Approximation Definition and 705 Threads

Hi. Please help me prove the approximation formula below given in my book. This is not homework question. Thanks.

Homework Statement The restoring force of a pendulum is F_\theta=-mg\sin\theta and is approximated to F_\theta=-mg\theta. The period is T=2\pi\sqrt{\frac{L}{g}}, but can be expressed as the infinite series: T=2\pi\sqrt{\frac{L}{g}}\left(...

Hi all I was trying to understand the Born-Bethe approximation related to cross sections for atomic and molecular collisions. All the stuffs that i got are explaining in complicated way which am not able to follow. Can anyone explain in simple terms what the theory explains? It will be of...

Hello, I've come across equations where people use the approximation \int_0^1 \exp(f(x))\, dx \approx \exp \left( \int_0^1 f(x)\, dx\right) I can see that this is correct if f(x) is small, one just uses exp(x) = 1+x+... However, it appears that this approximation has a broader validity...

Homework Statement Using the Eikonal approximation (1) Determine the expression for the total scattering cross section of a particle in a potential V(r) (2) Using this result, compute the total scattered cross section for the following potential. V(r)= \begin{cases} V_0, \text{for } r < a \\...

I'm looking for a way to write down an analytic approximation for the following integral: \int_0^\infty \frac{k \sin(kr)}{\sqrt{1+v^2(k-k_F)^2}}dk Let's assume that v kF >> 1, so that the the oscillating piece at large k doesn't contribute much uncertainty. Ideas? Thus far, Mathematica has...

Homework Statement If we have the following data T = [296 301 306 309 320 333 341 349 353]; R = [143.1 116.3 98.5 88.9 62.5 43.7 35.1 29.2 27.2]; (where T = Temperature (K) and R = Reistance (Ω) and each temperature value corresponds to the resistor value at the same position) Homework...

Homework Statement Show that ∫f'(x)dx/f(x) = ln|(f(x)|+C where f(x) is a differential function. Homework Equations First order Taylor approximation? f(x)=f(a)+f'(a)(x-a) The Attempt at a Solution Well, I'm not really sure how to approach the question. It's my Numerical...

Hey! :o I am looking at the identities of the optimal approximation. At the case where the basis consists of orthogonal unit vectors,the optimal approximation $y \in \widetilde{H} \subset H$, where $H$ an euclidean space, of $x \in H$ from $\widetilde{H}$ can be written $y=(x,e_1) e_1 +...

1. Consider a quantum well described by the potential v(x)=kx^{2} for \left|x\right|<a and v(x)=ka^{2} for \left|x\right|>a. Given a^{2}\sqrt{km}/\hbar =2, show that the well has 3 bound states and calculate the ratios between the energies and ka^{2}. You may use the standard integral...

Hey! :o I am looking at the following that is related to the existence of the optimal approximation. $H$ is an euclidean space $\widetilde{H}$ is a subspace of $H$ We suppose that $dim \widetilde{H}=n$ and $\{x_1,x_2,...,x_n\}$ is the basis of $\widetilde{H}$. Let $y \in \widetilde{H}$ be...

I have a equation which represents a nonlinear system.I need to linearize it to obtain a linear system.I have studied various notes and asked my teachers but they are unable to explain how the solution has been obtained.I have the solution but I want to know how it has been done.Please could...

I'd rather not post the exact problem since it's homework, I don't think my instructor in E&M would want me posting full problems but I will just ask relevant conceptual question... Let's say we I have a long cylinder with time-defendant surface current density \vec K(t)=K_of(t). So if I want...

I have to solve the integral equation $$y(x)= -1+\int_0^x(y(t)-sin(t))dt$$ by the method of successive approximation taking $$y_0(x)=-1$$. Sol: After simplification the given equation we have $$y(x)=-2+cos(x)+\int_0^x y(t)dt $$. So comparing it with $$y(x)=f(x)+\lambda\int_0^x k(x,t)y(t))dt$$...

I have to solve the integral equation $$y(x)=1+2\int_0^x(t+y(t))dt$$ by the method of successive approximation taking $$y_0(x)=1$$. Sol: After simplification the given equation we have $$y(x)=1+x^2+2\int_0^xy(t)dt$$. So comparing it with $$y(x)=f(x)+\lambda\int_0^x k(x,t)y(t))dt$$ we have...

Hello everyone, I'm working on a problem and it turns out that this equation crops up: 1 = cos^{2}(b)[1-(c-b)^{2}] where c > \pi Now I'm pretty sure you can't solve for b in closed form (at least I can't), so what I need to do is for some value of c, approximate the value of b to...

