Questions tagged [integration]

For questions about problems related to physics that involve evaluating integrals. Purely mathematical questions should be asked at math.SE.

Filter by
Sorted by
Tagged with
0
votes
0answers
106 views

Difficult integral in quantum mechanics (Fourier transform) [closed]

I am an undergrad that has quantum physics as a course. We just finished the chapter about black body radiation and x-ray etc... and we also got our first assignment which I am basically done with. ...
3
votes
1answer
68 views

Is this box integral divergent or finite when “pinched” at one point?

Let us define the following conformal integral: $$X_{1234} = \int \frac{d^4 x_5}{(2\pi)^8} \frac{1}{x_{15}^2 x_{25}^2 x_{35}^2 x_{45}^2}\tag{1}$$ This is the box integral in position space, and it ...
0
votes
1answer
38 views

Confusion about volume component in Gauss's Law for a cylinder

I am currently working on a problem in which we use Gauss's Law to find the electric field within an infinitely long cylindrical shell (inner radius r, outer radius R) of charge density $\rho = \rho_0 ...
0
votes
2answers
113 views

Evaluating the Coulomb's law integral $\int \frac{\vec{r}-\vec{r}^{\prime}}{\left|\vec{r}-\vec{r}'\right|^{3}} d \tau^{\prime}$

I am trying to evaluate the integral $$\int \frac{\vec{r}-\vec{r}^{\prime}}{\left|\vec{r}-\vec{r}'\right|^{3}} d^3 \tau^{\prime}$$ over the spherical volume $r'<R$ with $\vec{r}$ inside the ...
0
votes
1answer
47 views

What does the area under an acceleration-displacement curve represent?

Considering the equation where, $$ \frac {1}{2} \left (v^2_f - v^2_i \right) = \int_0^s ads\, $$ What does the left-hand side of the equation actually represent? Is there an intuitive explanation ...
1
vote
1answer
49 views

Lebesgue Integral in physics [duplicate]

I study physics and in this year I have to formule and write my bachelor thesis. I have a lot of ideas but some of them looks more interesting for me. A few days ago I thought about situations in ...
-1
votes
0answers
27 views

How do I evaluate the integral of the square root of a quartic equation? [migrated]

I'm currently trying to evaluate the integral $$\int^1_0 \text{d}t\sqrt{(1-t^2)(1-k^2t^2)}$$ where $k\in(0,1)$. Is it the case that this can be expressed in terms of elliptic integrals? I'm ...
0
votes
0answers
16 views

Is it possble to do this complex symbolic calculation with Matlab? [migrated]

Sorry it's bit abrupt, but recently I am caught up in some symbolic calcualtion which is tedious and almost impossible with mere human hands, so just wondering is it possible to solve the double ...
2
votes
3answers
134 views

Struggle in understanding the definition of voltage

I ask some help in understanding better the concept of voltage. The voltage is a difference in electric potential between two points $a$ and $b$. It is defined as $$V_{ab}=-\int^a_b\mathbf{E}\cdot d\...
1
vote
0answers
53 views

Deriving potential from central force [closed]

I've read in a book that for a central force of the form $$ f(r)= \frac{{-ke^ {-r/a}}}{r^2} $$ the adequate potential is $$ V(r)= \frac{{-ke^ {-r/a}}}{r} $$ I'm trying to understand why $$ -\frac{\...
-1
votes
1answer
88 views

Integration and average in physics? [closed]

Many applications of physics theory involve computations of integrals. Examples are voltage, force due to liquid pressure, surfaces... In some cases, when there is linear dependence between two ...
-3
votes
1answer
71 views

Bessel function of first kind [closed]

Can someone tell me how $$\frac{1}{T}\int_0^T e^{i(m-n)\omega t} e^{-ix\sin(\omega t)} e^{iy\sin(\omega t +\phi)}\, dt = J_{m-n}\left(\sqrt{x^2 +y^2 -2xy\cos(\phi)}\right)?$$
0
votes
1answer
66 views

$\phi^4$-theory: Feynman diagrams loop integral calculation [closed]

I am studying quantum field theory by myself, could anyone help me with this integral? How can I get this result? Be more specific?
0
votes
1answer
44 views

Equations of motion for an object with non-constant acceleration related to its velocity [duplicate]

If I have an object flying through space with an initial velocity $v_0$ and undergoing constant acceleration $a$, then I can easily compute its velocity or displacement at any point in time $t$ using ...
0
votes
2answers
154 views

QED integral is zero in dimensional regularization [closed]

Why is this integral zero in dimensional regularization? $$ \int\frac{d^Dk}{(2\pi)^D}\frac{1}{(k^2)^n} $$
0
votes
1answer
23 views

Converting density=mass/volume to relative rate equation for integration

In order to get the mass of an object from density, we might use \begin{equation} m = \int\rho(x)dx \tag{1} \end{equation} I understand why this works on a conceptual level, but I would like to be ...
2
votes
2answers
82 views

Riemann sum of completeness relation in continuous basis

Suppose I have a wave function $\psi $ we express it in a continous states as $$\psi= \int_{-\infty}^{\infty} dxC (x)\rvert x\rangle = \int_{-\infty}^{\infty} dx\rvert x\rangle \langle x \rvert \...
0
votes
0answers
38 views

Hypersurface four-vector, or a familly of four 3-forms?

