What is Approximation: Definition and 760 Discussions

An approximation is anything that is intentionally similar but not exactly equal to something else.

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  1. chwala

    Use Euler method to determine the approximation of given problem

    There is a mistake in my opinion on the text. In my working i have, ##y_1= 3 + 0.2 e^{\cos1} = 3+ 0.54357 = 3.54357## ##y_2 = 3.54357 + 0.2 e^{\cos 1.2} = 4.0871## ##y_3 = 4.0871 + 0.2 e^{\cos 1.4} = 4.6305## I also noted that we do not have an exact solution for this problem.
  2. walkeraj

    I Is the Idea of a Continuum Always an Approximation to the Physical?

    Question: When thinking of continuums the most notable seems to be space-time but they also mark a simplification to reality like in continuum mechanics, often taught when learning the tensor calculus needed for general relativity. The question is that for general relativity when a geodesic...
  3. H

    British physics olympiad problem: A ball bearing bouncing off a steel cylinder

    I am struggling to find correct approximation for the problem. Is momentum conserved at the immediate impact of... (1) Can I ignore gravitational force and potential for the spring which is connected to ground and vertically upholding a mass . ( using equilibrium)
  4. L

    Dirac delta function approximation

    Hi, I'm not sure if I have calculated task b correctly, and unfortunately I don't know what to do with task c? I solved task b as follows ##\displaystyle{\lim_{\epsilon \to 0}} \int_{- \infty}^{\infty} g^{\epsilon}(x) \phi(x)dx=\displaystyle{\lim_{\epsilon \to 0}} \int_{\infty}^{\epsilon}...
  5. L

    Plotting the approximation of the Dirac delta function

    Hi, I am not sure if I have solved the following task correctly I did the plotting in mathematica and got the following Would the plots be correct? What is meant by check for normalization, is the following meant? For the approximation for ##\epsilon > 0##, does it mean that for the...
  6. G

    Help me prove integral answer over infinitesimal interval

    In the book, I see the following: ##\int_{x_1}^{x_1 + \epsilon X_1} F(x, \hat y , \hat y') dx = \epsilon X_1 F(x, y, y')\Bigr|_{x_1} + O(\epsilon^2)##. My goal is to show why they are equal. Note that ##\hat y(x) = y(x) + \epsilon \eta(x)## and ##\hat y'(x) = y'(x) + \epsilon \eta'(x)## and...
  7. P

    I On error estimates of approximate solutions

    I'm reading Ordinary Differential Equations by Andersson and Böiers. They give an estimate for how the difference between an exact and an approximate solution propagates with time. Then they give an example where they encourage the reader to check that this estimate holds. When I do that, I get...
  8. ribella

    A What is the energy dependence of the Equivalent photon approximation?

    Hi, What is the energy dependence of the Equivalent photon approximation? For this approach to be valid, what is the maximum center of mass-energy. As know, this approach is an energy-dependent approach. Can this approach be used to calculate, for example, at a center of mass energy of 100...
  9. Lotto

    B For what elements does Born–Oppenheimer approximation fail the most?

    I would say that for the elements with the lowest atomic numbers, because these elements have their nuclei the lightest and so they can move more and their movement influence electrons more than in some heavier elements, whose nuclei move less. Am I right or not?
  10. yucheng

    A References: continuum approximation of discrete sums?

    Is there more references for how accurate is the continuum approximation to discrete sums? Perhaps more mathematical. What I've found: https://lonitch.github.io/Sum-to-Int/ https://arxiv.org/pdf/2102.10941.pdf Some examples are: Sum to integral $$\sum_{\mathbf{k}} \to 2 \left ( \frac{L}{2...
  11. C

    When is Torricelli’s Law an Approximation?

    Hi! For this problem, When Area 2 > Area 1, but not by much, is this phenomenon no longer called Torricelli's Law because the water is not approximately stationary at the top surface? What is this called now? Many thanks!
  12. P

    I Error of the WKB approximation

    hello everyone I tell you a little about my situation. I already found the approximate wavefunctions for the schrodinger equation with the potential ##V(x) = x^2##, likewise, energy, etc. I have the approximate WKB solution and also the exact numeric solution. What I need to do is to calculate...
  13. S

    B Error in approximation to log(223)/log(3) .... senior moment?

