Questions tagged [integration]

For questions about problems related to physics that involve evaluating integrals. Purely mathematical questions should be asked at math.SE.

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Evaluating the electrostatic potential at the centre of a ring

I was solving a problem which was stated "Find the potential difference between $2$ thin wire rings each of radius $R$, whose axes coincide. The charges of rings are $Q$ and $-Q$. Find the ...
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80 views

What is the intuition behind why $\vec{E} = \int \mathrm{d}\vec{E}$ works for summing up infinitesimal contributions of electric field?

Say you want to find the electric field, $\vec{E}$, at some point in space, $P$, which is induced by some uniformly charged rod, $Q$, of known length, $L$. What you would do is break this charged rod ...
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$\frac{0}{0}$ from Curvilinear Dirac Delta

The definition of the Dirac Delta in an arbitrary curvilinear coordinate: $$\delta(\vec{r})=\frac{\delta(x^1-x^1_0)\delta(x^2-x^2_0)\cdot \cdot\cdot \delta(x^N-x^N_0)}{h_1h_2\cdot\cdot\cdot h_N}$$ ...
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31 views

Klein-Gordon Propagator for spatial separation $x - y = (0, r)$

On pg. 27 of Peskin and Schroeder I would like to know how we get the first equality when deriving the Klein-Gordon propagator for $x^0 - y^0 = 0, \vec{x} - \vec{y} = \vec{r}$: $$ D(x-y) = \int \frac{...
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When to trust numerical integration in mathematica? [closed]

I have an integral that I need to solve for a research problem on scattering in scalar QED. It is a 5-D integral which (as far as I can see) can only be done numerically. In attempting to do this ...
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210 views

Calculating displacement from acceleration (intuitively) [closed]

If I say acceleration of car is constant at $4\; \rm m/s^2$. Then isn’t it that it covers $4\; \rm m$ in $1\; \rm s$ with velocity $4\; \rm m/s$. Then in $2\; \rm s$, the velocity is $8\; \rm m/s$. ...
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Integral of the product of 4 spherical harmonics

Recently, I saw a closed formula for the integral of the product of three spherical harmonics in two dimensions here Integral of the product of three spherical harmonics and I was wondering if someone ...
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2answers
31 views

Electric potential of disc [closed]

Find the potential $V$ at the edge of a thin disc of radius $R$ with a charge uniformly distributed over one of its sides with surface density $S$. The solution of the question is to consider an ...
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88 views

Do closed line integrals need to be evaluated in “the line's” rest frame?

I've seen it said that the definition of emf requires that the integral be carried out in the circuit's rest frame. \begin{equation*} \mathcal{E} =\oint \mathbf{f} \cdotp d\mathbf{l} \end{equation*} ...
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How to make integral substitution from 3d k-space to 1d energy domain with non-trivial dispersion relation

In solid state phyics, one often encounters a subsitution from the 3d k space to a 1d integral via the standard dispersion relation $E= \vec{k}^2/2m$ via \begin{align} \int d\vec{k} Z(\vec{k}) = \...
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Improper Integrals and Contour Integration

I am reading a physics paper which employs contour integration to evaluate some integrals and I am a little confused about something. According to the author if we want to integrate some function from ...
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How do I integrate this function? [closed]

I have an exercise where I have to calculate the potential energy function $U(x)$ of this force $F$. I know the function is given by integrating $-F$, but how do I do this? $c$ is a constant, and a ...
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1answer
38 views

Fourier transform of linear response function

I was studying Linear Response Theory from 'A modern course in statistical physics' by Reichl, and some doubts came up. The response function is defined as $$<\alpha(t)>_{F} = \int_{-\infty}^{+\...
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Calculating for total distance travelled vs position given $v(t)$ graph

It is currently to my understanding that the area under a $v(t)$ graph is the displacement of an object because $$\int v(t)dt = s(t).$$ However, some of the problems I have attempted recently give you ...
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278 views

What does it mean to integrate with respect to mass?

I've encountered many integrals that seem to integrate functions of distance with respect to mass, for example, $\int_0^Mr^2dm$ for the moment of inertia of continuous mass distribution. I'm not sure ...
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63 views

How can I solve an integral by using hypergeometric function of the second kind? [migrated]

This is an article that I've been studing for a while. I came across this integral equation $$\mathrm dm=\frac{\mathrm dr}{{r}^{z+1}\left[1-{\left(\dfrac{r_h}{r}\right)}^{n+z-1}\right]}$$ where $n$, $...
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Why does an exact differential mean a force is conservative?

