Questions tagged [integration]
For questions about problems related to physics that involve evaluating integrals. Purely mathematical questions should be asked at math.SE.
1,099
questions
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How do units work when integrating a distribution that has units?
In case some want more context, the source which I'm using (could be wrong...?) is:
https://arxiv.org/abs/physics/0202029 (Eqs. 2,5,7)
The equations of concern are:
$$\sigma_a(\omega) = \sigma_0 g(\...
2
votes
1answer
51 views
Truesdell integration
So I been reading "A first course in Rational Continuum Mechanics" by C. Truesdell and got the following confusion: torque is defined as the integral
$$F_{x_0} = \int_B (x - x_0) \wedge df_{...
1
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2answers
41 views
Confusion regarding the derivation of commutator relations using the radial ordered contour integrals in 2D CFT
I am a little confused about the way commutators are derived in the radial quantization of a 2D CFT. I am trying to derive the relation $$\int_w \mathcal{R} \left(a(z)b(w)\right)dz = \left[A,b(w)\...
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0answers
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Center of mass position for open string theory with Dirichlet typo in textbook
In the string theory textbook Basic Concepts of String Theory by Lust, Blumenhagen, in equation (2.100) is presented the mode expansion for an open string subjected to Dirichlet boundary conditions:
$...
3
votes
1answer
93 views
Continuum States in QM
In the Hilbert space of QM, in the finite dimensional case, for a complete orthonormal set of basis vectors, one writes the generic state vector as: $\psi=\sum_j(\phi_j,\psi)\phi_j$. When the complete ...
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1answer
49 views
Galactic Dynamics: spherical coordinates velocity integral help!
I have been studying galactic dynamics and the following is an extract from Binney and Tremmaine's 'Galactic Dynamics' book.
I have been having some trouble to understand how in the (4.37) integrals, ...
0
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1answer
46 views
Fourier Tranformation Identity
In our class we said for the following fourier transformation:
$$\phi(\vec r,t)=\int_{-\infty}^{\infty}d\omega\int d^3k \hat{\phi}(\vec k,\omega)e^{i(\vec k \vec r - \omega t)}$$
We said that $$\phi(\...
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2answers
80 views
According to Gauss's theorem the integral of $\vec \nabla⋅(\phi \vec E)$ taken over the whole space is equal to 0?
The following excerpt is from Space—Time—Matter by Hermann Weyl, starting on page 101. Among the things I'm not understanding is Weyl's assertion that the integral of $\nabla\cdot(\phi\mathbf{E})$ ...
2
votes
1answer
162 views
Trace identity for $SU(N)$ matrix integral
I would like to know if there's a nice way to compute the following:
$$ \int_{SU(N)} \underbrace{ dU}_{\text{Haar Measure}} \mathrm{tr} \left(U^n \right)~?$$
The following is necessary:
$U \in SU(N)$
$...
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4answers
75 views
Why are areas of graph taken with respect to $t$-axis in velocity time graphs?
If the following graph is given then, why is the displacement equal to the area of the shaded triangle above the axis minus the are of the shaded triangle below the $t$ axis.
Why can it not be area of ...
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2answers
42 views
Help on integrating differential dynamic pressure (kinetic energy per unit volume) for 1D radial flow towards a line sink
A quick introduction to my question and then the question asked at the end. For this problem the cross-sectional area normal to flow is the surface of a cylinder, $A=2\pi r L$, where $r =$ radial ...
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1answer
61 views
Proving that the normalization is independent of time
The function that I want to normalize represents an Airy wave-packet:
$$\psi(x,t)=\mathrm{Ai}[q(x-ut+ivt-\tfrac12at^2)]e^{i\frac{mat}{\hbar}(x-ut-\frac13at^2)}e^{\frac{mv}{\hbar}(x-ut+\frac i2vt-at^2)}...
0
votes
1answer
85 views
Definite integral over a vector field
This article on Wikipedia showed that if the force field is conservative, then the work done on a mass between $t_1$ and $t_2$ is $$\int_{\vec{x}(t_1)}^{\vec{x}(t_2)} \vec{F} \cdot d\vec{x} $$
where $\...
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1answer
33 views
Can I partial integrate contravariant derivatives?
Let us assume I have the integral
$$\int_{\mathcal{M}}\text{d}r\ r^2\phi(r)\partial_r^2\phi(r).$$
If I partial integrate I get
$$\int_{\mathcal{M}}\text{d}r\ r^2\phi(r)\partial_r^2\phi(r)=r^2\phi(r)\...
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3answers
78 views
Why does this integral calculating the electrostatic energy converge?
I came across the problem of calculating the interaction energy of two point charges separated by some distance a in Griffith's Introduction to Electrodynamics.
Here and everywhere else that I look, I ...
3
votes
2answers
147 views
Fourier transform in Minkowski space
Recently, I encountered a difficulty in proving the equation,
$$\int \mathrm d^4x\, \frac{e^{-ipx}}{x^4} =\pi^2 \ln(p^2+i\epsilon)\quad .$$
Here, $x$ is the coordinate, $p$ is the momentum in ...