    This is probably a silly question, but I am really stuck. A possible senior moment, is my only excuse. Here is an approximation: ##log(223)/log(3) \approx 10818288 / 2198026 ## So we have: ##log(223)/log(3) - 10818288 / 2198026 = 0.0399292## which is OK but not great -- the error shows up...
  14. S

    I Diophantine approximation via Lattice Reduction

    With some help from another thread, I learned how to solve a simultaneous diophantine approximation involving log(2), log(3), log(5) etc. This method is based on Mathematica's LatticeReduce function. At first, I was quite happy to use it as a black box to work on some hobby math exploration, but...
  15. yucheng

    I Validity of Fresnel Approximation

    Wikipedia says that Fresnel diffraction is valid if the Fresnel number is approximately 1. What Fresnel number then is the Fresnel approximation for paraxial-paraboloidal waves valid? It's not mentioned... Oh I just realized that $$\frac{N_F \theta_m^2}{4} \ll 1$$ So it depends on the maximum...
  16. T

    A Secular Approximation of Dipole-Dipole Hamiltonian

    Hey folks, I'm looking for a derivation of the secular approximation of the dipole-dipole Hamiltonian at high magnetic fields. Does anybody know a reference with a comprehensive derivation or can even provide it here? Given we have the dipolar alphabet, I'd like to understand (in the best...
  17. Arman777

    Comoving Distance in LCDM - Understanding an Approximation

    I am trying to find the comoving distance, $$\chi = c\int_0^z \frac{dz}{H(z)}$$ for the ##\Lambda##CDM model (spatially flat universe, containing only matter and ##\Lambda##). $$H^2 = H_0^2[\Omega_{m,0}(1+z)^3 + \Omega_{\Lambda, 0}]$$ When I put this into integral I am getting, $$\chi =...
  18. M

    MacKay Textbook Example: Laplace Approximation

    Hi, I was attempting example 27.1 question from the book: 'Information Theory, Inference, and Learning Algorithms'. It is about the Laplace approximation. I was confused about part (b) of the question and wanted to check my method if possible. [EDIT]: The link to the book website (official) is...
  19. C

    I Difficulty in understanding step in Deriving WKB approximation

    In Zettili book, it is given that ## \nabla^2 \psi \left( \vec{r} \right) + \dfrac{1}{\hbar ^2} p^2 \left( \vec{r} \right) \psi ( \vec{r} ) =0 ## where ## \hbar## is very small and ##p## is classical momentum. Now they assumed the ansatz that ## \psi ( \vec{r} ) = A ( \vec{r} ) e^{i S( \vec{r} )...
  20. K

    I Exploring Continuous Approximation of 1D Random Walk Steps (Reif, pg 14)

    Reif,pg 14. ##n_1## is the number of steps to the right in a 1D random walk. ##N## are the total number of steps "When ##N## is large, the binomial probability distribution ##W\left(n_{1}\right)## ##W\left(n_{1}\right)=\frac{N !}{n_{1} !\left(N-n_{1}\right) !} p^{n_{1}} q^{N-n_{1}}## tends to...
  21. Arman777

    A Relativistic Redshift and understanding it's approximation

    I was reading an article, and I saw this expression. $$ 1+z=\frac{(g_{\mu\nu}k^{\mu}u^{\nu})_e}{(g_{\mu\nu}k^{\mu}u^{\nu})_o} $$ Where ##e## represents the emitter frame, ##o## the observer frame, ##g_{\mu\nu}## is the metric, ##k^{\mu}## is the photon four-momentum and ##u^{\nu}## is the...
  22. C

    I Finding a Rational Function with data (Pade approximation)

    Dear Everybody, I need some help understanding how to use pade approximations with a given data points (See the attachment for the data). Here is the basic derivation of pade approximation read the Derivation of Pade Approximate. I am confused on how to find a f(x) to the data or is there a...
  23. P

    I Notation of the approximation in quantum phase estimation algorithm

    I'am interested in the notation of the approximation in quantum phase estimation algorithm. In the literature there are different definitions, which I divide into two cases here. Both different in their definition of the ##\delta##. In both cases I start with a quote of the source and show an...
  24. R

    Stirling's Approximation for a factorial raised to a power

    Using log identities: ##log((\alpha - 1)!^2) = 2(log(\alpha - 1)!)## Then apply Stirling's Approximation ##(2[(\alpha - 1)log(\alpha - 1) - (\alpha - 1)## ## = 2(\alpha -1)log(\alpha -1) - 2\alpha+2## Is this correct? I can't find a way to check this computationally.
  25. S

    B Continuity correction when using normal as approximation for binomial

    What if the value of X is not integer, such as P(X < 1.2)? a) Will the continuity correction be P(X < 1.2 - 0.5) = P(X < 0.7)? or b) Will the continuity correction be P(X < 1.2 - 0.05) = P(X < 1.15)? or c) Something else? Thanks
  26. SilverSoldier

