Approximation Definition and 705 Threads

Hi guys, I've made a Mathematica n-body simulation of the first few planets in our Solar System and thought it would be a good idea to try and simulate a spacecraft transfer from Earth to Mars. I've thought about using a patched conic approximation, but I was wondering if there is anything...

How can I prove that, for N\gg n \frac{N!}{(N-n)!}\approx N^{n} I've tried doing \frac{N!}{(N-n)!}=\exp\left(\ln\frac{N!}{(N-n)!}\right)=\exp\left(\ln N!-\ln\left(N-n\right)!\right) \underset{stirling}{\approx}\exp\left(N\ln N-N-\left(N-n\right)\ln\left(N-n\right)+N-n\right)...

Homework Statement In order to simplify problems in physics, we often use various approximations. For example, when we investigate diffraction and interference patterns at small angles θ, we frequently approximate sinθ and tanθ by θ (in radians). Here you will calculate over what range these...

Hi, Lets say that I have a 4x4 matrix, and am interested in projecting out the most important information in that matrix into a 2x2 matrix. Is there an optimal projection to a lower dimensional matrix where one keeps most of the matrix intact as best as possible? Thanks.

Hi everyone, I was wondering if you guys could suggest me some good books in cosmology with finely explained WKB method and Perturbations especially in Structure formation area. I have "The early universe" by Klob and Turner and "Cosmology" by Weinberg , but they seem unpalatable at first...

Here is the question: I have posted a link there to this topic so the OP can see my work.

Hello, I am trying to find the effective resistance of the NLR in the attachment (to the first order). It is given that IL = gVL2 + I0. I understand that this is normally achieved via ∂g/∂V at V=V0, but when I do so I get that R should be 1/(2gV0), and not 1/2g as shown in the solution. Could...

Hi there. At first I tought of posting this thread on the homework category, but this is a conceptual doubt rather than anything else. While revisiting Heat Transfer I stumbled upon a simple problem, that yet got me thinking. It is as follows: Before anything else, let me show...

I am in process of designing a homemade capacitance sensor and I'd like to have an approximation of the resulting capacitance of the following geometry. The plates are placed on the outer surface of a food grade plastic cylinder. The distance z between their edges is many times smaller than the...

There is no analytical solution of the integral below. Can we approxiamate the analytical solution? \int_{k}^{K} \frac{exp(-log^2 (x))}{x(x-A)}dx

Hi Guys, I have just started studying about this field. Can you give me some ideas about some best eigen value estimators? Both for SPD and non-SPD matrices. Thanks you. :-)

When I started learning Multivariable calc first we went back and developed a new notion of derivative as a linear approximation. And what we came up with was F(a+h)=F(a)+mh+E(h) * where m is the derivative. Basically the function minus the line tangent to point a. However there is a...

Hello community, I'm new here. I'm using FORTRAN to model the motion of electrons and ions when accelerated under a high voltage potential. I'm using a fluid approximation and MHD-like equations (conservation of mass, energy, momentum) and a finite volumes numerical method to solve the...

What is the most general method of approximating arbitrary systems of ODEs of 4 variables(x,y,z,t) that fit these conditions? The conditions that are assumed true of the ODEs are: 1) that I require differentials to be explicitly defined (but they can be defined in terms of other...

Dirichlet's Approximation Theorem not working for n=8 and α= pi? I am reading a number theory textbook that states Dirichlet's Approximation Theorem as follows: If α is a real number and n is a positive integer, then there exists integers a and b with 1≤ a ≤ n such that |aα-b|< 1/n . There...

Hello, I have read several articles/websites which talk about modelling white dwarfs, In all of these papers they state that it can be assumed the electrons have temperature zero, i.e. T<<T_fermi. I haven't been able to find a solid explanation of why this is approximation is possible...

Hi everyone I have trouble with this task Homework Statement the specific heat cv is given by c_v =\frac {N_A k_b \hbar^2}{{\Omega_D}^3 {k_b}^2 T^2} \int \limits_{0}^{\Omega_D} \! \frac {\Omega^4 exp\frac{\hbar \Omega}{k_b T}}{{(exp\frac{\hbar \Omega}{k_b T}-1})^2} \, d\Omega I...

Homework Statement Take 50 free throws in basketball, and record the number of successful shots. 1) Use this data to estimate your free throw percentage. 2) Construct a 95% confidence interval for this estimated quantity.Homework Equations X ~ Binomial(n,p) The Attempt at a Solution I...

