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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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15 views

Contour integrals

Someone can check if those integrals are correct? $$\int_0^{\infty}d\omega\frac{\Theta(q-k_F)}{(\omega-\omega_q+i\eta)^2}\approx2\pi i\Theta(q-k_f)\frac{d}{d\omega}(1)=0$$ $$\int_0^\infty d\omega_2\...
0
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0answers
34 views

Field Theory: Converting $\int_0^{x_0} d^dx$ to $\int_0^{x_0} dr$, where $r=^{\textrm{def}}\|x\|$

For my Statistical Field Theory class (http://www.damtp.cam.ac.uk/user/tong/sft/sft.pdf), the prof converts integrals over each element of a vector $x$ into a single integral over the magnitude of the ...
8
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1answer
137 views

How Do We Define Integration over Bra and Ket Vectors?

I'm having trouble understanding the completeness condition for bra and ket vectors in Hilbert space, especially in the continuous case. The discrete case makes a fair amount of sense; given any ...
0
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2answers
73 views

Feynman's proof for Newton's shell theorem

I have two questions concerning this proof: Firstly, what is the difference between the increments ds and dx? Are they not just the same thickness of the strip? Secondly, why can the integral ...
0
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0answers
45 views

An Integration about error function [migrated]

An integration which Mathematica gives the result: $$ \int_0^{+\infty} \frac{e^{-\lambda k^2}}{a+b k^2}\mathrm{d}k = \frac{e^{\frac{a\lambda}{b}}\pi}{2\sqrt{ab}}\mathrm{Erfc}\left(\sqrt{\frac{a\lambda}...
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0answers
13 views

Magnetic field from a current arc and its limiting behaviour

For my sanity and the sake of book keeping, I have been going over the very well known and documented calculation for the magnetic field of a current loop. I have been trying to verify the limit ...
1
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1answer
37 views

Problem in the continuum limit of a Kronecker delta

I am having troubles in understanding how to correctly perform the continuum limit of a double sum containing a Kronecker delta. Imagine to integrate a function depending on $t$ and $t'$, both ...
1
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1answer
52 views

Regularising the Green's function in 2D

The Green's function for the 2D Helmholtz equation satisfies the following equation: $$(\nabla^2+k_0^2+\mathrm{i}\eta)\,{\mathsf{G}}_{2\mathrm{D}}(\mathbf{r}-\mathbf{r}',k_o)=\delta^{(2)}(\mathbf{r}-\...
2
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1answer
42 views

How to calculate the gravitational binding energy of a uniform cube of length $L$ and mass $M$?

The functional form is known already (as attached). But what is the solution for this integral?
0
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0answers
16 views

Finding Line Integral Of Magnetic Field over Quarter Circle

This is one of the question in my workbook and I tried solving it in above way. However I am not able to get the answer. Can anyone suggest a method to solve this question ? Am I doing a mistake in ...
0
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0answers
22 views

Literature result for generic one-loop triangle integral

I've never really asked a literature based question here before, but was wondering if anyone knows where I may find a reliable source for symbolic expressions for generic one-loop triangle diagrams ...
0
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1answer
42 views

Properties of Dirac delta function in Integral

I was reading commutation relation of canonical momentum in KG Field from Lectures of Quantum Field Theory by Ashok Das. In page 179, He has used Integration to derive the result where he expressed ...
0
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1answer
20 views

What is the volume charge density (in spherical coordinates)?

What is the volume charge density (in spherical coordinates) of a uniform, in finitesimally thin spherical shell of radius $R$ and total charge $Q$, centered at the origin? Give your answer in terms ...
0
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3answers
57 views

How to derive kinematics equations using calculus? [closed]

I read derivation of kinematics equations using calculus: $$a=\frac{\text dv}{\text dt}$$ $$\implies \text dv=a\text dt$$ $$\implies \int_{v_0}^v\text dv=\int_0^t a\text dt$$ $$\implies v-v_0=at$$ $$\...
1
vote
1answer
47 views

Calculating electric potential — denominator going to zero [closed]

Calculate the potential inside a uniformly charged solid sphere of radius $R$ and total charge $q$. My attempt: There are several ways to solve this problem but I'm curious as to whether this ...
0
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0answers
22 views

Action variable integral

I am solving an action angle variable problem and I'm stuck at the point where I have the following expression for the integral $$ I = \frac{b\sqrt{mE}}{\pi} \int_\theta^{2\pi-\theta}\sqrt{(1-cos \...
0
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0answers
16 views

Bessel differential equation from integral [migrated]