    Approximations in Chemical Equilibrium (add a weak acid HA into pure water)

    Suppose we add a weak acid HA into pure water, so that upon addition its initial concentration is c. The following equilibria should establish in the system. $$\text{HA}+\text{H}_2\text{O}\rightleftharpoons\text{H}_3\text{O}^++\text{A}^-$$...
  27. M

    MHB Approximation of eigenvalue with inverse iteration method

    Hey! :giggle: We have the matrix $\begin{pmatrix}2 & 1/2 & 1 \\ 1/2 & 3/2 & 1/2 \\ 1 & 1/2 & 2\end{pmatrix}$. We take as initial approximation of $\lambda_2$ the $1.2$. We want to calculate this value approximately using the inverse iteration (2 steps) using as starting vector...
  28. LCSphysicist

    Variational approximation and QM

    Hello. I should find the energy aproximatelly using the variational approximation for this physical hamiltonian: ##bx^4 + p²/2m## Imediatally, i thought that the better trial wave function would be the one correspondent to the ground state of the harmonic quantum oscilator. THe problem is, in...
  29. Poetria

    Good approximation - multivariable calculus

    I tried to use a Taylor series expanded at 3 and set to 3.01: https://www.wolframalpha.com/input/?i=27+++9+(-3+++x)^2+++(-3+++x)^3+++3+y^2+++y^3+++(-3+++x)+(27+++y^2)=3.01 I got the vector ## (\Delta x, \Delta y)= (0.37887, -0.54038)## It does give a desired result but it is marked as wrong...
  30. alwaystiredmechgrad

    I Defect concentration formula w/o Stirling approximation

    In many cases, the concentrations of defects or charges are quite big enough to use SA, due to a big number of Avogadro's number. The derivation for the well-known formula of a defect concentration is followed. If the n_v is expected to be lower than 1, then it would be impossible to use SA...
  31. P

    Methodology / Philosophy of Science

    Summary:: When experimenting to improve a theory, account for the fact that your experimental equipment is made using the very same theory which you are trying to improve. 1.) It would take many decades (~ 80 years?) to design and make equipment entirely using a proposed new theory which has...
  32. Arman777

    A Understanding an Approximation in Statistical Physics

    In a book that I am reading it says $$(V - aw)(V - (N-a)w) \approx (V - Nw/2)^2$$ Where ##V## is the volume of the box, ##N## is the number of the particles and ##w## is the radius of the particle, where each particle is thought as hard spheres. for ##a = [1, N-1]## But I don't understand how...
  33. S

    Why a normal distribution is not a good approximation for these exam scores?

    I am not really sure what the reason is but my argument would be if normal distribution is appropriate, then almost all the score will fall in the range of μ - 3σ to μ + 3σ For this case, the range of μ - 3σ to μ + 3σ is 26.6 to 118.4 and all the score is unlikely to be within the range. I...
  34. B

    Python Laplace approximation in Bayesian inference

    Hello everybody, I am working on a Python project in which I have to make Bayesian inference to estimate 4 or more parameters using MCMC. I also need to evaluate the evidence and I thought to do so through the Laplace approximation in n-dimensions: $$ E = P(x_0)2\pi^{n/2}|C|^{1/2} $$ Where C...
  35. B

    4th order Taylor approximation

    So I just followed Taylor's formula and got the four derivatives at p = 0 ##f^{(0)}(p) = (1 + \frac {p^2} {m^2c^2})^{\frac 1 2} ## ##f^{(0)}(0) = 1 ## ## f^{(1)}(p) = \frac {p} {m^2c^2}(1 + \frac {p^2} {m^2c^2})^{\frac {-1} 2} ## ## f^{(1)}(0) = 0 ## ## f^{(2)}(p) = \frac {1} {m^2c^2}(1 +...
  36. R

    Adiabatic Approximation for Infinite Square Well

    I took the w derivative of the wave function and got the following. Also w is a function of time, I just didn't notate it for brevity: $$-\frac{\sqrt{2}n\pi x}{w^{3/2}}cos(\frac{n\pi}{w}x) - \frac{1}{\sqrt{2w^3}}sin^2(\frac{n\pi}{w}x)$$ Then I multiplied the complex conjugate of the wave...
  37. boniphacy

    I Problem with function approximation

    We have a function: ## f(x,y)=\sqrt{\frac{1−2x}{1−y^2}} = \frac{\sqrt{1−2x}}{\sqrt{1−y^2}}## for small x and y, we can use standard approximations: ## 1/\sqrt{1−x}=1+x/2+... ## and ##\sqrt{1−x}=1−x/2−... ## Ok. Now we can approximate the whole function f(x,y) First method: ##...
  38. GrimGuy