This feels silly asking but I have a question about units using the double harmonic approximation to determine IR spectral intensities. In the double harmonic approximation the intensity is given by the square of the derivative of the dipole with respect to a normal mode coordinate times a...

I'm working through a solved problem in a fluid mechanics textbook. In it, the group velocity of a dispersive wave is calculated as: $$c_g = \frac{1}{2}c\left (1 + 2kh\ \text{cosech} (2kh) \right)$$ Where k is the angular wavenumber, and h is the depth of the water, which is fine. Now for...

Hello all, Which of the following asymptotic equations (as rho goes to infinity) correct: 1-\prod_{m=1}^M(1-\rho^{-x_m})\doteq \sum_{m=1}^M \rho^{-x_m} or 1-\prod_{m=1}^M(1-\rho^{-x_m})\doteq \rho^{-\underset{m}{\min}x_m} Thanks

Hello, I'm trying to solve for the allowed energies with the WKB approximation of the Schrodinger equation, using the Morse potential. So I have (as per equation 35 at http://hitoshi.berkeley.edu/221a/WKB.pdf), \int_a^b \sqrt{2m(E-V(x))}dx=\left(n+\frac{1}{2}\right)\pi\hbar However, how do I...

Homework Statement The IPA potential-energy function ##U(r)## is the potential energy "felt" by an atomic electron in the average field of the other ##Z-1## electrons plus the nucleus. If one knew the average charge distribution ##p(r)## of the ##Z-1## electrons, it would be a fairly simple...

\frac{\sqrt{1-a}}{\sqrt{1-b}}\approx \left ( 1-\frac{1}{2}a\right )\left ( 1+\frac{1}{2}b\right ) I know that the binomial approximation is first used, \frac{\sqrt{1-a}}{\sqrt{1-b}}\approx \frac{1-\frac{1}{2}a}{1-\frac{1}{2}b} But how does one approximate...

Homework Statement Given this data: hours / value ----------- 2 | 1.6 4 | 1.5 6 | 1.45 8 | 1.42 10 | 1.38 12 | 1.36 fit a curve of the form Y ≈ ae^{-bx} What value can you predict after 15 hours? The Attempt at a Solution So i can rewrite the equation as Y ≈ log(a)-bx...

Homework Statement Given that the scattering amplitude off of a single atom is f_{1}(\vec{q}), find the scattering amplitude for 1) four atoms each placed in the corner of a square of length a, and 2) two atoms a distance d apart Homework Equations The total scattering amplitude can...

Hi, I have been looking for rigorous mathematical conditions for when the WKB approximation may be applied. Here is my understanding of the topic. We start with the most general form that the wavefunction could take, i.e. exp[if(x)/h] , Where "i" stands for square root of -1, f(x) is...

Here is the question: Here is a link to the question: Approx. series help please? - Yahoo! AnswersI have posted a link there to this topic so the OP can find my response.

Hello, If tau<<T which of the following relations are true: \int_{\tau/(1+a)}^{(T+\tau)/(1+a)}v(t)e^{-j2\pi f_0 at}e^{-j2\pi\frac{k}{T}t[1+a]}\,dt=\int_{0}^{T/(1+a)}v(t)e^{-j2\pi f_0 at}e^{-j2\pi\frac{k}{T}t[1+a]}\,dt or \int_{\tau/(1+a)}^{(T+\tau)/(1+a)}v(t)e^{-j2\pi f_0...

Here is the question: Here is a link to the question: Need help with calculus word problem? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Hi all, i have a problem to solve that i want maybe to solve with MATLAB o excel. I have a numerical samples and with linear approsimation i have a function, but now i want to use less samples for example only 20 and i want to find the best set of samples to approsimate in the best way the...

Homework Statement Find a value for n for which the nth partial sum is ensured to approximate the sum of the alternating harmonic infinite series to three decimal places. Homework Equations Sn = Ʃ(-1)^k+1*1/k = 1 - 1/2 + 1/3 - 1/4 + 1/5 - . . . S1 = 1 S2 = 1 - 1/2 S3 = 1 - 1/2 + 1/3 S4...

I've been working on a Theremin Excel simulation for the past couple of months. For those who don't know what a Theremin is, it was one of the very first electronic instruments to be invented, and has two "antennas" that independently change the pitch and amplitude of a tone via hand...

