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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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12 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},...
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1answer
15 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
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0answers
26 views

How to apply the Gauss law when charge density is a function of not only $r$? [on hold]

How to apply the gauss law when charge density is a function of not only r? Caution : I am not asking anyone to do the sum fully . I just wanna know how to apply gauss law in this problems only the ...
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0answers
34 views

The integral of a multivariable Dirac delta function [migrated]

Does anybody know how to perform the following integral analytically: \begin{equation} I=\int^\infty_{eV/2} d\epsilon_1 \int_{-\infty}^{-eV/2} d\epsilon_2 \int^\infty_{0} d\epsilon' \int_{-\infty}^{0} ...
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3answers
51 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=...
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1answer
45 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
49 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
9 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 ...
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1answer
28 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
20 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 ...
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1answer
34 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 ...
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1answer
55 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}{|\...
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0answers
25 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
57 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
30 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
31 views

Integrals formulae in two dimension [migrated]

I come across the following two integral formulae The first integral formula is \begin{equation} \int_C d^2z |z|^{2a}|z-x|^{2c}|z-1|^{2b} = \frac{S(a)S(c)}{S(a+c)}|I_{0x}|^2+\frac{S(b)S(a+b+c)}{S(a+...
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0answers
32 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 ...
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0answers
16 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 ...
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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}(...
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0answers
30 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 ...
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1answer
87 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
54 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 ...
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1answer
66 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
32 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 ...
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0answers
52 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
46 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 ...
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2answers
41 views

Accuracy of Continuous Charge Distributions [duplicate]

In E&M, it is common to represent the charge on a body with a continuous, scalar charge density function. In reality though, the body contains discrete charges. I understand why this approximation ...
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1answer
33 views

Integral limits in phase space

If I am calculating the partition function for $H=cp$, ultrarelativistic gas in three dimensions. And by breaking down $d \Gamma$ into $dq$ and $dq$ and further using spherical coordinates I will get $...
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1answer
46 views

If kinetic energy is mass times the integral of velocity, isn't it just a product of mass times distance? [closed]

I'm still learning Calculus at the moment and I'm currently on integration. The moment I realized the "$1/2$" and square value in $v^2$ are just products of integration, can't one just use ...
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4answers
153 views

Different expressions for distance & displacement : $\int$$d$$|\vec r|$, $\int$$|$$d$$\vec r$|, and $|$$\int$$d$$\vec r|$

I came across these expressions in my book. And the book says that all these are different from each other. The expressions are : $\int$$d$$|\vec r|$, $\int$$|$$d$$\vec r$|, and $|$$\int$$d$$\vec r|$ ...
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1answer
75 views

How to integrate pressure over a sphere?

if I had a function for pressure $P(y)$ dependent on $y$ (vertical position) and I had a sphere in a stationary position, how would I integrate the pressure differential over the surface of the sphere ...
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0answers
48 views

How to find density of a wire when we know information of it in polar form

We know that the wire has the "path" in the shape of the polar graph $$r = \theta, \quad 0 \leq \theta \leq \pi/2 $$ and we know at point $(\theta,\theta)$ the density of the wire is $2\theta$. I ...
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1answer
25 views

Faraday's Induction Law Notation

I am confused as to the notation used in a course I'm taking on physical optics. I have presented 2 variants of Faraday's Law, combined with the full set of Maxwell's equations. The first ...
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1answer
35 views

Is the Jacobian different for different ${\cal L}^p$ norms?

(I posted this to the math stackexchange, but I've yet to receive an answer so I figured I should post here too, as this forum seems faster to respond and is full of knowledgable people.) Because the ...
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1answer
73 views

Electric field at center of a uniformly charged hemispherical shell [closed]

My attempt : I divided the shell into charged rings by whom I know the electric field at the center by the following relation : $\hskip2in$ Integrating the electric fields by all the rings I get ...
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2answers
34 views

Anomalous value of electric field due to a uniformly charged disc at a point on its central axis

Upon trying to find the electric field using integration (as done below) we get the following result : According to me there are two problems with this result : This function is not of odd parity. ...
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0answers
22 views

The microcanonical ensemble approach to calculating the entropy of an ideal gas [duplicate]