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

I am trying to follow the reasoning of the last problem in the set linked below. I can't figure out what approximation they used in step 24. Thanks. http://www.physics.fsu.edu/courses/spring08/phy5524/sol1.pdf It looks like it might be an approximation based on a geometric series, but there is...

Homework Statement In a spherical well in which.. V= \begin{cases} 0,\text{for }0 \le r < R \\ ∞, \text{for } r > R \end{cases} the s-wave eigenstates are \phi_n(r)=\frac{A}{r}\sin\left( \frac{n\pi r}{R} \right) where A is a normalization constant. If a particle is in the ground state and...

Homework Statement Question: is the charge sum an approximation? Homework Equations E.g. consider 0.10 M Na_{2}SO_{4} solution. Charge sum appears to be 0.20 from elementary stoichiometric considerations. The Attempt at a Solution The charge sum, however, seems to be ignoring...

Triangle ABC is an equilateral triangle with side 4 cm long which is measured corrected to the nearest cm. Find the percentage error of the perimeter of triangle ABC.The Attempt at a Solution Is [(0.5 x 2 x 3) / 12] x 100% correct? the '2' here is the measurement errors of the starting pt and...

Hey! :o Given the following two-point problem: $$-y''(x)+(by)'(x)=f(x), \forall x \in [0,1]$$ $$y(0)=0, y'(1)=my(1)$$ where $ b \in C^1([0,1];R), f \in C([0,1];R)$ and $ m \in R$ a constant. Give a finite element method for the construction of the approximation of the solution $y$ of the...

Homework Statement Apply saddle point approximation to the following integral: I = ∫0∞xe-ax-b/√xdx a,b > 0 Recall that to derive Stirling formula from the Euler integral in class we required N >> 1. For the integral defined above, identify in terms of a and b appropriate parameter that...

Hi, I have to use a wide 5 point stencil to solve a problem to fourth order accuracy. In particular, the one I'm using is: u'' = -f(x + 2h) + 16f(x + h) - 30f(x) + 16f(x - h) - f(x - 2h) / 12h2 or when discretized u'' = -Uj-2 + 16Uj-1 -30Uj + 16Uj+1 -Uj+2 / 12h2 In addition to...

Hello! I have a nifty set of problems (or rather one problem, gradually building itself to be a great problem) that I like to collectively call "The final problem" as it is the last thing I need before I can take the exam in Numerical Methods.Information There is given a Laplace equation...

The explanation below illustrates why I think the method of successive approximations is merely a sneaky way of working with power series when you're not formally allowed to use a Taylor series expansion for a function (i.e. when it doesn't exist, as in proving the existence theorem on ode's for...

Homework Statement A coaxial cable with inner radius a and outer radius b lies on the z-axis (such that the cable's axis merges with the z-axis). its length (along z-axis) is L. at z=-L there are voltage sources that are distributed uniformly connecting the inner wire to the outer one. at...

I'll look over your other posts tomorrow but I'd like to post the 3rd problem for now in case you got the time to answer somewhere between that time. So the third and final (for now, anyway) problem. Problem III Approximate the differential equation $$\frac{d^3 u}{dx^3}=g(x)$$ on a model*...

Hi, I need help determining which of the statements are true and false. http://imgur.com/9BC1Wqf I know this involves steady state approximation, but I find that when I try it I am never able to rid of all the intermediates. Please help Thanks

Here is another problem I should understand of differential approximation. Maybe I shouldn't have posted it before solving the first one but I'm really anxious to learn them so I can pass the class. Problem II Using the Taylor series and the undetermined coefficient method, approximate the...

So in this thread I plan to present 3 problems and my takes on them. I think I'll post each next one after I get some help and solution with the current one. Note that this is a translated problem from my native language. Task Nr. 1 Write the difference analogue (I mean in discrete form) to a...

I'm looking for a concise description of an algorithm for low-rank approximation via SVD. I've seen a number of articles referring to Lanczos method and finding eigenvalues but nothing going all the way to determining all the matrices involved in the low-rank SVD of a given matrix. Any...

Use a linear approximation to find a good approximation to $$\sqrt{100.4}$$ $$x = 100.4$$ $$x1 = 100$$ $$y1 = 10$$ $$y - 10 = \frac{1}{20}(100.4 - 100) $$ $$y = 10.20 $$

I'm studying low rank approximation by way of SVD and I'm having trouble understanding how the result matrix has lower rank. For instance, in the link the calculation performed on page 11 resulting in the so-called low rank approximation on page 12. This matrix on page 12 doesn't appear to me to...

