Approximation Definition and 705 Threads

Hey there, I have two questions - the first is about an approximation of a central gravitational force on a particle (of small mass) based on special relativity, and the second is about the legitimacy of a Lagrangian I'm using to calculate the motion of a particle in the Schwarzschild metric...

Homework Statement .[/B] The full range input of a12-bit, successive-approximation type ADC is 1 volt. Determine: a) the maximum input change required to give a one bit change in output of the ADCb) The number of approximations made to complete the conversion of an input signal of 0.8125...

You have an infinitesimally small mass in the center of octahedron. Mass is connected with 6 different springs (k_1, k_2, ... k_6) to corners of octahedron. Equilibrium position is in the center, you don't take into account gravity, only springs. Find normal modes and frequencies. Relevant...

Is Differentiation exact or just an approximation? I am wonder whether this question is meaningful or not. Slope is expressed as "it is approaching to a value as x is approaching 0" so it is inappropriate to ask such question. But when I deal with uniform circular motion, it is very confusing...

I studied Taylor series but I would like to have an answer to a doubt that I have. Suppose I have ##f(x)=e^{-x}##. Sometimes I've heard things like: "the exponential curve can be locally approximated by a line, furthermore in this particular region it is not very sharp so the approximation is...

I am examining the polynomial approximation for $e^x$ near $x = 2$. From Taylor's theorem: $$e^x = \sum_{n = 0}^{\infty} \frac{e^2}{n!} (x - 2)^n + \frac{e^z}{(N + 1)! } (x - 2)^{N - 1}$$ Now, I don't get the next part: We need to keep $\left| (x - 2)^{N + 1} \right|$ in check so we can...

Hello, I would like to ask, if somebody knows anything about comparison post-Newtonian approximation of gravitational waves and these which were detected. Or generally post-Newtonian predictions vs. facts found in detection. I tried find some article but I didn't find. Please let me know what...

Homework Statement Suppose that the number of asbestos particles in a sam- ple of 1 squared centimeter of dust is a Poisson random variable with a mean of 1000. What is the probability that 10 squared cen- timeters of dust contains more than 10,000 particles? Homework Equations E(aX+b) =...

Hello, Can someone explain this to me? In the above case ct=yt-gt I tried to solve it as a three variable taylor approximation but got a few extra terms that weren't included in the above. So I am a little confused now. I only need to understand how the first line was derived because I get...

I am calculating the strain in a cantilever beam with a point load for a given deflection. The deflection is around .5mm for a beam that is just over 5mm long (width is 3.8mm and height is 0.15mm). I was told for the assumption of small deflections to be valid, deflection should be 2% of the...

Hello, could anyone please explain me some logic or derivation behind the approximation: Found it in the Hull Derivatives book without further explanation. Thanks for help

To solve the full many-body electron problem one often uses the approximation that the dynamics of the electron system, is that of N interacting electrons living in a periodic lattice of positive ion cores. What justifies this approximation and does it have a particular name?

Homework Statement Homework Equations ΔE = hf The Attempt at a Solutionf = ΔE/h = ##k\left [ \frac{1}{(n-1)^2} - \frac{1}{n^2} \right ]## = ## \frac{k(2n-1)}{n^2(n-1)^2}## = ## \frac{k}{n^4}(2n-1)(1-\frac{1}{n})^{-2}## = ## \frac{2kn}{n^4}(1-\frac{1}{2n})(1+\frac{2}{n} +## higher powers...

using binomial theorem can I write sqrt(1+(d^2)/(x^2)) = 1+ .5(d^2)/(x^2)? d is a variable. X known constant.

Homework Statement In regards to linearization of a nonlinear system in differential equations. What does it mean for a linear approximation to be reliable to describe the long term behavior of the non-linear system around the equilibrium point? Homework Equations jacobian matrix The Attempt...

Ever since the discovery of Pi, Mathematicians have been obsessed with finding methods of approximating Pi. I think I've a unique way of doing so via the Newton-Raphson. Newton-Raphson Formula: Let ## ƒ(x)=Sin(x) ⇒ ƒ'(x)=Cos(x) ⇒ X_n= X_{n-1} - tan(X_{n-1})## For example: Let ##X_0=X ⇒ X_3=...

X = # of cars that pass in one hour E(X) = λ = n * p λ cars/1hour = 60min/hour * (λ/60) cars/min In this old video (5:09) on poisson process Sal asks: "What if more than one car passes in a minute?" "We call it a success if one car passes in one minute, but even if 5 cars pass, it counts as 1...