If you can express an integrand as an expression of just one variable i.e. $xdy + ydx = d(xy) = df$ then why does that mean that a loop integral on that will equal 0? Is it because if it is just a ...
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Is there any example from physics, supporting $\int_0^1\frac1xdx=\int_1^\infty\frac1xdx$?

This question is related to the potential possibilities of classification of divergent integrals more precisely than just "divergent to infinity" and the like. Improper divergent integrals ...
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1answer
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Two formulas for current: how are they the same? [duplicate]

I am struggling to reconcile the two definitions for current density. Definition one: the current is the flux of the current density vectors through a surface: $$I = \iint_{S} \vec{J} \cdot d\vec{S}$$ ...
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Conservation of Energy / Poynting vector

On this page, it states: "The only fields' couples (a,b) for which we can get a non-zero value of the Poynting vector for a large distance r₀ over the sphere are (R, R); (radiation, radiation) ...
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Divergence theorem & Kinetic model of plasmas

In one section in my plasma physics notes (see below) on the Vlasov equation, 6D phase space & the kinetic model for plasma, I can't quite understand how (via Gauss's divergence theorem) this ...
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2answers
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Can we always integrate numerically?

I dont know if its suitable here or on Math SE, Most of the times, when I watched online lectures most lecturers say that if we cant solve a integral exactly we can always numerically integrate it. (E....
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70 views

How do I get the equations of motion starting from this Hamiltonian?

So I have a very complicated Hamiltonian given by: $$ H_R(\psi, P_\psi) = \frac{-B_0 R^3}{2 \pi} (2 \psi - sin(2 \psi) + \sqrt{B_0^6 R^6 V(\psi)^2 - P_\psi^2} $$ where $$V(\psi) = \frac{1}{\pi} \sqrt{...
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Help in understanding line and surface integrals

I am studying multivariable calculus, and I have not studied physics in depth yet in my high school career. I'm struggling to understand the real-world uses of line and surface integrals, especially, ...
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Simplifying an integral in Gravitational Statistical Mechanics

I want to know if there is any way to simplify the following integral into a product or a power of a certain simpler one. The integral is of the form: $$ Z_0=\int_{V^N} e^{-\beta Gm^2 \displaystyle\...
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1answer
62 views

How to choose the sign of the differential?

I know this is a very simple question, and I have searched it too. How to avoid incorrect symbols in calculation results.I don’t understand how to choose the sign of $ds$. An object moves from a to b,...
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26 views

Integral of displacement [duplicate]

Can somebody explain to me what is the integral of displacement $x$ with respect to time (if it is anything significant)? In high school physics, we are thought derivatives of $x$ which are velocity, ...
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34 views

Particular case of Green's theorem

Suppose we have $u(r)=\sum_{\lambda=1}^{\infty} a_{\lambda} u_{\lambda}(r), \, 0 \leq r \leq a$ in this article Introduction to R-matrix theory in atomic physics they say that $$\int_{0}^{a}\left[u_{\...
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Path integral calculation in complex scalar field theory

I have some trouble understanding a particular expansion in my QFT lecture. Consider a complex scalar field $\phi$, with the Lagrangian $$\mathcal{L} = \partial_\mu\phi^*\partial^\mu\phi-m^2\phi^*\phi....
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90 views

Centre of mass of a frustum of a cone [closed]

How do I find the centre of mass of a solid frustum of a cone? Do I try it in a manner similar to finding the centre of mass of a solid cone? However I'm not sure what limits of integration do I take ...
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Suitable choice of surfaces for integrals

Especially in electrodynamics, the integral theorems of Gauss and Stokes (in connection with the Maxwell equations) are often used to compute electric or magnetic fields. To compute the resulting ...
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Calculating electric potential

When caculating electric potential using $dV = -E\cdot dl$ when the distribution of $E$ is known, how do we determine the upper and lower limit for the integration?
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Integrating amplitude for electron scattering from stationary proton

I am to replicate a computation that Zee does on page 134 of his QFT book, 2ed. We are using the canonical formalism to compute the amplitude for an electron to scatter from a staionary proton. I ...
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Integrating amplitude for electron scattering from Coulomb potential