While reading my old personal notes on forms in relativity, I got confused about some aspects of the mathematical formalism (integration on tensors and p-forms). The energy-momentum flux across some ...
2
votes
1answer
46 views

Is the output of a line integral over a scalar field a vector?

In my physics book of "mathematical methods for physics", the author writes that line integral of a scalar function $\phi$ over a curve $C$ can be written as the following: $$\int_C\phi\,\text d{\...
1
vote
0answers
38 views

Why does integrability imply compatibility?

In mechanics, we have the so called compatibility conditions, which quarantee that when a body deforms, the strains are "compatible" in such a way to no discontinuities or gaps for inside the body as ...
-1
votes
3answers
63 views

Derive gravitational potential energy for this system [closed]

This is on a study guide for my Physics 221 final. I feel like I almost got it but I am off by a sign error. Here is the question: Here is what I got so far: Known: $$F_g = \frac{GMm}{r^2}$$ $$U_g =...
2
votes
1answer
135 views

Peskin QFT Contour Integral — Chapter 6

On page 178 of Peskin's QFT, they have the vector potential $$A^\mu(x)=\int\frac{d^4k}{(2\pi)^4}e^{-ik\cdot x}\frac{-ie}{k^2}\left(\frac{p'^\mu}{k\cdot p'+i\epsilon}-\frac{p^\mu}{k\cdot p-i\epsilon}\...
-1
votes
1answer
49 views

Solving a two variable integration [closed]

I was going through the solid state book by Philip Phillips. I came across an integral similar to: $$\int_{0}^{\beta}d\tau d\tau^{'}e^{-E_c|\tau-\tau^{'}|}$$ where $\beta E_c >> 1$. I am not ...
2
votes
2answers
114 views

Integral form of work during an irreversible process?

Question Why can't the work during an irreversible process be integrated? Where is my understanding amiss? Motivation (for this question) A lot of my physics background seems to say this is a math ...
0
votes
3answers
153 views

Why does the solenoidal term vanishes in a barotropic fluid?

In fluid dynamics, and in particular in atmospheric dynamics, the so-called solenoidal term is the line integral: $$\oint \frac{\vec{\nabla p}}{\rho}\cdot d\vec l$$ where $p$ and $\rho$ are the ...
3
votes
1answer
114 views

Why is $(-\frac{e^2}{4\pi \epsilon_0}) = (-\frac{\hbar ^2}{ma})$?

Note: No, this is NOT a homework question. I am struggling to understand how two physical concepts are related and truly think this could be helpful to a broader audience. Also, I already have the ...
1
vote
1answer
52 views

How can I relate this integral to dimensional regularization?

In the paper "Scattering into the Fifth Dimension of $\mathcal{N}=4$ Super Yang-Mills", the authors give the following result for an integral: $$\begin{align} I^{(1)}(x_{13}^2,x_{24}^2,m) =& \...
3
votes
1answer
68 views

Torque due to continuous force distribution / pressure [closed]

In my fluid mechanics course, I was exposed to some cases where I need to calculate the torque due to the pressure and all solutions manuals or online tutorials take it as a known fact that $d\tau=rdF$...
1
vote
1answer
27 views

Torque experienced by a coplanar loop of current in a uniform magnetic field

There are a lot of posts on this already, but apparent all of them just consider some special case. I am now struggling with this more general case. Let there be a magnetic field with strength $B$. ...
2
votes
2answers
69 views

Electric fields in continuous charge distribution

My question may be very basic, but I can't think of a reasonable explanation for this. Consider a solid charged sphere. Now, we have an electric field inside the solid sphere, but at any particular ...
3
votes
1answer
63 views

Regulating a divergent integral in QED

When we try to regulate a divergent integral, we introduce another parameter, say $\lambda$ and then compute the integral. We finally take a limit (either $\lambda \rightarrow 0, \infty $) to restore ...
1
vote
0answers
73 views

Can I use dimensional regularization with this integral?

I would like to extract the divergence of this integral in 4d Euclidean space: $$\int d^4z \frac{1}{(x-z)^4}\tag{1}$$ This divergence is expected to cancel with other divergences, which I got using ...
1
vote
1answer
116 views

When can I set $d=4$ in dimensional regularization?

