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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Electric field of an infinite sheet of charge [closed]

I am trying to derive the formula for E due to an infinite sheet of charge with a charge density of $ \rho C/m^2$. I assumed the sheet is on $yz$-plane. I used Coulomb's law to get an equation and ...
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2answers
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Derivation of Rotational Motion Equations using Calculus

How are the equations for rotational motion derived using calculus and the following general equations ? $$\mathbf{v}(t) = \mathbf{v}_0+\int_{t_0}^t \mathbf{a}(t')dt'$$ $$\mathbf{r}(t) = \mathbf{r}...
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1answer
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Integrating $1/x$ in radioactive decay derivation

I have a question concerning, for example, the derivation of the equation for radioactive decay. You start with the following differential equation $$-\lambda \cdot N=\frac{\mathrm dN}{\mathrm dt}$$ ...
1
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1answer
1k views

Why is displacement equal to the area of velocity-time graph? [duplicate]

why is the distance of a body equal to the area of its speed-time graph? the general formula of speed(v) is v=distance(s)÷time taken(t) so the formula of distance(s) should be s=v×t so if the speed-...
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2answers
480 views

Massive versus Massless $\phi^4$ Sunset Diagram - does $\frac{1}{\epsilon^2}$ term vanish for $m=0$?

In a real scalar massive $\phi^4$-interacting theory consider the amputated sunset diagram. This is the integral out of Kleinert and Schulte-Frohlinde Critical Properties of $\phi^4$-Theories: The ...
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2answers
393 views

Moment of inertia: why $\mathrm dI=r^2\mathrm dm$ instead of $\mathrm dI=m\mathrm dr^2$?

When computing the moment of inertia, I observed that people usually use the following logic: $d I=r^2 dm,\ \therefore I=\int r^2 dm$ My question here is, why not use $dI=m ~d(r^2)$? I ...
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0answers
50 views

Closed form of Iterated integrals arising in Fredholm's integral equation solution in the context of Nonequilibrium Quantum field Theory?

While solving a Non-equilibrium quantum field theory problem I came across this class of $2n_{}^{\text{th}}$ order iterated integral : $$F(T_{}^{},T_{0}^{},\epsilon)=\int_{T_{0}^{}}^{T_{}^{}}dt_{1}^{}...
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4answers
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Is dimensional analysis valid for integrals

Can we apply dimensional analysis for variables inside integrals? Ex: if we have integral $$\int \frac{\text{d}x}{\sqrt{a^2 - x^2}} = \frac{1}{a} \sin^{-1} \left(\frac{a}{x}\right),$$ the LHS has no ...
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0answers
64 views

Vector potential of a partially-known magnetic field

let's consider a three-dimensional space permeated by a known magnetic field $\vec{B}$. Let's consider in this space a topologically spherical surface $\mathcal{S}$ centred in the origin. I put a ...
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1answer
219 views

Find velocity using integration method or relative velocity approach [closed]

In the diagram shown below, jeep moves with a speed of 60kmph and the car's velocity as observed from the moving jeep is 20kmph. we need to find the velocity of the car. I used relative velocity ...
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1answer
144 views

Help with proof in Griffiths QM book [closed]

So, I'm having trouble with this proof in Griffiths' QM text. I don't get how Griffiths exactly goes from the text(circled in read) on page 47 to the next step(also circled in red). He says that he ...
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76 views

Get rid of the derivatives and relativistic mass in Feynman lectures

i have a problem with get rid of the derivatives in Feynman lectures (chapter 15, Equivalence of mass and energy). The problem: we have $\frac {d(mc^2)}{dt} = v\cdot \frac {d(mv)}{dt}$, then we ...
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1answer
95 views

Approximating sums as integrals and divergent terms

I have the following sum (notice that the sum starts from 2, i.e. there's no divergence): $$\sum_{i=2}^{N}C_i\dfrac{\exp{\left(-k| \mathbf{R}_i-\mathbf{R}_1| \right) }}{| \mathbf{R}_i-\mathbf{R}_1|}$$...
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73 views

How to solve this complex Gaussian integral? [duplicate]

I have a complex Gaussian Integral for my QFT course to solve without looking at the integral table but I don't know how I should do it The integral is: $$\int_{-\infty}^{+\infty} \exp\left(\frac{...
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1answer
574 views

Divergent integrals in QFT

I am starting to learn about QFT and something that I noticed is that integrals who would diverge otherwise are assigned a value if we do it by contour integration using the residues theorem and the ...
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0answers
27 views

First order expansion in D dimension in an integral

I'm trying to understand the following passage, from A. Schakel Boulevard of broken symmetries (page 135). He's evaluating the normal mass current density in a superfluid. It starts with the ...
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0answers
40 views

Zero Temperature non relativistic fermionic propagator

As an assigmente i am trying to prove the following result: $$\hbar \int \frac{d\omega}{2\pi}\frac{e^{i\omega0^+}}{\hbar\omega-\zeta(\vec{k})+isgn(\omega)\eta}=i\theta(-\zeta(\vec{k}))$$ Where $\...
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2answers
43 views

Is there anything wrong with my Euler's method equations for a pendulum outside of small angles?