    Lagrangian of system of bodies in PN approximation [Landau Textbook]

    Hey guy, I'm having problems to understand the final part of this section. The book says we have the lagrangian from one particle (106.16), then we have some explanation and then the total lagrangian is given(106.17). For me is everything fine until the 106.16, then i couldn't get what is going...
  39. A

    Teaching approximation techniques in basic courses

    Not sure how universal it is, but my experience through half of my undergrad education gave me the impression that 90% physics was about exactly solvable problems. Off the top of my head, the only approximation we ever did in introductory courses was the binomial expansion to get the electric...
  40. R

    I The Central Field Approximation for Many-Electron Atoms

    Attached is my book's section on many-electron atoms. It says that in the central field approximation, an electron's potential energy is a function of its distance from the nucleus. Later on it says there is an effective atomic number. Does this mean that in this approximation, all charges...
  41. B

    I Molecules when Born-Oppenheimer approximation doesn't work

    Hello! I am trying to do some molecular physics calculations, involving the calculation of the expectation value of certain vector operators (such as the electric dipole moment of the molecule) in given molecular states. In most cases assuming the Born-Oppenheimer (or adiabatic) approximation...
  42. Leo Liu

    Midpoint Riemann sum approximation

    Can someone please explain why the formula for midpoint approximation looks like the equation above instead of something like $$M_n=(f(\frac{x_0+x_1}2)+f(\frac{x_1+x_2}2)+\cdots+f(\frac{x_{n-1}+x_n}2))\frac{b-a}n$$? Thanks in advance!
  43. Leo Liu

    How does this approximation work?

    My physics textbook does the approximation in the homework statement. Here, x and y are variables and are much smaller than h. I attempted to figure out why it is valid with ##(1+x)^-1\sim 1-x##. However, after trying to convert the initial equation into 1+x form, I obtained ##h(1-(h+x-y-1))##...
  44. Leo Liu

    B Approximation of the equation of ellipse

    My physics textbook does the approximation that $$r=\frac{r_0}{1-\frac{A}{r_0}\sin\theta}\approx r_0\left( 1+\frac A r_0\sin\theta\right)$$ when ##A/r_0 \ll 1##. Can someone please explain how it is done?
  45. L

    Thermodynamics: gas expansion formula or approximation error?

    FIRST TYPE: REVERSIBLE PROCESS At the temperature of 127 ° C, 1 L of CO2 is reversibly compressed from the pressure of 380 mmHg to that of 1 atm. Calculate the heat and labor exchanged assuming the gas is ideal. Q = L = - 34.95 J CONDUCT 380 mmHg = 0.5 atm L = P1 * V1 * ln (P1 / P2) = 0.5 * 1...
  46. S

    Integral as approximation to summation

    Writing down several terms of the summation and then doing some simplifying, I get: $$\sum_{r=1}^n \frac{1}{n} \left(1+\frac{r}{n} \right)^{-1}= \frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+...\frac{1}{2n}$$ How to change this into integral form? Thanks
  47. M

    MHB Approximation of the integral using Gauss-Legendre quadrature formula

    Hey! :giggle: Let $\displaystyle{I_n(f)=\sum_{i=0}^na_if(x_i)}$ be a quadrature formula for the approximate calculation of the integral $I(f)=\int_a^bf(x)\, dx$. Show that a polynomial $p$ of degree $2n+2$ exists such that $I_n(p)\neq I(p)$. Calculate the approximation of the integral...
  48. VexCarido

    Runge-Kutta Projectile Approximation From Initial Conditions

    Hi everyone. I'm a new member, great to be here:) I have a few questions that I wanted to ask you guys regarding the method by which we implement the Runge-Kutta approximation of Projectile Motion if we should do it using a numerical iterative method with a Spreadsheet like Excel. I have...
  49. J

    Debye Approximation of Heat Capacity in 1D

    So really i am just unsure how to answer the last part of the question. I am unsure how to apply the low and high temperature limits the way i have done it. Do i set upper/lower limits on the integral and solve? If so i am not sure what to put Here is what he book has for 3d
  50. U

    I What is the approximation used here?

    I'm reading a paper (Beamwidth and directivity of large scanning arrays, R. S. Elliott, Appendix A) in which the author starts from this expression: ##\frac{\sin\left [ \left ( 2N+1 \right ) u_0\right ]}{\sin(u_0)}\sum_{p=-P}^Pa_p\cos(p\pi)\left [\frac{\sin(u_0)}{\sin(u_p)} -1+1 \right ]##...