I am using dimensional regularization to extract the divergence of some complicated integral. I work in $d=2\omega$ dimensions, with $\omega\approx 2$. After I extract the divergence, I have an ...
0
votes
1answer
46 views

Work done to compress a liquid in container [closed]

Into a compressed container containing water with pressure p and volume V we want to pump additional water. What is the work done? Unlike in the ideal gas, the work cannot be simply found out using ...
0
votes
0answers
42 views

Integrating a product of Gaussian distributions

I'm stuck at this relatively easy looking integral where I have gaussian distributions \begin{equation} \sigma(x,y)=\frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{x^2+y^2}{4\sigma^2}} \end{equation} and the ...
0
votes
4answers
109 views

Confusion about this simple electrostatics line integral

Suppose we want to find the electrostatic potential $\phi_{0}$, with reference to infinity, at $r_{0}$ resulting from a positive charge $q$ located at the origin. For simplicity, let us assume we are ...
0
votes
2answers
70 views

Evaluation of contour integration [closed]

Consider the integral $\int^{\infty}_{-\infty}\frac{q\exp(iqR)}{q^2-k^2}dq$. This kind of equation appears in evaluation of Green's function in scattering theory.We use contour integration to evaluate ...
1
vote
3answers
72 views

Find velocity from acceleration equation [closed]

Suppose the acceleration of a particle is a function of $x$, where $$a(x) = (2.2 s^{-2})x$$ (a) If the velocity is zero when x = 1.0 m, what is the speed when x = 3.4 m? (b) How long does it take ...
0
votes
0answers
77 views

What am I not understanding about this double integration of acceleration to get position?

Brilliant.org has a module on classical mechanics and I'm having difficulty with a mathematical step. They want you to represent position in terms of acceleration and then to solve the double integral ...
0
votes
0answers
16 views

“Euclid’s test” , Negative Pressure and Measure Theory

I don't understand what does it mean for 'Euclid's Test' when they talk about negative pressure Using Euclid’s test for a hypothesis of examining its implications, one finds that negative pressure ...
0
votes
0answers
47 views

Monte Carlo integration - convergence

I have a 5D integral being calculated with a Monte Carlo uniform random sampling. The issue is that the region of integration is very small and for 100000 points I get only around 20-30 points every ...
1
vote
1answer
104 views

A triple integral in Spherical coordinates from Jackson's book on Electrodynamics [closed]

I have been trying a solution for the following integral from Jackson but i do not seem to go anywhere. Please help. The problem is to compute interaction energy due to 2 charges. Compute following ...
0
votes
1answer
52 views

Oscillator integral for frequency

If, for a (not necessarily simple harmonic) oscillator I have that $$\frac{dx}{dt} = G(x)$$ then I can express the period of motion as $$\int_{0}^{T/4} dt = \int_{0}^{X_{max}} \frac{dx}{G(x)}.$$ What ...
0
votes
2answers
55 views

Area under a velocity graph

If I took the definite integral of a velocity graph from 0 to 10 seconds, the answer would be the change in position over those 10 seconds correct? I am told by my teacher the area is change in ...
0
votes
0answers
60 views

Gaussian oscillatory integral evaluation using regularization

To evaluate the Gaussian integral $$ \int_{-\infty}^\infty dx e^{iax^2} = \sqrt{\frac{\pi i}{a}}, $$ one can use an appropriate contour as here, or use the method of "regularization", contained for ...
0
votes
1answer
67 views

Gauss's law for magnetism : double integral

Gauss's law for magnetism is stated as followed with the beautiful closed surface double integral (by wikipidia): $$ \mathop{\vcenter{ \huge\unicode{x222F}\, }}_{S} \mathbf{B} \cdot \text d\...
3
votes
2answers
89 views

Field of a uniformly charged disk: integration question

In my book (University Physics by Young and Freedman), during solving the common example of finding the electric field along the x-axis from a uniformly charged disk, they arrive at this differential ...
1
vote
2answers
51 views

Integral evaluation

I am reading this paper where I have encountered the following integral: $$ I_2 = \lim_{a\rightarrow \infty}\int_0^\infty e^{-\beta k^2} \frac{\cos(2ka)}{\kappa^2 + k^2}dk.$$ where $\beta>0$ is ...
0
votes
0answers
7 views

Angular positions of points on a pole figure

I have to integrate a certain physical property of a crystal within a section of orientation space, Say between the space bound by directions <100>, <110> and <111> and so on. How can I ...
1
vote
1answer
48 views

How to take the Fourier transform of the propagator of a vector field?

In the paper Wilson Loops in N=4 Supersymmetric Yang--Mills Theory, the authors give the following generalized Fourier transform for a propagator in $d=2\omega$ dimensions: $$\int \frac{d^{2\omega}p}{...