It is a relatively well-known fact that $$\int_{0}^{2\pi}e^{-ikr\cos\theta}d\theta=2\pi J_{0}(kr),$$ where $J_{0}$ is the Bessel function of the first kind and order zero. I'm trying to show that this ...
0
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4answers
62 views

Deriving the formula of potential difference duo to a solid conducting sphere

The doctor in university was deriving a formula and I can't understand how it works A sphere with charge Q The Sphere's radius is R, and we are trying to derive a formula for potential difference at ...
0
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1answer
40 views

Nonlinear GPE of a solid block seems wrong

I am trying to calculate the gravitational potential of a solid block, and I have a nonlinear answer which strikes me as wrong. A block with horizontal surface area $A [m^{-2}]$ and uniform density $\...
0
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0answers
40 views

How to prove that the limit of surface integral exists?

My book "Electromagnetic Fields" says in $\text{Section}\ 3.4$: Question Why does the limit in equation $(3.42)$ exist (convergent)? Why is the contribution from $(S-S_{\delta})$ remains ...
0
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0answers
11 views

Find the time required to travel pulse from bottom to the top of string .(as shown in the figure)

Given that μ is the mass per unit length of string , $M_{b}$ is the mass of block hanged to the bottom of the string and M is the mass of string . I tried like this we know that $V=\sqrt\frac{T}{μ}$...
0
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0answers
36 views

A. Zee Contour Integral

In A.Zee's book I have come a cross an integral which I found difficult to solve.
3
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2answers
77 views

Can someone provide to me an intuitive explanation of the second integral of position with respect to time?

I am aware of what the first integral of position, absement means (at least to a very superficial level). However, I can find nothing regarding the physical intuitive meaning of absity, the second ...
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0answers
18 views

Amperian Loops vs Gaussian Surfaces

I am wondering what makes an amperian loops different from a gaussian surface? I know they are used the calculate different things, but when I take the integral how are they different? This is in ...
0
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0answers
20 views

Problem with an exchange integral

I'm reviewing the chapter on molecules from Gasiorowicz's Quantum Physics book (Ch. 20 of 1st edition) and it gets to a part where it solves the $H_2^+$ ion using the variational principle. There are ...
0
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1answer
37 views

Anomalous magnetic moment of the electron - integration problem

In Schwartz's QFT book (eqn 17.31), to find the anomalous magnetic moment of the electron from the form factors, near the end of the calculation the following integral needs to be evaluated: $$ F_{2}(...
3
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2answers
542 views

Integral Notations in Quantum Mechanics [duplicate]

I've been learning about Quantum Dynamics, time evolution operators, etc. I am confused about the notation used in integrals. Normally I am used to integrals written in this way (with $dx$ on the ...
0
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1answer
67 views

How to calculate the derivative of scale factor as a function of conformal time from the solution of Friedmann equation

For the flat geometry of lamda CDM model, the solution for Friedmann equation is $$ a(t) = \left\{ \frac{Ω_{m,0}}{Ω_{Λ,0}} \sinh^2 \left[\frac{3}{2} \sqrt{Ω_{Λ,0}} H_0(t - t_0)\right] \right\}^{1/3},...
1
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2answers
23 views

element of surface area versus vector element of surface area

In the context of calculating electric flux, is there a difference between element of surface area versus vector element of surface area? Thanks
1
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3answers
55 views

Deriving an integral in a mechanics problem (massless string holding up a disk)

I am told that a massless string is holding up a disk of mass $M$ and radius $R$. I want to find out the value of the tension $T$ in the string. The textbook does this trivially by stating that $2T=...
0
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1answer
50 views

UV divergence integral

Could anyone please explain how to calculate integral such as $$\frac{\Omega}{2}\int_{-\infty}^{+\infty} \frac{d^3k}{(2\pi)^3}\ln\left[{1+\frac{a^2}{k^2}}\right]=-\frac{\Omega a^3}{12\pi}+I_0~?$$ ...
1
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1answer
54 views

Integral of the divergence of a vector field multiplied by the component of another vector field

In Forces in Molecules by Richard Feynman (Phys. Rev. 56, 340 (1939)), eq. (5) implies that $$\int(\nabla\cdot \textbf{F})E_\mu^\alpha dv=-\int F_\mu(\nabla\cdot E_\mu^\alpha)dv,$$ being $\textbf{F}...
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0answers
10 views

Need a sample of a Probability density function in Oblate Spheroidal Coordinates

I need to develop a probability density function in Oblate Spheroidal Coordinates. That is, the volume under a this function surface is equivalent to 1 . Any idea how to propose this ? In a two ...
0
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1answer
35 views

Why don't we use Leibniz integral rule when solving Diffusion equation using the Fourier transform?