Homework Statement Good day all! I'm studying for finals and i'd like to know how to do this problem (its not homework): "Using the WKB method, find the bound state energies E_n of a particle of mass m in a V-shaped potential well: V(x)= \begin{Bmatrix} -V_0 (1- \begin{vmatrix}...

I have noticed that in a lot of theoretical modelling of semiconductors you assume that the electrons living in the bottom of the conduction band obey a free particle Hamiltonian: H = p^2/2m* , where m* is the effective mass in the conduction band and p^2 is the usual differential operator. I...

Homework Statement Consider a D dimensional Ising model with N sites, defined by the Hamiltonian $$\mathcal H = -J \sum_{\langle i j \rangle} \sigma_i \sigma_j - h \sum_i \sigma_i$$ where the sum extends over nearest neighbours and each spin variable ##\sigma_i = \pm 1##. For a given spin...

Hi. I can't for the life of me understand the math behind the SVEA. I graphically/intuitively understand what it means that the envelope varies slowly, but I can't connect that with the mathematical expression: $$ \left \vert \frac{\partial ^2 E_0}{\partial t ^2} \right \vert << \left \vert...

Homework Statement Find the least squares approximation of cos^3(x) by a combination of sin(x) and cos(x) over the interval (0, 2pi) Homework EquationsThe Attempt at a Solution I know how to find a least squares approximation with vectors, but I don't even know how to start with a function...

Any good book for starting wkb approximation except Griffth... Please suggest some...

I am across this equation with no known analytical solutions: x^5 - x + 1 = 0. I asked this before and the answer was that you can get as good an approximation you want by approximation methods. It is possible that when applying approximation methods, you will get singularities. These...

Homework Statement Suppose that T4 is used to approximate the ∫ from 0 to 3 of f(x)dx, where -2 ≤ f ''(x) ≤ 1 for all x. What is the maximum error in the approximation? Homework Equations |ET|≤ (K(b-a)^3)/(12n^2) The Attempt at a Solution So I know how to find the error of the trapezoidal...

1. The question is. Show that if |nx| <1, the following is exact up to (and including) the x^2 order. The hint giving says to use the Taylor Expansion for both sides of the equation2. (1+x)^n = e^n(x-(1/2)x^2) ; the n(x-(1/2)x^2) is all an exponent3. My first attempt was to take the taylor...

For some small parameter ##\epsilon##, how would one go about making an approximation such as ##\sqrt{k^2-\epsilon^2}\approx k-\frac{\epsilon^2}{2k}##? I was thinking that these types of approximations came from truncating Taylor series expansions, but I can't see how it would be obvious which...

I am studying in IGCSE and I learned simple techniques to find, say, approximate change in Area of a circle for a small change in its radius, making use of : δy/ δx ≈ dy/dx . or δA/ δr ≈ dA/dr . or δA ≈ dA/dr x δr so what I basically have to do is find the derivative of A ( πr​2 ) , which...

Hi all, I am reading a book on spacecraft engineering in the section about trajectory dynamics. They define linear and angular momentum as: ##I = \int_{0}^{\tau}{F}dt## (Linear Momentum) ##L = \int_{0}^{\tau}{T}dt## (Angular Momentum) But they (and so many other sources) always mention the...

Hi all, hopefully this is in the correct section here. Any help is really gratefully received. 1. Homework Statement I have a coursework, one question asks us to use a 2nd order approximation of the transfer function to..."estimate the settling time (5% of the settling value of output, peak...

The following comes from Griffiths Intro. to QM (2nd Ed) page 53. We want to solve the Schrödinger Equation for the harmonic oscillator case using a power series method. The details aren't important but you want to solve ##h''(y)-2yh'(y)+(K-1)h=0## whose recursion formula is...

Consider the Dirac equation for bounded electron in hydrogen atom. I am trying to get a clear physical explanation for all mathematical terms that appear in the Hamiltonian and energy spectrum. Kinetic and Coulombic potential and rest energies are the first terms and easy to identify. Then we...

Homework Statement There are ##N = 10000## clients of an insurance company. One-half of them will file claims with probability ##p_1 = .05##, another half of them will file claims with probability ##p_2 = .03##. Each claim is worth ##$1000##. Find the Value-at-Risk at the level α = 0.99, that...

Good evening. Is there a way to take a decimal approximation and see if there is a relatively simple expression? I'm guessing there might be software for this, but I'm not sure I'm even asking the appropriate question. If it matters, the number I'm after is...