I would like to set up the following problem. Assume I have a box of volume $V$ with $N$ noninteracting particles in it. The energy of each particle can be $\mathcal{E}_i$ such that $\sum_i \mathcal{E}...
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1answer
24 views

Kinematics Problem requiring Calculus [closed]

Let the Instantaneous Velocity of a rocket just after launching be given by v={ 3t for 0<= t <2 2t+ 3t^2 for 2<= t<=3 t^3 for t>3 Find the ...
2
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1answer
62 views

How is the complex integration done for the Wigner function in coherent state representation?

$$W(\alpha)=\frac{1}{\pi^2}\int e^{\lambda\alpha^*-\lambda^*\alpha} \operatorname{Tr}\left[ \hat{\rho}e^{\lambda\hat{a}^\dagger} e^{-\lambda^* \hat{a}} \right] e^{-\frac{|\lambda|^2}{2}} \, d^2\lambda....
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1answer
41 views

Manipulation of momentum factors in Feynman integrals (solved)

My current understanding: Consider a massless Feynman graph with propagators of the form $1/p^2$ and some momentum factors in the numerator. Now consider a vertex, which for simplicity of notation is ...
4
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1answer
120 views

Complex Gaussian integral with different source terms

Do the source terms multiplying a complex field and its conjugate need to be conjugates for the Gaussian identity to hold? E.g. is $$\int D({\phi,\psi,b}) e^{-b^\dagger A b +f(\phi, \phi^\dagger,\...
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2answers
184 views

Explicit computation of singular part of two-loop sunrise diagram

For $\phi^4$, there is two-loop self-energy contribution from sunrise (sunset) diagram. The integration is $$ I(p)=\int\frac{d^D p_1}{(2\pi)^D}\frac{d^Dp_2}{(2\pi)^D}\frac{1}{(p_1^2+m^2)(p_2+m^2)[...
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2answers
75 views

How is the wave function Lebesgue integrable?

Let's assume we have a plane wave $\psi(x,t)= A_{0}e^{i(kx-wt)}$ in position space. To find the momentum representation of this wave we'd apply the Fourier transform. However, I don't see how this is ...
2
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0answers
48 views

Polylogarithmic integrals

NB: I was sent here from Math.SE, stating that polylog integrals are more common in physics and someone here might have an answer. I have asked in Phys.SE chat whether it was okay to post here but no ...
0
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1answer
64 views

How to know what the area under curve represents?

Is there a way to find out what the area under the curve represents? For eg. If i gave you a graph of $v$ with respect to $t$ would you be able to tell me what the area under the curve represents ...
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2answers
27 views

Trying to get moment of inertia of a disc using moment of inertia of a rod

I know how moment of inertia of a disc is calculated using the usual way, but just for fun, I tried this way which is rather giving incorrect answer. I don't know what's the flaw in this and thus ...
2
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0answers
79 views

An involved Feynman integral

Working with QCD, I have found the following integral from Feynman diagrams to solve $$ I(p)=\int\frac{d^4p_1}{(2\pi)^4}\int\frac{d^4p_2}{(2\pi)^4}\frac{1}{p_2^2-m_0^2} \left(\frac{p\cdot p_1-p_1\cdot ...
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2answers
96 views

How is $\int \frac{d^{3}\mathbf{p}}{(2\pi)^3}\frac{1}{2\sqrt{|\mathbf{p}|^2+m^2}}$ manifestly Lorentz-Invariant?

When writing integrals that look like $$ \int \frac{d^{3}\mathbf{p}}{(2\pi)^3}\frac{1}{2\sqrt{|\mathbf{p}|^2+m^2}} \ = \int \frac{d^4p}{(2\pi)^4}\ 2\pi\ \delta(p^2+m^2)\Theta(p^0) $$ it is often said ...
2
votes
1answer
70 views

Solving the rocket differential equation

I'm trying to derive the rocket equation. I'm pretty sure that the differential equation for the rocket equation is $$v(t)\delta t =\frac{m(t)\delta t }{m(t)} V_e$$ where $v(t)\delta t$ is the ...
0
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1answer
37 views

Follow up on understanding path integral measures

A while ago I asked the following question: Understanding Measure in Path integrals and got to the conclusion that path integral measures are infinite products of $d\phi(x_i)$ for some scalar field $\...