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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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What is the role of constant in 1st equation of motion [closed]

In the proof of the equation of motion , while we integrate both sides.so, the constant is produced on both sides which cannot be always same then what about the constant in the equation
-3 votes
0 answers
37 views

Why can't we find electric field of a uniformly charged hollow hemisphere by using vertical strips which are perpendicular to the flat surface? [closed]

I tried finding electric field of hollow hemisphere by using integration on vertical strips which are at angle theta but the answer comes wrong but I get the answer when I integrate horizontal strip? ...
3 votes
1 answer
98 views

Confusions on The Gravitational Energy of a Point P in a Cube

I have been working, quite tirelessly, to try and find an answer to a question that has been bothering me for some time now. I have been working over some proofs, in the Newtonian Mechanics world, to ...
5 votes
1 answer
64 views

Net particle number density for relativistic particles at finite chemical potential (tricky integral)

Question: How does one show that the chemical potential of relativistic fermions is negligible at high energies? In particular, I would like to show that the difference between the particle density $n$...
2 votes
1 answer
196 views

How to use a piecewise acceleration function to get a position function?

This should be a relatively easy problem but I think I am missing something somewhere. This problem consists of a object that is being thrown into the air at $t = 4s$ at a velocity $v_0$ here is my ...
0 votes
0 answers
11 views

$p(z)$ polynomial with $p(0) \neq 0 \neq p(1)$; $\int_{\partial R} \frac{p(z)}{(z-1)z^2}dz$; $R=[-1,2] \times [-1,3]$ [migrated]

Consider $p(z)$ a polynomial such that $p(0) \neq 0 \neq p(1)$ and the rectangle $R=[-1,2] \times [-1,3]$, calculate $\int_{\partial R} \frac{p(z)}{(z-1)z^2}dz$ The poles $P=\{0,1\}$ are in the ...
0 votes
1 answer
100 views

What does the superscript $3$ in $d^3x$ mean in an integral?

At the risk of seeming ignorant, please explain what does the superscript $3$ in $d^3x$ mean in the integrals 5.12, 5.13, 5.14, 5.15? Why 3? Why there is no such in the 5.10 integral?
0 votes
1 answer
311 views

Do we put negative sign while calculating inward flux by Gauss Divergence theorem?

Suppose we have a vector field $\vec A=ax\hat i + by\hat j+cz\hat k$ and we want to calculate its inward flux $\int \vec A\cdot\vec {dS}$ over the spherical surface $x^2+y^2+z^2=1$(with area vector ...
3 votes
3 answers
178 views

Closed form expression of 2D CFT integral

I am currently working on a 2d CFT and am wanting to compute a complex plane integral, making sure I take into consideration potential contact terms as well. The integral in question is $$ \int_{\...
0 votes
0 answers
13 views

Describing force accumulation trend of an infinite volume with evenly distributed radiative sources

I am looking for confirmation if I've built my equation properly. My goal is to describe the change in force over time at a given point if evenly distributed radiators (in-phase or cumulative energy/...
0 votes
1 answer
269 views

Partial integration of dot products and the $\nabla$ operator

I am working my way through Alexander Schekochihin's lecture notes on the kinetic theory of plasmas (section 5.1.), and I'm having trouble with the derivation of the law of energy conservation in a ...
2 votes
0 answers
34 views

Confused with the product of Feynman propagators [closed]

While reading Thomas Rauh's 'Introduction to Feynman integrals and multiloop techniques' I came across the following identity $$\frac{1}{A^aB^b}=K\int^1_0 \frac{dx x^{a-1}(1-x)^{b-1}}{[xA+(1-x)B]^{a+b}...
2 votes
1 answer
227 views

Trick for integration into a plane wave basis

I'm reading the article https://arxiv.org/abs/hep-th/9705200 and part of it has left me very confused. In order to speak about their equation (1) the authors make the following statement: \begin{align*...
-1 votes
1 answer
52 views

Difference of $p^0$ and $E_p$

In QFT when I learn about Feynman-propagator, I see such an expression: $$ \frac{1}{2E_p}e^{-iE_p(x^0-y^0)}=-\frac{1}{2\pi i}\int_Cdp^0\frac{e^{-ip^0(x^0-y^0)}}{(p^0-E_p)(p^0+E_p)}. $$ I know that ...
1 vote
1 answer
40 views

Proof for Moment of Inertia of Triangle (with respect to the axis out of the page at a origin) [duplicate]

I found a formula for a "Triangle with vertices at the origin and at $P$ and $Q$, with mass $m$, rotating about an axis perpendicular to the plane and passing through the origin" given on ...
1 vote
2 answers
3k views

Volume integral of current density

I'm currently studying magnetostatics and have a simple question : What is the volume integral of the current density over the whole space in magnetostatics $$\int_{V} \textbf{j} \space d^3\textbf{r}$...
5 votes
3 answers
4k views

Contour integral of Feynman propagator

I am now reading the David Tong's lecture note on Quantum Field theory. I have some questions about the contour used in the integral \begin{equation} \int \frac{d^{4}p}{(2\pi)^{4}} \frac{i}{(p^{0})^{...
1 vote
3 answers
94 views

Why can you integrate with different bounds in thermal expansion differential equations?