I'm trying to write a program to calculate the angle, angular speed and energy of a pendulum at different times using Euler's method. The equation I started with was:$${\rm d}^2θ/{\rm d}t^2 = - g\sin(...
3
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1answer
1k views

Electric field at the apex of a cone

Consider a hollow cone with uniform charge distribution over its surface. When one finds the electric field at its apex it comes out to be an infinite value. However, when a solid cone with uniform ...
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1answer
45 views

Thickness of a $3N$ dimensional spherical shell (entropy of classical gas)

I'm brushing up on statistical mechanics and calculating the entropy of a classical gas (i.e. particles in a box). Working through the calculation, we end up with an integral of the form: $$ \int^{'} ...
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3answers
1k views

Contour integral of Feynman propagator

I am now reading the David Tong's lecture note on Quantum Field theory. I have some questions about the contour used in the integral \begin{equation} \int \frac{d^{4}p}{(2\pi)^{4}} \frac{i}{(p^{0})^{...
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4answers
372 views

Transformation of $d^4x$ under translation disregarded?

Under a translation in spacetime i.e., $$x\mapsto x^\prime=x+a,\tag{a}$$ a scalar field $\phi(x)$ $$\phi(x)\mapsto\phi^\prime(x)=\phi(x-a).\tag{b}$$ My aim is to verify the invariance of an action of ...
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1answer
57 views

Solid State Phonons at the edge of Brillion zone

Bose-Einstein statistics tell us that the number of phonons with energy k at temperature T via the bose einstein distribution. My question is regarding phonons in the low energy regime. My notes ...
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2answers
122 views

Gravitational force of point mass on a rigid body - Integral proof

Assume a point mass $m$ located at $\vec{x}$. Assume also a solid body whose coordinates $\vec{x}'$ belong to a connected subdomain $\vec{x}' \in \Omega$. The solid body has a non-uniform mass density ...
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2answers
55 views

The EM field action for interaction with particles

The action for an EM field that interacts with a set of point particles is $$ S_{mf} = -\sum_i \frac{e_i}{c} \intop A_k dx^k \tag{27.3} $$ (ref - Landau & Lifshitz, eq. 27.3.) In the book ...
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1answer
71 views

Space (distance) as a derivative [duplicate]

We are all used to the mechanical definitions of velocity, as the variation in time of the distance, the acceleration as the variation in time of the velocity, and even more other quantities like the ...
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1answer
85 views

Unusual Feynman Integrals at Two-Loops

I'm studying a very particular conformal field theory where unusual Feynman integrals appear when I'm trying to evaluate a two-loop correlator (in position space). These integrals are on the form $\...
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1answer
108 views

Gauss Theorem for Electrostatics : Why is $D_r$ considered constant?

Gauss Law states that $\int \vec D. d\vec s = Q_{enclosed}$ Consider the spherical case with $\rho_v = cst$ -- for the sake of not doing an integral on other side. Now the proof I was reading says ...
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1answer
246 views

How to calculate the pressure force on Magdeburg hemispheres?

I have tried as below: I choose surface element of hemisphere as $dA=r^2 d\theta d\varphi$,and force exerted on that is $dF=\Delta pdA\cos\theta\cos\varphi$,integrating leads to $$r^2\Delta p\int_\...
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2answers
227 views

Capacitance of tilted circular plate to ground

I am trying to calculate the capacitance between a circular plate of radius $r$ and infinite ground plane, where the circular plate is tilted at an angle $\theta$ to the ground plane. The aim is to ...
5
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1answer
321 views

Wick-rotating the Fourier transform of $\mu+1$ propagators

In Equation (8) of this paper by Groote et. al., we are given the following Euclidean identity: $$ \int \frac{d^{4}\mathbf{p}_{\mathrm{E}}}{(2\pi)^{4}} \frac{e^{ i \mathbf{p}_{\mathrm{E}} \cdot \...
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0answers
208 views

Feynman rules for this perturbative expansion in Grassmann variables

I'm given the integral $$ Z[w] = \frac{1}{ (2 \pi)^{n/2}} \int d^n x \prod_{i=1}^n d \overline{\theta}_i d \theta_i \exp \left( - \overline{\theta}_i \partial_j w_i (x) \theta_j - \frac{1}{2} w_i(x) ...
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1answer
193 views

Solving the quantum field propagator for 1+1 dimensional spacetime at $t=0$

I just started studying Quantum Field Theory in a Nutshell By Zee to quench my thirst for physics. But I got stuck on one of the early exercises. In exercise 1.3.1 (...
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1answer
83 views

What's the result of this integral? [closed]

$$\int_{|\vec k|<k_F} \frac{d^3k}{(2\pi)^3} e^{i\vec k\cdot \vec r} $$ it's not a Fourier transformation since the integrand is not infinite.
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2answers
265 views

Calculation of a net electric field for a charged ring - weird integration

I am just reading book "University physics with modern physics 14-th edition (Young & Fredman)". And on page 702 there is an example 21.9 which says: Charge $Q$ is uniformly distributed ...
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1answer
224 views

How do physicists integrate?

