounded by y = 5/x (already did this one), and y =...

1. The relationship Lf = Li(1 + α ΔT ) is an approximation that works when the average coefficient of expansion is small. If α is large, one must integrate the relationship dL/dT = α L to determine the final length. a)Assuming that the average coefficient of linear expansion is constant as L...

Homework Statement Even at low density, real gases don't quite obey the ideal gas law. A systematic way to account for deviations from ideal behavior is the virial expansion, PV=nRT(1+B(T)/(V/n) + C(T)/(V/n)^2+...) where functions B(T), C(T) and so on are called the virial...

I am reading quantum mechanics (Messiah) now. And I get confused about the condition for the validity of the sudden approximation in CH. XVII. The author use perturbation theory to derive the result T<<\hbar/\delta \overline{H} ,when the Hamiltonian change over time T. The condition tells...

How to prove that this formula is correct: \lim_{x\to\infty} \frac{\Gamma(x+1)}{\sqrt{2\pi x}\big(\frac{x}{e}\big)^x} = 1 I have seen a proof for this: \lim_{n\to\infty} \frac{n!}{\sqrt{2\pi n}\big(\frac{n}{e}\big)^n} = 1 but it cannot be generalized easily for gamma function. The proof...

Homework Statement Find the linear approximation of tan(44) Homework Equations f(x) = tan(44 + x), let x = a = 1 f`(x) = sec2(44+x) The Attempt at a Solution L(x) = f(a) + f`(a)(x-a) L(x) = 1 + 2(x-1) L(0) = 1 + 2(0-1) = -1 (Answer should be approx. 0.965) Where am I going wrong? X should...

The function f is continuous on the closed interval [2,8] and has va;ues that are given in the table below. Using sub intervals [2,5]. [5,7], and [7,8], what is the trapezoidal approximation of the anti derivative from 2 to 8 of f(x)dx? |x| |2|5|7|8| f(x)| |10|30|40|20| (a) 110 (b)...

I am asked to make the near zone approximation instead of the far zone (radiation zone) approximation, that is to assume kr<<1 instead of kr>>1 for both the magnetic and electric fields. We are told that the B and E field before making the near zone approximation is given by: \vec{B}=k^2...

Homework Statement Homework Equations http://books.google.co.uk/books?id=4NXHYg70qqIC&pg=PA85&lpg=PA85&dq=paraxial+approximation+wave+equation&source=web&ots=6PbKKzSEz6&sig=bspXdKfxc-IiMV6AmoifMSJTHuk&hl=en&sa=X&oi=book_result&resnum=10&ct=result The Attempt at a Solution I...

Homework Statement There is a fluid flowing over a hot plate. We non-dimensionalized the problem from three partial diff eq's to two ode's. I am modeling I have two coupled differential equations that are a system of initial value problems. I am supposed to numerically integrate the two...

Homework Statement The tangent line through given points (pi/4,pi/4) m=1 y= cos(y)cos(x)/sin(x)sin(y) The Attempt at a Solution dy/dx= d/dx[cos(y)cos(x)/sin(x)sin(y)] First use quotient rule ? vu'-uv'/v^2 v= sin(x)sin(y) v'= do i need to use product rule? for product rule...

Let f(x) = x^(1/3). The equation of the tangent line to f(x) at x = 8 can be written in the form y = mx+b where m is: and where b is: Using this, we find our approximation for 8.1^(1.3) is: I found the slope to be 1/12 I found b to be 1.3333333333333333333 I still can't get the answer...

Homework Statement I am given a table of data derived from experiment. A force (F) is applied to a spring and the extension (x) is measured and recorded. An additional column of data for the derivative (dF/dx) is also provided. Here is the data: x(m) F(kN) df/dx (kN/m) 0.0...

Find an asymptotic approximation as p goes to infinity: f_{\lambda}(p)=\oint_{C}exp(-ipsinz+i\lambda z)dz where C is a square contour and p, lambda are real. Taking C to be of side length pi and centered at the origin, I applied the method of steepest descent at the point z=-pi/2...