I am just an independent student and was learning thermal expansion with differential equations and i saw someone on the internet solving the differential equation for the law like below: $$\frac{1}{L}...
-1 votes
1 answer
100 views

How to Find Trajectory of Particle?

Let’s say I have a particle, and I know all the forces acting on it at every position. (Let’s say the particle is in an electric/gravitational field to simplify the mathematics involved.) Now, is ...
5 votes
3 answers
1k views

Finding the illuminance from a triangular light source

Since most light sources in games are point-like, it's pretty difficult to approximate area light sources with point sources. As triangles are a universal form to represent 3D models (thus area light ...
-1 votes
1 answer
64 views

On causality for space-like intervals and the Klein-Gordon field: integral pg 27 of Peskin and Schroeder's Introduction to Quantum Field Theory [closed]

Note: this is not a duplicate: I am not interested in the issue of the contour, but in methods of integration. I am desirous to integrate the following: $$\int_m^{\infty}{\rho e^{-\rho r}\over\sqrt{\...
0 votes
2 answers
390 views

How to lead Bessel $K$ function from eq. (4.12) in Srednicki?

In Srednicki, $$\begin{align*} \left[\varphi^{+}(x),\varphi^{-}(x^\prime)\right]_\mp=&\int\widetilde{\mathrm{d}k}\int\widetilde{\mathrm{d}k^\prime}\mathrm{e}^{\mathrm{i}(kx-k^\prime x^\prime)}\...
0 votes
0 answers
56 views

Second-Order Perturbation Correction for Helium atom

I am trying to calculate the second order perturbation correction for helium atom: $$E^{(2)}_n = \sum^\infty_{i, i \neq n} \frac{|\langle \Psi_n | V | \Psi_i \rangle|^2}{E_n - E_i}$$ with the ...
0 votes
0 answers
52 views

When can I commute the 4-gradient and the "space-time" integral?

Let's say I have the following situation $$I = \dfrac{\partial}{\partial x^{\alpha}}\int e^{k_{\mu}x^{\mu}} \;d^4k$$ Would I be able to commute the integral and the partial derivative? If so, why is ...
0 votes
0 answers
61 views

How do I integrate the Maxwell-Boltzmann-Distribution in terms of the 4-velocity?

I want to calculate the interaction rate of a 2->2 process where the distribution function of the particles is given by the Maxwell-Boltzmann distribution. The result has already been published in ...
0 votes
0 answers
76 views

Ghost-free quadratic gravity

This is an question about how to write an equivalent of "energy-squared" in terms of a gravitational metric. i.e. a spin-2 term that approximates to the spin-0 term $\int (\nabla^\mu \phi \...
4 votes
3 answers
107 views

Complex coordinates $ds^2 = dzdz̄$ in 2d

I have a very elementary question about complex coordinates in two dimensions. When we have a 2D Euclidean space, $$ds^2 = dx^2 +dy^2$$ and we go to complex coordinates: $$z = x + iy$$ $$z̄ = x - iy$$ ...
0 votes
4 answers
2k views

Work done by electric force when moving like charges together

Consider two positive charges. I would like to find the work done by the electric force on charge +q as it is brought closer to charge +Q from radius a to b. I first define electric force as a ...
0 votes
1 answer
208 views

Electrostatic potential of finite charged wire

So I was trying to find the electric potential at any point $\boldsymbol{x}$ of a charged wire of length $L$ at the $z$ axis, from $-L/2$ to $L/2$, and I had to write it down in terms of elliptic ...
0 votes
0 answers
57 views

Why is Dirac Delta Function term in QFT's momentum operator not infinite?

In Weigand's Quantum Field Theory notes, he has the following equations for the spatial components of the momentum operator. $$ \begin{align} P^i &= \int \frac{d^3p}{(2\pi)^3} \dot{\phi(\vec{x})}\...
1 vote
1 answer
109 views

Lorentz-invariant phase space integral

Consider the following Lorentz invariant integral associated to a $2\to 2$ scattering: \begin{equation*} I = \int \frac{d^3\mathbf{p_3}}{(2\pi)^3 2E_3} \int \frac{d^3\mathbf{p_4}}{(2\pi)^3 2E_4} \...
3 votes
2 answers
126 views

Some integrals in QED Renormalisation

I am currently leaning the renormalisation of QED and I have met some tricky integral that seems unsolvable. The integrals are shown in Quantum Field Theory and the Standard Model by Schwartz, page ...
1 vote
1 answer
40 views

How to derive the magnetic field at a distance $x$ (in the same plane) from the centre of a circular loop with current $I$ and radius $R$? [closed]

how to get the expression for magnetic field at any point in a circular loop (plane) with current $I$ and radius $R$, I can't seems to find it anywhere and all I'm getting is the derivation for ...
0 votes
1 answer
777 views

How to calculate the pressure force on Magdeburg hemispheres?