My question concerns the solution to the diffusion equation: $$\frac{\partial{p(x,t)}}{\partial{t}}=D\frac{\partial^2{p(x,t)}}{\partial{x}^2}~.\tag{1}\label{1}$$ I have a question about the solution ...
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0answers
31 views

Does Jackson's result for the vector potential of current loop correct?

General form of Maxwell equation is given by $$ \nabla_\mu F^{\mu\nu} = 4\pi J^\nu $$ where $F_{\mu\nu}=\nabla_\mu A_\nu-\nabla_\nu A_\mu$ is the tensor of EM field. Then Maxwell equations can be ...
1
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1answer
41 views

Solving Lagrangian given initial and final coordinate

Consider a Lagrangian $$L=L\left(q, \dot{q}\right)$$ I can use the Euler-Lagrange equation to find an expression $$\ddot{q}=A\left(q,\dot{q}\right).$$ Let's assume that the equation can be ...
0
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1answer
57 views

Help integrating potential of interaction

I'm having trouble integrating a potential that is very present in many theories regarding Condensed Matter Physics. The potential I'm trying to integrate is $$\int_{0}^{\alpha}d^3\textbf{r}\frac{1}{|\...
2
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0answers
45 views

Integrating to find net pressure and force on a sphere by a fluid

How would I go about solving for the net force on a sphere exerted by air pressure as a result of pressure being dependent only on the vertical position of the ball? When the sphere it is placed in a ...
1
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1answer
64 views

Integrating Laplace's equation over a sphere

The Wikipedia page on Laplace's equation states that if the Laplacian of $u$ is integrated over any volume that encloses the source point, $$\iiint_V \nabla \cdot \nabla u \, d^3V =-1.$$ I can'...
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1answer
34 views

Question about explicit notation of averaged energy conditions integrals

Beyond the basics of general relativity, we rapid encounter the so called Averaged energy conditions. The mathematics of these quantities are related to line and volume integrals. As given by [1], ...
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0answers
48 views

Fourier transform of variable in path integral

In Sredinicki's QFT given below, he changed the integration variables in eq(174). This step confuses me. I only know some basics about path integral. In my opinion, when he used fourier transform of ...
0
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0answers
18 views

How do I calculate the integral for the point spread function of an optical system without the fresnel approximation?

For an object located at the $(x_o, y_o)$ plane, and a lens located at the $(x, y)$ plane, the image is produced at the $(x_i, y_i)$ plane. We can consider the image field $U_i$ to be the sum of the ...
4
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2answers
106 views

Denoting the antiderivative of velocity

With simple Newtonian laws (and in a specific context), I learned that the speed $\vec{v}$ of an object is the derivative of the corresponding position vector $\vec{OM}$. So that means that $$\vec{v}(...
0
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0answers
32 views

Looking for high dimensional integral problems in statistical physics

I've currently working on a Monte Carlo Estimation of high dimensional integrals. I'm wondering if there is a list of well-studied high dimensional integral (50-100 dimensions) integrals in ...
2
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1answer
95 views

What is the closed form of the following integral?

I want to know the closed form of the following master integral in (any) $D$ dimension \begin{equation} \int\frac{d^D k}{(2\pi)^D}\frac{1}{k^2(k+r)^2(k+p)^2}. \end{equation} The references that I can ...
2
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1answer
64 views

Contour Integration in Schwartz

In Matthew Schwartz's QFT text, on page 39, he has the following contour integral: $$\int_{-\infty}^{\infty}dk\frac{e^{ikr}-e^{-ikr}}{k+i\delta }.\tag{3.63}$$ This can be split into two terms, one ...
0
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1answer
68 views

Proving a theorem about the average value of a function over a specific region

Let's say transient phenomenon in a function. A transient phenomenon is defined as: "A transient event is a short-lived burst of energy in a system caused by a sudden change of state." So, for ...
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0answers
44 views

Feynman Parameters vs Passarino-Veltman reduction

I have computed the following one-loop integral: $$\int \frac{d^dp}{(2\pi)^d} \frac{p^{\mu}p^{\nu}}{(p+k)^2p^2}.$$ Using both Feynman Parameters and the Passarino-Veltman reduction. However, while I ...
0
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1answer
73 views

Integral and Wick rotation (Srednicki ch75)

I was reading chapter 75 of Srednicki's QFT book and I ran into this statement. To determine the value of its integral, we make a Wick rotation to euclidean space, which yields a factor of i as ...
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2answers
53 views

Confusion about Change in Integration Variable [closed]

I'm working through example 3.2 of Zangwill's Modern Electrodynamics and have come across a change in integration variables that I just can't seem to get. The example has two different change ...