I am following Zee's QFT book in Section II.6. I have found the amplitude for an electron to scatter from a static Coulomb potential as \begin{align*} \mathcal{M}&=ie\!\int\!d^4x\,\big\langle ...
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Use of Cauchy's integral formula in the derivation of the Feynman propagator

In deriving the Feynman propagator in Timo Weigand's 2014 QFT2 notes, at the top of page 37, (equation 1.170), we use Cauchy's integral formula: $$g(z_0)=\frac{1}{2\pi i}\oint_{C_1}\frac{g(z)}{z-z_0}\...
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Why can we multiply by $dt/dt$ to change variable of integration? Please look at equation 5-20 [closed]

I am struggling to understand why can we just multiply by $dt/dt$. I was thinking it was just a change of variables, but I cannot come up how that works. Can someone explain why we are allowed to do ...
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Fourier Transform of Coulomb potential in QFT

I am master student of particle physics and I want to find coulomb potential $V(r)$ from $\tilde{V}(p)$ in Schwartz-Quantum Field Theory and the Standard Model what I have as $\tilde{V}(p)$ from 16....
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Conceptual question on the force experienced by a plate on one of its sides

I imagined a situation like this: Let's say dip a plate in water so that it is just submerged. I Tried to calculate the force experienced by the plate in one of its sides. To do that I must find total ...
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Justification for “reducing integrals” in the virial expansion for gases

I have been following this document (https://sites.chem.utoronto.ca/chemistry/jmschofi/chm427/gases.pdf) regarding the virial expansion of gases and on finding the virial coefficients. On Page 7, they ...
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Gaussian Integral with Complex Parameters — Divergence and Convergence

Although this is more of a mathematical question, I will in what follows refer to an answer of @Qmechanic that has been posted in this forum (I am sorry for creating a new post for this, but I ...
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How come the formula $W=Fd$ doesn't apply for energy stored in springs?

I always thought that work is like the energy transferred and it is given by $W=Fd$, but this concept gets problematic for springs. If the force $F$ is applied to a spring which compresses it by a ...
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3answers
165 views

Line Integral of electric field through the nucleus

In the case of electrostatics, We know that for electric field $\mathbf{E}$, $$\nabla \times \mathbf{E}=0$$ which in the line with the integral form $$\oint \mathbf{E}\cdot d\mathbf{l}=0$$ Now Can I ...
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31 views

Planck`s law for an interval of wavelengths

Planck's law for spectral radiance of a black body for a wavelength: $$P=\frac{2hc^2}{\lambda^5}\frac{1}{e^{hc/\lambda k_bT}-1}$$ If I want to know the spectral radiance for an interval of wavelengths,...
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Please help me formulate (and solve) a differential equation for a practical DIY problem concerning air sterilization (for covid)

Here I present the DIY air sterilisation project which this is about. I want to calculate the overall sterilization rate of the device under some simplifying assumptions. (UVC is the hard ultra violet ...
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3answers
99 views

The force of gravity between a shperical shell and a particle

I am trying to understand the proof of why the force acting on a spherical shell and a particle is $$\frac{GMm}{r^2}$$ Where M is the mass of the sphere and m is the mass of the particle. I am looking ...
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35 views

Calculating space average of an operator in quantum mechanics

I am trying to rederive the results of L Smrcka and P Streda's paper about transport coefficients in strong magnetic fields. I wonder how they calculate space averages of current density operators in ...
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Equations of motions vs Verlet: Collision ignored?

I initially coded my simulation using the standard equations of motions, but as is known, it ended up being quite unstable, even if it technically worked. If we take x(t) to be the position-function ...
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54 views

Unusual one-loop Feynman diagram

I encountered the following integral for the one-loop Feynmann diagram of a non-local massive scalar field theory in four dimensions, $$ I(s,t,m)=\int {d^4 k\over (2\pi)^4} {1\over (|k|+m)(|k+p_1|+m)(|...
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95 views

Absolute entropy: is it valid to write $Q=ST$?

All the basic texts say $dQ = T dS$. Some say it is possible to define absolute entropy. That source gives an equation integrating C dT which should reduce to $Q=ST$, if $S(0) = 0$ and $Q$ is ...
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Matching Two Point Function in momentum space using spherical coordinate

Background of the problem: The problem I am currently struggling is related to the momentum representation of Fourier transform. Briefly speaking, the integral in Minkowski under Cartesian coordinate ...

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