Homework Statement consider a capacitor with circular plates of radius a, separated by a distance d (d<<a) and V(t)=V_{0}sin(wt) a)Considering the z axis to be the capacitor axis, verify that the electric field between the plates is , in good approximation, given by \vec{E}(t)\approx E_{0}...

So \sqrt[5]{306} is a pretty good approximation for Pi (=3.14155). If you add 1/51, so that you have \sqrt[5]{306+1/51} you get 3.1415925 (last digit is 6 for actual Pi.) If you add 1/12997, \sqrt[5]{306+1/51+1/12997} you get 3.141592653587 (vs 3.141592653589 for actual Pi.) And so on. As you...

Hi all, I try to understand the difference which can made by using or not using NWA .. I have a process have cross section (p p > x x) ~ 10^-5 pb , where x is a paricle have mass mx = 2 TeV and dominant decay channel (x > b b~) with Gamma (x > b b~) ~ 6 * 10^2 GeV , while sigma ( p p > x x ...

[FONT=arial]Prove that a set $A\subset\mathbb{R}^n$ is (Lebesgue) measurable $\iff$ there exist a set $B$ which is an $F_{\sigma}$ and a set $C$ which is a $G_{\delta}$ such that $B\subset A\subset C$ and $C$~$B$ (C without B) is a null set. $F_{\sigma}$ is a countable union of closed sets, and...

Hi, could someone please help me with this one: I'd need to form a sequence of rational functions ##R_{n}(x)## such that ##\lim_{n \to \infty} R_{n}(x)=\theta(x)##, where ##\theta(x)## is the Heaviside step function. The functions ##R_{n}(x)## should preferably be limited in range, i.e. for some...

I have stumbled upon an approximation to the average of integer square roots. \sum^{n}_{k=1}{\sqrt{k}/n} \approx \sqrt{median(1,2,...,n)} Sorry I am not very good at LaTeX, but I hope this comes across okay. Could anyone explain why this might be happening? In fact, I just discovered that...

Homework Statement https://scontent-b-mia.xx.fbcdn.net/hphotos-prn2/v/1456973_10201043975243279_1765184125_n.jpg?oh=05b39611ad70d28d837ed219e1b0f2aa&oe=52838593 Homework Equations The area can be approximated by using the sum of the areas of the rectangles. Area of rectangle = change...

If I have a polynomial function f(x) and I want to find an approximate polynomial g(x). I can apply the Nth order Taylor Polynomial of f centered at some value a. So suppose I have f(x) = 5 - 6x + 20x^3 + 10x^5 and I want to find an approximate function, g(x), centered at 5. From what I...

Are Local Linear Approximation, Linear Approximation, and Linearization all the same thing? Question is, I learned about something called Local Linear Approximation in Calc 1. Now in Calc 2, the topic of Linearization from Calc 1 was mentioned. But I never did anything that was referred...

https://www.physicsforums.com/attachments/1614 never done this before so this is an intro problem it mentioned that LA is used in Physics a lot hopefully correct no ans in bk(Speechless)

Homework Statement Problem taken from Boas Mathematical Methods book, Section 14 page 35. Prove that if ##S=\sum_{n=1}^{\infty} a_n## is an alternating series with ##|a_{n+1}|<|a_n|##, and ##\lim_{n \to \infty} a_n=0##, then ##|S-(a_1+a_2+...+a_n)|\leq|a_{n+1}|##. The Attempt at a...

Homework Statement Find the linear approximation of (xy)/z at the point (-3,2,1) The Attempt at a Solution So the example my book gives has 2 variables so I'm struggling a bit with this, But I started off by taking the partial derivative with respect to each variable and solving for...

Statistics problem---exponential approximation Homework Statement A box contains 2n balls of n different colors, with 2 of each color. Balls are picked at random from the box with replacement until two balls of the same color have appeared. Let X be the number of draws made. a) Find a...

Homework Statement The question requires me to approximate binomial distribution to get poisson distribution. Show that N!/(N-n)!=N^n. Homework Equations N!/n!(N-n)! p^n q^(N-n)=Binomial distribution The Attempt at a Solution I expanded N!/(N-n)! and got...

Hello everyone, my first post :shy: I'm reading Zee's 'QFT in a Nutshell' and I came to one thing that bothers me - he's short discussion of steepest-descent approximation. I've known this thing for quite a long time now, but I've never seen the approximation of the corrections. Here is what...

I just need some help with some basic questions I can't remember from a long time ago, just started up school again... 1) Given a function f(x) = (quadratic on top)/(quadratic on bottom) When at x=1, I am given a tangent line to the function f(x), and also given the equation of the tangent...