The Simple Approximation Lemma Let f be a measurable real-valued function on E. Assume f is bounded on E, that is, there is an M \geq 0 for which |f|\leq M on E. Then for each \epsilon > 0, there are simple functions \phi_{\epsilon} and \psi_{\epsilon} defined on E which have the following...

So are Newton's laws also an approximation to quantum phenomena. Can it be derived from quantum laws?

My questions are from lecture 9, MIT OCW SV Calculus, Jerison, 2009; At 27:50 he is deriving the linear approximation for the function e^(-3x)(1+x)^(-1/2)≈(1-3x)(1-1/2x)≈1-3x-1/2x+3/2x^2≈1-7/2x, for x near 0. In the last step he drops the x squared term since it is negligible(no questions so...

Homework Statement \mu = \frac{mM}{m+M} a. Show that \mu = m b. Express \mu as m times a series in \frac{m}{M} Homework Equations \mu = \frac{mM}{m+M} The Attempt at a Solution I am having trouble seeing how to turn this into a series. How can I look at the given function...

I'm currently teaching myself some QFT trough Peskin and Schroeders Introduction to QFT and I've noticed that in several arguments they rely on appealing to the Born approximation of non-relativistic QM scattering theory. For example on page 121 equation (4.125) they appeal to the scattering...

Homework Statement What is the probability that an electron in the ground state of hydrogen will be found inside the nucleus? a) First calculate the exact answer, assuming the wave function \psi(r,\theta,\phi) = \frac{1}{\sqrt{\pi a^3}} e^{-r/a} is correct all the way down to r=0. Let b be the...

Homework Statement obtain the number r = √15 -3 as an approximation to the nonzero root of the equation x^2 = sinx by using the cubic Taylor polynomial approximation to sinxHomework Equations cubic taylor polynomial of sinx = x- x^3/3!The Attempt at a Solution Sinx = x-x^3/3! + E(x) x^2 =...

Hello! In order to prepare for an exam I have started solving exercies problems and have gotten most of them right but have quetion a regarding a solution. In this probelm I used the first order Born approximation in order to calculate the differential and total cross section for the...

Homework Statement consider the function f(x) = aln(x+2). Given that f'(1) = a/3, what is the approximate value of f(0.98)?Homework Equations f(x1) = f(x0) + f'(x0)x(x1-x0)The Attempt at a Solution I solved it and get f(.98) = aln(1+2) + (.098-1) = aln(3) - (.02)(a/3) <= not an answer the...

I am confused with a couple of terms usually used in the context of non-radiative transitions. I believe that I understand the concept of diabatic and adiabatic states described in http://en.wikipedia.org/wiki/Adiabatic_theorem. The basic finding is that the coupling terms in the Hamiltonian...

Here is the question: Here is a link to the question: CALCULUS HELP PLEASE!? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Homework Statement I just stumbled upon an approximation I don't get where comes from Homework Equations F(x+dx) -F(x) = dF/dx dx The Attempt at a Solution My textbook just stated it out of nothing, so I have no idea where to start.

Hi I'm trying to understand how the sum over spin polarizations is treated when using the Narrow Width Approximation(NWA) with a spin 1 resonance. For a spin zero resonance there is no such problem and the factorization is quite straightforward. I'll go through some details to explain where...

Hi, I'm reading about the Born-Oppenheimer approximation for a solid and they're doing the formalism of it. They say that we can basically consider the ions stationary with respect to the electrons because they move so little and so slowly in comparison to them. They say that ##R_i## are the...

Homework Statement Find, by comparison with exact trigonometry, the angle,  (provide a numerical value in degrees), above which the small angle approximation departs from the exact result by more than 1 percent. Homework Equations Approx.: d = s = rθ Exact: d = 2*r*Sin(θ/2) The...

Homework Statement f(x,y) = y' = \frac{y+x^2-2}{x+1} , y(0) = 2 Write the formula for the 2nd order Taylor approximation I just want to ask a question Homework Equations Taylor seriesThe Attempt at a Solution Taylor: y(x) = y(x_0) + y'(x_0)(x-x_0) + \frac{y''(x_0)(x-x_0)^2 }{2} = \\...

Is there a way to "crudely" approximate PDEs with Fourier series? By saying crudely, I meant this way: Assuming I want a crude value for a differential equation using Taylor series; y' = x + y, y(0) = 1 i'd take a = 0 (since initially x = 0), y(a) = 1, y'(x) = x + y; y'(a)...