I've always thought that the integration notation in physics is weird, but I understood it nevertheless for a single variable, until I started reading Zee's QFT in a nutshell, where 4 dimensions are ...
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1answer
143 views

Integration over phase space for a one-dimensional harmonic oscillator

The problem asks for a proof of the following equation, and I have no idea on how to approach this: $$\int dx dp \delta(E-\frac{p^2}{2m}-\frac{m \omega^2 x^2}{2})f(E) = \frac{2\pi}{\omega}f(E) ,$$ ...
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0answers
101 views

is it true that $\lim\limits_{\epsilon \to 0^{+}} \ln( x \pm i \epsilon ) = \mathscr{P}\ln|x| \pm i \pi \Theta(-x)$?

Is the following statement true? $$ \lim\limits_{\epsilon \to 0^{+}} \ln( x \pm i \epsilon ) = \mathscr{P}\ln|x| \pm i \pi \Theta(-x) $$ where $\mathscr{P}$ is the Cauchy principal value. The above ...
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1answer
362 views

Commutator of canonical fields in Quantum Field Theory

Let $\phi(\vec{x},t)$ denote the canonical fields and $\pi(\vec{x},t)$ denote the canonical impulses where they're given by: \begin{equation} \phi(x)=\int\frac{d^3\vec{p}}{(2\pi)^3\sqrt{2\omega_{\...
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3answers
122 views

Issue with deriving the work-energy theorem

I'm a little confused regarding the way Total work = Change in kinetic energy is derived using calculus. My issue can be seen at 3:26 of this video: https://youtu.be/2dqO4sy4Njg?t=3m20s Why can the ...
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1answer
77 views

Natural coordinates and time integration

So I have this physics mechanics dynamics textbook with an example and there is a step I couldn't understand in one of the solution examples. Starting with, $$ mR \ddot\theta = mg\sin\theta, \tag{1}$$...
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1answer
279 views

Gaussian integral in momentum space

My question is related to p. 353 of Altland and Simon (section 6.7) which concerns about the following field integral where $\beta = 1/T$ and $V_n$ is defined in the following way: It seems to be a ...
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3answers
91 views

Is the formula for work $W= \vec{F}\cdot \vec{s}$ or $W=\int_C \vec{F} \cdot d\vec{s}~$?

I'm pretty much not so much introduced to calculus (I am grade 11 of India and they teach integration part of basic calculus by the end of grade 12) so I would be glad if the answer will be much more ...
1
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1answer
112 views

Basic cut-off regularization

I've been reading these notes on regularization by Hitoshi Murayama here, and on page 3 there's a few lines of calculations on a quick method of regularizing an integral. But I can't follow the steps ...
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1answer
191 views

RC Circuit Step Response Derivation

This is the step response of an RC circuit: $$V(t)=V_S+(V_0-V_S)e^{-\frac{t}{RC}}$$ Where $V(t)$ is the voltage across the capacitor with respect to time, $V_S$ is the height of the step in supply ...
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1answer
977 views

How to derive the moment of inertia of a thin hoop about its central diameter?

For lack of a better image, I am searching for the moment of inertia of this where$\ r_1 = r_2$ (negligible thickness), and where the object would be rotating around its central diameter, which is ...
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1answer
58 views

Integration of Acceleration to Get Delta Velocity

How do you get delta velocity if you have times t1 and t2 and their velocities v1 and v2, but you only know their accelerations a1 and a2. If you integrate over accelerations a1 and a2, do you get a "...
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2answers
81 views

3D volume integral and changing sign

Let $k$ be 3d momentum. What happens with $\int_{-\infty}^{\infty}d^{3}k$ when I change $k$ to $-k$? I thought that \begin{align} \int_{-\infty}^{\infty}d^{3}k&=\int_{\infty}^{\infty}dk_{x}dk_{y}...
3
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2answers
135 views

Cauchy-Schwarz inequality for Grassmann Integrals?

For square integrable functions $f,g$ of a real variable, the Cauchy-Schwarz inequality states that $$ \left(\int f(x)g(x)\,dx \right)^2 \le \int f(x)^2\,dx \int g(x)^2\,dx. $$ My question is: are ...
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1answer
65 views

Mathematically Describing Auto-unraveling systems [closed]

I am trying to mathematically describe the motion of a auto-unravelling system: systems comprised a material (string, chain, cloth etc.) wound around a cylinder and left to unwind under the weight of ...

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