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0answers
57 views
Calculating UV radiation dose in a moving airstream [closed]
I am trying to calculate the UVC irradiation dose absorbed by an airstream passing around a set of multiple (3) UVC lightbulbs.
Here is a brief description of the scenario:
$3000$ CFM of $55F$ air is ...
1
vote
1answer
36 views
Integral for the sum of $y$-component of an infinite number of forces
I'm new to integrals, so I'm kind of trying to understand how it works
Let's say you are in the first quadrant and you have a infinite number of forces with intensity $F$ which create with the ...
1
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1answer
69 views
Does the abstract wavefunction change in this following example?
Suppose, we have a basis $|u\rangle$, described by the function $u=g(x)$. We can normalize this basis, using our standard $|x\rangle$ basis using the following :
$$\hat{I}=\int |x\rangle\langle x|dx=\...
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2answers
47 views
Electric field in the center of hemisphere shell without double/triple integrals
I'm trying to derive the electric field in the centre of a solid hemisphere of radius $ R $ where the charge is distributed uniformly. I have seen different methods involving double/triple integrals ...
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votes
2answers
54 views
Moment of inertia integration error [closed]
I am trying to calculate moment of inertia of a curve. I tried two methods both of which should yield the same result, but only the first method provides the correct solution.
I'd appreciate it very ...
0
votes
1answer
43 views
Theoretical Magnetic Monopole (USAPHO 2012) [closed]
I'm trying to solve question B2 on the 2012 USAPHO semifinal exam (page 14). The solutions are here on pages 15-16.
I can follow everything until the last line on page 15:
The change in speed in one ...
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0answers
38 views
Error when calculating velocity by numerical integration
I'm trying to do a consideration about the error obtained when we choose to approximate a discrete signal of acceleration (and the relative discrete signal of velocity) using a continuous function.
...
0
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1answer
49 views
Gravitational potential of a disc [closed]
The question says
Find the potential at the center of a disc whose surface area density varies as $$σ = σ_0(1+\cosθ)r $$ where theta is the angle made by the radius with the horizontal and $r$ is the ...
1
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2answers
133 views
How limit is changed while deriving $E = mc^2$?
While deriving $E = mc^2$ I found following in book by Arthur Beiser :
In non relativistic physics, the kinetic energy of an object of mass m and speed v is $\mathrm{KE} = \frac {1}{2} \ mv^2$ . To ...
0
votes
1answer
75 views
Why is the sample interferogram given by the following integral of the intensity pattern?
I am trying to understand equation (2) in the paper by J. E. Greivenkamp with the title Generalized data reduction for heterodyne interferometry from year 1984. If you dont have acces to the article, ...
2
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0answers
77 views
Why are physists using intergrals without specifying limits? [closed]
Take an integral like this as an example:
$$\int U(\vec{r})e^{i\vec{q}\cdot\vec{r}/\hbar}d^3\vec{r},$$
which has somethng to do with scattering. The integration is over all space so correctly it ...
3
votes
2answers
110 views
Euclidean propagator expression for massless particle
Let $\Delta_F(\tilde{x})$ denote the Feynman propagator in the Euclidean variable $\tilde{x}$, in $D$ dimensions,
$$\Delta_F(\tilde{x}) = \int \frac{\text{d}^D\tilde{p}}{(2\pi)^D}\frac{e^{i\,\tilde{p}\...
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0answers
108 views
Given the mass density $\rho(x):=1/|x|$, a ball centered in the origin contains a finite amount of mass
Consider the following charge/mass distribution:
$$\mathbb{R}^3\ni x\mapsto\rho(x):=\frac{1}{|x|}$$
One can show that a ball centered in the origin contains a finite amount of charge/mass, i.e. that
$$...
2
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0answers
89 views
Dirac delta function representations in physics
The most common representation of the Dirac delta function in physics is
$$\delta(x)= \frac{1}{2\pi}\int_{-\infty}^{\infty}dk \,e^{ikx}.$$
My question is in which sense is it a faithful representation ...
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0answers
20 views
Calculating the surface charge density as a function of $X,Y$
$x$-$y$ plane is a conducting plate stretching to infinity. A charged line is placed from $(0,0,0)$ to $(0,0,L)$. It has charge density function of
${\lambda=\lambda_L\frac{z^2}{L^2}}$.
Calculate the ...
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1answer
30 views
Deriving force between continuous distributions of two volume charges without using infinitesimals
We know that force between two point charges is:
$$\vec{F}=k\ q\ q'\ \dfrac{\hat{r}}{r^2}\tag1$$
From here how shall we derive the equation for force between continuous distributions of two volume ...
2
votes
1answer
196 views
Dirac notation, integral and change of basis
Suppose, I have some operator $\hat{A}$, such that in the $x$-basis, it is written as $f(x).$ I'm trying to calculate the expectation value of this operator in integral form. That is given by the ...