Hi I'm having trouble visualizing why in a function such as 1/(1-x2) linear approximation of 1/(1-u) where u = x2 is the same as quadratic approximation of 1/(1-x2) The linear approximation is 1+u or 1+x2 Quadratic approximation is the same, 1+x2 Can someone explain to me why this happens...

Consider E>V(x). WKB states the wavefunction will remain sinusoidal with a slow variation of wavelength $ \lambda $ and amplitude given that V(x) varies slowly. From the equation \begin{equation} k(x)=\frac{\sqrt{2m(E-V(x))}}{\hbar} \end{equation}, I can see that the k(x) is directly...

Hello, I don't understand the following. I have this function: V(x,y,z)=\frac{(A+B+C)r^2-3(Ax^2+By^2+Cz^2)}{r^5} with r=\sqrt{x^2+y^2+z^2} and on the textbook they say that if x,y,z are approximately equal or comparable as order of magnitude to r, and if they are all "big" enough (they are...

Homework Statement Find linear approximation of the surface z2 = xy + y + 3 at the point (0,6, -3) and use it to approximate f(-0.01, 6.01, -2.98) Homework EquationsThe Attempt at a Solution [/B] so this means the total surface has decrease by -0.17?

Hi there, in my notes for Heun's method for solving an ODE, I have y(new) = y(old) + 0.5(k1 + k2)Δh And k1 is supposed to be f(y(old)) while k2 is f(y(old) + q11k1Δh) and q11 is 1 So if for example I have a simple differential equation like du/dt = au It would be du/dt = 0.5(k1 + k2) du/dt...

Hi, I've attached an image of an equation I came across, and the text describes this as an approximation to the second derivative. Everything seems to be exact to me (i.e. not an approximation) if the limit of h was taken to 0. Is that the only reason why it's said to be an approximation or is...

I recently read a paper titled : " The Saros cycle: obtaining eclipse periodicity from Newton's laws " My question is, more or less: " Is it possible to obtain an approximation of G by observing Saros periodicity ? " I'm currently studying the derivations of the Lunar , Solar, and Stellar...

I have a problem with static/non-static spacetime. The problem is that the notion of spacetime includes time itself, so how can it change with time? Imagine an asteroid approaching the Earth-Moon system. The Earth-Moon system is a non-static spacetime, so presumably is giving off gravitational...

Homework Statement Hello, I have a question about using Eulers Method to approximate a solution to a differential equation. The problem lists forces that would be applied on an object and influences its velocity and therefore its position. I believe I am doing the Euler method correct to...

Hi! I read from Perry Green's ChE Handbook that the friction factor for Re ≤ 2,100 can be approximated by ƒ = 16/Re. But there was this question that I encountered (though I don't know the source) and according to it, ƒ = 64/Re for laminar flow. Can someone clarify which is which? Thank you!

Homework Statement y'+y=3+x y(0)=1 (a) Find approximate values of the solution of the given initial value problem at t = 0.1, 0.2, 0.3, and 0.4 using the Euler method with h = 0.1. Homework Equations yn+1 = yn + f(x0, y0)(x-x0). Adjusting 0 for the next number as we go up The Attempt at a...

My apologies for having to post in an image, my latex skills are not good enough for the question at hand :( a) There is no solution since the system has more unknowns than equations (the equations are equal giving 1=2 which does not make sense). b) I get a solution of \begin{bmatrix}1 \\1 \\...

According to rules for working with approximate data, why the final result of a multiplication or division involving approximation data is round off so that the result has as many significant digits as the given data with the fewest significant digits? How is this rule established? For example...

Dear forum people in the Slater-Koster approximation interatomic matrix elements is a function of the cosine direction. How to calculate cosine direction for silicene by sp2 hybrid.

The authors of a physics textbook want to determine the number of grains, N in a beach of 500 m long, 100 m wide, and 3 m deep. They assumed that each grain is 1-mm-diameter sphere. They also assumed that the grains are so tightly packed that the volume of the space between the grains is...

Homework Statement I put up the image so that you can see the hints if you're curious. I am supposed to prove that if ## S=\sum_{n=0}^{\infty}a_{n}x^{n}## converges for ##|x|<1##, and if ##|a_{n+1}|<|a_{n}|## for ##n>N##, then $$|S-\sum_{n=0}^{N}a_{n}x^{n}|<|a_{N+1}x^{N+1}|\div (1-|x|)$$...