# Questions tagged [approximations]

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### Taylor expanding a vector function of many variables

How does one Taylor expand a vector function of many variables? The question arises in the context of deriving the geodesic deviation in Newtonian gravity, where we need to subtract as follows, and ...
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### Approximations in deriving infinitesimal operator commutation relationship

I'm having trouble understanding some derivation from Sakurai's QM, chapter 1. To derive $[x,J(dx)]=dx$ he claimed that $dx|x+dx\rangle \simeq dx|x\rangle$ because the approximation is of second order ...
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### Differential Equation & MacLaurin Series for Newton’s Second Law

I am currently working with a differential equation, where I think I need to take the derivative of $ma$ (corrected as per comment). I am trying to write $F = ma$ as a MacLaurin series and eventually ...
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### Laws of Physics and Resolution of Measurement

Any 'physical' quantity is expressed as (generally) a Real Number. Real Numbers are abstract mathematical constructs. Laws of Physics are written as mathematical equations; where these real numbers ...
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### How is it possible to differentiate or integrate with respect to discrete time or space?

As far as I have understood, the case is that there is nothing that argues that time or space is continuous, but at the same time we must assume this in order to be able to calculate derivatives or ...
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### Physical interpretation of the nil ADM mass of gravitational waves

The ADM mass is defined for any asymptotic flat spacetime. Using cartesian coordinates: \begin{equation}\tag{1} E_{\text{ADM}} = -\: \frac{1}{16 \pi G} \, \lim_{r \, \rightarrow \, \infty}\oint_{\...
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### Is it possible to make a tiny approximation in the equations of General Relativity so that they collapse to Newton's law of gravity? [duplicate]

GR and Newton give almost exactly the same result for the orbits of planets and the acceleration of falling bodies. Is it an incredible coincidence, or does GR have some tiny term (e.g., ict) that ...
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### In what sense do we say that the Earth surface is almost inertial in Newtonian mechanics?

From what I understand, inertial frames are the ones in which the momentum of every particle in the universe gets well accounted for. Like if there's any particle losing momentum, another particle ...
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### Quantum factorization approximation for first order Coulomb energy

I'm working through "Advanced Quantum Mechanics" by Franz Schwabl, and he uses this G-correlation function to estimate the first order correction to the ground state energy in a Coulomb ...
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### Why is Schrodinger equation taught while it does not describe an electron?

Strictly speaking, it is "wrong" because it does not describe spin-1/2 particle like an electrons. Why in every QM textbook is it taught, not as a historical equation, but as a current ...
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### Finding second virial coefficient from a potential

Given the potential $\phi = \alpha/r^n$, I want to find the second virial coefficient of my system. My integration limits are from some cut-off length $D$ because the particles don't experience a self-...
### At which distance $r$ from a black hole does its gravity become Newtonian?
The force near a black hole (outside event horizon $r=3r_s/2$) onto a mass $m$ can be calculated by General Relativity: $$F=\frac{GMm}{r^2}\frac{1}{\sqrt{1-\frac{2GM}{c^2r}}}.$$ However, there must be ...