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2answers
74 views
Mathematical Derivation of Expression for the Total Power Received By Earth from Sun, via Integration [closed]
I recently tried to derive an expression for the total power received by the earth from the sun, using integration. However, I am stuck at an integration step. Would appreciate if anyone could help ...
0
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2answers
77 views
Calculating the gravitational field on a point mass at central axis of a uniform ring [closed]
Consider a uniform ring of mass $M$ and radius $r$ and centre $O$. Let $P$ be a point on the central axis of the given ring at a distance $a$ from the centre of the ring (line passing through the ...
0
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1answer
47 views
Physical validity of taking limits inside an integrand
Consider an integral expression for the spectrum, $$I(\omega)=\int_{-\infty}^{\infty}G(\omega,t)dt\tag{1}$$
with
$$G(\omega,t)=g(\omega,t)e^{t/\tau}.$$
Here $t$ is time, $\omega$ is frequency and $\...
0
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2answers
41 views
Negative Electrostatic Potential Energy for a system of two point charges
This thing has been confusing me for some time now. When I try to find the potential energy of a system of two charges by evaluating the work done to bring them into a certain configuration, I get a ...
0
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1answer
55 views
Inertia Tensor of an Ellipsoid [closed]
I tried calculating the Inertia Tensor for a symmetric ellipsoid given by the equation;
$ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$
I had no trouble finding the diagonal elements for ...
2
votes
1answer
94 views
Is the constant of integration for the scalar potential in Jackson chapter 1 a mistake?
In Jackson's Classical Electrodynamics, section 1.5 he represents the electric field in terms of the gradient of a scalar:
$$
\begin{align}
\mathbf{E} &= -\nabla\Phi(x),\\
\Phi(x) &\equiv \...
-1
votes
1answer
93 views
Find Potential energy of a hanging chain
A chain of Length L is fixed at one end to a point and the rest of the chain is hanging such that the other end of the chain just touches the ground. Find the potential energy of the chain given its ...
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1answer
42 views
Integral of Liouville-Von Neumann [closed]
When integrating the Liouville-Von Neumann equation:
$$
\frac{\partial\rho}{\partial t} = -\frac{i}{\hbar}[H, \rho]
$$
you get:
$$
\rho(t) = \rho(0)-\frac{i}{\hbar}\int^{t}_0d\tau[H(\tau),\rho(\tau)]
$...
1
vote
1answer
34 views
Marginal Hilbert Spectrum, Does my integration scheme match the analytical expression?
According to the original paper by Huang
https://arxiv.org/abs/1401.4211
The marginal Hilbert spectrum is given by:
$$h(\omega)=\int_0^\infty p(\omega,\mathcal A)\mathcal A^2\mathrm d\mathcal A$$
...
2
votes
1answer
56 views
How to interpret "integrals with operators" like $\int p(\rho) \rho^{\otimes N} d\nu$?
In the last weeks, I came across expressions of the kind $$\int p(\rho) \rho^{\otimes N} d\nu,$$ where $\rho$ is a density operator and $\nu$ is "some appropriate" measure. It also often ...
2
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2answers
230 views
How do I get around this problematic $\log$ term?
This question came while solving for another question on the site.
Suppose we have a body of mass $m$ with $\vec F_2$ and $\vec F_1$ acting on it as shown above. Let $ \vec F_2= c.(\hat F_2)$ be a ...
0
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1answer
66 views
Doubt in finding center of mass of extended bodies [closed]
While finding center of mass of extended bodies, we generally write the coordinate of the starting point of the infinitesimal mass element. Why not the ending or starting point? How does the error ...
0
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0answers
31 views
Trouble with understanding step in derivation of Newton's Shell Theorem
I am currently reading a paper on Dyson Spheres (DOI). When deriving the fact that the net gravitational force on an empty hollow sphere would be $0$, the author reasons the following. I know calculus ...
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0answers
30 views
How do I prove this integral solution in terms of modified Bessel function?
I came across this equation while studying about Dark Matter relic density but I can't seem to prove it.
In the early universe, the number density of dark matter in the thermal bath is approximately ...
2
votes
1answer
97 views
Berezin integral of a Grassmann field
Consider a time dependent Grassmann field i.e. $\theta(t)$. Now, consider the following Berezin integral $$\int [\mathcal{D}\theta] ~\prod_{t}\dot{\theta}\tag{1}$$
where $\dot{\theta}$ time derivative ...
0
votes
1answer
44 views
Equation of infinitesimal ring when finding $ \vec{E}$ of a disc?
When trying to find the electric field created by a uniformly charged disc at a point P on axis of the disc, it can be done by integration.
We start by finding the electric field dE created by each ...
0
votes
1answer
18 views
Heat kernel expression for index calculation/Nakahara Exercise 12.5
A quick question. I was reading this paper, on the last page, the authors wrote
$$
\lim_{\mu^2\rightarrow\infty} \text{Tr} \int_0^\infty dT\ e^{-T} \Big(e^{-(T/\mu^2)LL^\dagger}-e^{-(T/\mu^2)L^\dagger ...