I have tried as below: I choose surface element of hemisphere as $dA=r^2 d\theta d\varphi$,and force exerted on that is $dF=\Delta pdA\cos\theta\cos\varphi$,integrating leads to $$r^2\Delta p\int_\...
0 votes
0 answers
26 views

Density of States, Photoemission and Integrating the Number of States

I'm reading Fowler's theory on photoemission. I'm stuck on a part which Fowler helpfully identifies as "obvious". Fowler sets up the free electron model, suggesting that electrons need a ...
3 votes
2 answers
362 views

Proving a Grassmann integral identity

How to prove the following identity $$ \begin{align} \int {\rm d} \eta_{1} {\rm d} \bar{\eta}_{1} \exp\left(a \left(\bar{\eta}_{1}-\bar{\eta}_{0}\right)\left(\eta_{1}-\eta_{0}\right) + b \left(\bar{\...
1 vote
2 answers
218 views

Fermionic measure in path integral

When writing the fermionic path integral one arrives at an expression containing $\mathcal{D}\bar{\psi}$ and $\mathcal{D}\psi$: $$ \int \mathcal{D}\bar{\psi} \mathcal{D}\psi e^{iS} $$ Usual ...
0 votes
0 answers
18 views

Integration over the complex plane and the completeness relation of the coherent states [duplicate]

I am studying some of the properties of coherent states using the book "Introductory Quantum Optics" by C. Gerry & L. Knight. (C. Gerry & L. Knight, Chapter 3, Section 5) And when I ...
2 votes
0 answers
28 views

Integration by parts to switch sign on anti-Hermitian Louivillian

I am self studying non-equilibrium stat mech and a common theme is that I am unable to reproduce the "this follow from integration by parts" parts of derivations. Here is the most recent: In ...
2 votes
2 answers
341 views

Help in understanding line and surface integrals

I am studying multivariable calculus, and I have not studied physics in depth yet in my high school career. I'm struggling to understand the real-world uses of line and surface integrals, especially, ...
0 votes
0 answers
25 views

Integral identity for thermal states

Context: In this paper, the authors establish a relationship between quantum Fisher information and the imaginary part of the dynamic susceptibility of a system. To do so they utilize the identity: $$...
3 votes
1 answer
82 views

What is the homogeneous charge density (vs inhomogeneous)?

Books/exams often write something like "a homogeneously charged sphere"/"a homogeneously charged rod" - I am unsure what exactly is meant. Take the example of a sphere with radius $...
1 vote
1 answer
481 views

Why can we do Wick rotations even if $\Delta<0$?

The above page is from Schwartz QFT. Why can we do Wick rotations even if $\Delta<0$?
1 vote
0 answers
50 views

Integral of Theta function [closed]

I'm trying to compute the following integral, useful to calculate Amplitudes in String theory \begin{equation} \int \frac{d^2z}{\tau_2} \;\partial^2_z \log \vartheta_1\left(z\right) = - \frac{\pi}{\...
0 votes
0 answers
36 views

Residue theorem application

First I want to provide a little bit of context: I finished my undergrad degree in physics in 2008 and after that I moved into strategic consulting and into the financial world. Right now, at 41 years ...
3 votes
2 answers
87 views

Hooke's law and derivation of work done by spring confusion

For simplicity I'll do this in 1 direction so I can let the sign dictate direction and ditch the vector notation. The force done by a spring can be written as $$F = -k\Delta x , $$however this ...
0 votes
1 answer
34 views

How to know the position of an object when calculating the center of mass, without using integrals? [closed]

For example, if I have a 1/4 piece of pizza, what is the position?
0 votes
1 answer
91 views

Exponential decay of propagator outside lightcone

In Tong's lecture notes (http://www.damtp.cam.ac.uk/user/tong/qft.html) page 38, he calculates the following propagator: $$D(x-y) = \int \frac{d^3 p}{(2\pi)^3} \frac{1}{2E_\vec{p}} e^{-ip \cdot (x-y)}....
1 vote
1 answer
1k views

Orthogonality condition for momentum eigenfunction - integral

Let's consider the orthonormality condition for an eigenfunction $\Psi_p(x)=(2 \pi \hbar)^{1/2}e^{\frac{i}{\hbar} px}$ (plane wave) of the momentum operator with eigenvalue $p$. Then the ...
1 vote
1 answer
40 views

Electric field at a point created by a charged object (derivation/integration process)

I was hoping someone can help me understand the math behind the electric field (electrostatics). I have gaps in my knowledge about integrals and derivatives (university moves very quickly and it has ...

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