Questions tagged [integration]
For questions about problems related to physics that involve evaluating integrals. Purely mathematical questions should be asked at math.SE.
-2
votes
0answers
15 views
Contour integrals
Someone can check if those integrals are correct?
$$\int_0^{\infty}d\omega\frac{\Theta(q-k_F)}{(\omega-\omega_q+i\eta)^2}\approx2\pi i\Theta(q-k_f)\frac{d}{d\omega}(1)=0$$
$$\int_0^\infty d\omega_2\...
0
votes
0answers
34 views
Field Theory: Converting $\int_0^{x_0} d^dx$ to $\int_0^{x_0} dr$, where $r=^{\textrm{def}}\|x\|$
For my Statistical Field Theory class (http://www.damtp.cam.ac.uk/user/tong/sft/sft.pdf), the prof converts integrals over each element of a vector $x$ into a single integral over the magnitude of the ...
8
votes
1answer
137 views
How Do We Define Integration over Bra and Ket Vectors?
I'm having trouble understanding the completeness condition for bra and ket vectors in Hilbert space, especially in the continuous case. The discrete case makes a fair amount of sense; given any ...
0
votes
2answers
73 views
Feynman's proof for Newton's shell theorem
I have two questions concerning this proof:
Firstly, what is the difference between the increments ds and dx? Are they not just the same thickness of the strip?
Secondly, why can the integral ...
0
votes
0answers
45 views
An Integration about error function [migrated]
An integration which Mathematica gives the result:
$$
\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}...
0
votes
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
vote
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
vote
1answer
52 views
Regularising the Green's function in 2D
The Green's function for the 2D Helmholtz equation satisfies the following equation:
$$(\nabla^2+k_0^2+\mathrm{i}\eta)\,{\mathsf{G}}_{2\mathrm{D}}(\mathbf{r}-\mathbf{r}',k_o)=\delta^{(2)}(\mathbf{r}-\...
2
votes
1answer
42 views
How to calculate the gravitational binding energy of a uniform cube of length $L$ and mass $M$?
The functional form is known already (as attached). But what is the solution for this integral?
0
votes
0answers
16 views
Finding Line Integral Of Magnetic Field over Quarter Circle
This is one of the question in my workbook and I tried solving it in above way. However I am not able to get the answer. Can anyone suggest a method to solve this question ? Am I doing a mistake in ...
0
votes
0answers
22 views
Literature result for generic one-loop triangle integral
I've never really asked a literature based question here before, but was wondering if anyone knows where I may find a reliable source for symbolic expressions for generic one-loop triangle diagrams ...
0
votes
1answer
42 views
Properties of Dirac delta function in Integral
I was reading commutation relation of canonical momentum in KG Field from Lectures of Quantum Field Theory by Ashok Das. In page 179, He has used Integration to derive the result where he expressed ...
0
votes
1answer
20 views
What is the volume charge density (in spherical coordinates)?
What is the volume charge density (in spherical coordinates) of a uniform, in finitesimally thin spherical shell of radius $R$ and total charge $Q$, centered at the origin? Give your answer in terms ...
0
votes
3answers
57 views
How to derive kinematics equations using calculus? [closed]
I read derivation of kinematics equations using calculus:
$$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
votes
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
$$ I = \frac{b\sqrt{mE}}{\pi} \int_\theta^{2\pi-\theta}\sqrt{(1-cos \...
0
votes
0answers
16 views
Bessel differential equation from integral [migrated]
It is a relatively well-known fact that $$\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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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 ...
0
votes
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
votes
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
votes
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
votes
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
votes
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
$$ a(t) = \left\{ \frac{Ω_{m,0}}{Ω_{Λ,0}} \sinh^2 \left[\frac{3}{2} \sqrt{Ω_{Λ,0}} H_0(t - t_0)\right] \right\}^{1/3},...
1
vote
2answers
23 views
element of surface area versus vector element of surface area
In the context of calculating electric flux, is there a difference between element of surface area versus vector element of surface area?
Thanks
1
vote
3answers
55 views
Deriving an integral in a mechanics problem (massless string holding up a disk)
I am told that a massless string is holding up a disk of mass $M$ and radius $R$. I want to find out the value of the tension $T$ in the string.
The textbook does this trivially by stating that $2T=...
0
votes
1answer
50 views
UV divergence integral
Could anyone please explain how to calculate integral such as
$$\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~?$$ ...
1
vote
1answer
54 views
Integral of the divergence of a vector field multiplied by the component of another vector field
In Forces in Molecules by Richard Feynman (Phys. Rev. 56, 340 (1939)), eq. (5) implies that
$$\int(\nabla\cdot \textbf{F})E_\mu^\alpha dv=-\int F_\mu(\nabla\cdot E_\mu^\alpha)dv,$$
being $\textbf{F}...
0
votes
0answers
10 views
Need a sample of a Probability density function in Oblate Spheroidal Coordinates
I need to develop a probability density function in Oblate Spheroidal Coordinates. That is, the volume under a this function surface is equivalent to 1 . Any idea how to propose this ? In a two ...
0
votes
1answer
35 views
Why don't we use Leibniz integral rule when solving Diffusion equation using the Fourier transform?
My question concerns the solution to the diffusion equation:
$$\frac{\partial{p(x,t)}}{\partial{t}}=D\frac{\partial^2{p(x,t)}}{\partial{x}^2}~.\tag{1}\label{1}$$
I have a question about the solution ...
0
votes
0answers
31 views
Does Jackson's result for the vector potential of current loop correct?
General form of Maxwell equation is given by
$$
\nabla_\mu F^{\mu\nu} = 4\pi J^\nu
$$
where $F_{\mu\nu}=\nabla_\mu A_\nu-\nabla_\nu A_\mu$ is the tensor of EM field.
Then Maxwell equations can be ...
1
vote
1answer
41 views
Solving Lagrangian given initial and final coordinate
Consider a Lagrangian $$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).$$
Let's assume that the equation can be ...
0
votes
1answer
57 views
Help integrating potential of interaction
I'm having trouble integrating a potential that is very present in many theories regarding Condensed Matter Physics. The potential I'm trying to integrate is $$\int_{0}^{\alpha}d^3\textbf{r}\frac{1}{|\...
2
votes
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
vote
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'...
1
vote
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], ...
0
votes
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
votes
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
votes
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
$$\vec{v}(...
0
votes
0answers
32 views
Looking for high dimensional integral problems in statistical physics
I've currently working on a Monte Carlo Estimation of high dimensional integrals. I'm wondering if there is a list of well-studied high dimensional integral (50-100 dimensions) integrals in ...
2
votes
1answer
95 views
What is the closed form of the following integral?
I want to know the closed form of the following master integral in (any) $D$ dimension
\begin{equation}
\int\frac{d^D k}{(2\pi)^D}\frac{1}{k^2(k+r)^2(k+p)^2}.
\end{equation}
The references that I can ...
2
votes
1answer
64 views
Contour Integration in Schwartz
In Matthew Schwartz's QFT text, on page 39, he has the following contour integral:
$$\int_{-\infty}^{\infty}dk\frac{e^{ikr}-e^{-ikr}}{k+i\delta }.\tag{3.63}$$
This can be split into two terms, one ...
0
votes
1answer
68 views
Proving a theorem about the average value of a function over a specific region
Let's say transient phenomenon in a function. A transient phenomenon is defined as:
"A transient event is a short-lived burst of energy in a system caused by a sudden change of state."
So, for ...
1
vote
0answers
44 views
Feynman Parameters vs Passarino-Veltman reduction
I have computed the following one-loop integral:
$$\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
votes
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
...
0
votes
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 ...
