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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17
votes
5answers
106k views

How to get distance when acceleration is not constant?

I have a background in calculus but don't really know anything about physics. Forgive me if this is a really basic question. The equation for distance of an accelerating object with constant ...
27
votes
4answers
5k views

Complex integration by shifting the contour

In section 12.11 of Jackson's Classical Electrodynamics, he evaluates an integral involved in the Green function solution to the 4-potential wave equation. Here it is: $$\int_{-\infty}^\infty dk_0 \...
12
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3answers
1k views

Why we use $L_2$ Space In QM?

I asked this question for many people/professors without getting a sufficient answer, why in QM Lebesgue spaces of second degree are assumed to be the one that corresponds to the Hilbert vector space ...
17
votes
3answers
1k views

Why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$?

I'm preparing for my exam, but I have difficulties in perceiving why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$?
7
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3answers
343 views

Understanding the differential in integrals

After years, I still find myself having trouble really internalizing the meaning of various differentials in integrals—specifically, when they come about via reasoning regarding physical phenomena. ...
8
votes
1answer
1k views

Basic Grassmann/Berezin Integral Question

Is there a reason why $\int\! d\theta~\theta = 1$ for a Grassmann integral? Books give arguments for $\int\! d\theta = 0$ which I can follow, but not for the former one.
8
votes
4answers
5k views

Integrating radial free fall in Newtonian gravity [duplicate]

I thought this would be a simple question, but I'm having trouble figuring it out. Not a homework assignment btw. I am a physics student and am just genuinely interested in physics problems involving ...
18
votes
3answers
5k views

Why is the functional integral of a functional derivative zero?

I'm reading Quantum Field Theory and Critical Phenomena, 4th ed., by Zinn-Justin and on page 154 I came across the statement that the functional integral of a functional derivative is zero, i.e. $$\...
29
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3answers
3k views

When is Lebesgue integration useful over Riemann integration in physics?

Riemann integration is fine for physics in general because the functions dealt with tend to be differentiable and well behaved. Despite this, it's possible that Lebesque integration can be more ...
3
votes
4answers
409 views

Transformation of $d^4x$ under translation disregarded?

Under a translation in spacetime i.e., $$x\mapsto x^\prime=x+a,\tag{a}$$ a scalar field $\phi(x)$ $$\phi(x)\mapsto\phi^\prime(x)=\phi(x-a).\tag{b}$$ My aim is to verify the invariance of an action of ...
19
votes
2answers
6k views

Gaussian integral with imaginary coefficients and Wick rotation

Although this question is going to seem completely trivial to anyone with any exposure to path integrals, I'm looking to answer this precisely and haven't been able to find any materials after looking ...
2
votes
1answer
427 views

Recovering QM from QFT

Reading through David Tong lecture notes on QFT. On pages 43-44, he recovers QM from QFT. See below link: QFT notes by Tong First the momentum and position operators are defined in terms of "...
3
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3answers
913 views

S. Weinberg, “The Quantum theory of fields: Foundations” (1995), Eq. (9.2.15)

In Weinberg's book The Quantum Theory of Fields, Volume 1 on p.388 (Chapter 9), the following identity is used (with $f$ being any "reasonable" function): $$f(+\infty) + f(-\infty) = \lim_{\epsilon \...
4
votes
2answers
119 views

Do integrals of position make any sense? Do they have an application? [closed]

I know that taking the derivative of position with respect to time defines what we call velocity, but I've never heard of physicist going in the opposite direction with position. Is there any ...
5
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5answers
30k views

Wrong calculation of work done on a spring, how is it wrong?

So I would have thought that this would be how you derive the work on a spring: basically the same way you do with gravity and other contexts, use $$W=\vec{F}\cdot \vec{x}.$$ If you displace a spring ...
41
votes
3answers
24k views

How is the Saddle point approximation used in physics?

I am trying to understand the saddle point approximation and apply it to a problem I have but the treatments I have seen online are all very mathematical and are not giving me a good qualitative ...
10
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3answers
3k views

Three integrals in Peskin's Textbook

Peskin's QFT textbook 1.page 14 $$\int_0 ^\infty \mathrm{d}p\ p \sin px \ e^{-it\sqrt{p^2 +m^2}}$$ when $x^2\gg t^2$, how do I apply the method of stationary phase to get the book's answer. 2....
7
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3answers
1k views

What is the meaning of the double complex integral notation used in physics?

In Altland and Simons' condensed matter book, complex Gaussian integrals are introduced. Defining $z = x + i y$ and $\bar{z} = x - i y$, the complex integral over $z$ is $$\int d(\bar{z}, z) = \int_{-\...
11
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3answers
1k views

Why is the $dx$ right next to the integral sign in QFT literature?

I've noticed that in QFT literature, integrals are usually written as $\int \!dx ~f(x)$ instead of $\int f(x) dx$. Why?
4
votes
1answer
2k views

Newton's original proof of gravitation for non-point-mass objects

Suppose we have two bodies, one very large (Earth), and one very small (a cannon ball). If the cannon ball is some distance away from the Earth, to find out the force produced on the cannot ball, we ...
11
votes
4answers
2k views

A four-dimensional integral in Peskin & Schroeder

The following identity is used in Peskin & Schroeder's book Eq.(19.43), page 660: $$\int\frac{d^4k}{(2\pi)^4}\,\frac{1}{(k^2)^2}e^{ik\cdot\epsilon}=\frac{i}{(4\pi)^2}\log\frac{1}{\epsilon^2},\...
9
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4answers
14k views

How do you do an integral involving the derivative of a delta function?

I got an integral in solving Schrodinger equation with delta function potential. It looks like $$\int \frac{y(x)}{x} \frac{\mathrm{d}\delta(x-x_0)}{\mathrm{d}x}$$ I'm trying to solve this by ...
7
votes
2answers
1k views

Exponential decay of Feynman propagator outside the lightcone

In Chapter three (I.3) of A. Zee's Quantum Field Theory in a Nutshell, the author derives the Feynman propagator for a scalar field: $$ \begin{aligned} D(x)&=\int \frac{\operatorname{d}^4 \mathbf{...
6
votes
1answer
2k views

Switching from sum to integral

I'm specifically asking about an equation in An Introduction to Quantum Field Theory, by Peskin and Schroeder. Example from page 374: $$\mathrm{Tr} \log (\partial^2+m^2) = \sum_k \log(-k^2+m^2)$$ ...
5
votes
1answer
285 views

What it means to integrate over $n$ variables out of $N$, where $N>n$?

I was reading Theory of Simple Liquids, when I came across BBGKY hierarchy. In deriving the expression for the hierarchy, they integrate an integration of N variables over N-n variables to make the ...
2
votes
1answer
613 views

Physical intuition on the integral contained in D'Alembert's Formula for the wave equation

If $\phi(t,x)$ is a solution to the one dimensional wave equation and if the initial conditions $\phi(0,x)$ and $\phi_t(0,x)$ are given, then D'Alembert's Formula gives $$\phi(t,x)= \frac 12[ \phi(...
1
vote
3answers
45k views

How to calculate the moment of inertia of a solid cube? [closed]

How do I calculate the moment of inertia of a uniform solid cube about an axis passing through its center of mass? I also wanted to know if the moment of inertia ...
-1
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5answers
792 views

Question about the use of integration in physics

I've always thought of integration as a way to solve differential questions. I'd solve physics problems involving calculus by finding the change in the function $df(x) $when I increment the ...
3
votes
3answers
3k views

Electrostatic energy integral for point charges

The electric energy stored in a system of two point charges $Q_1$ and $Q_2$ is simply $$W = \frac{1}{4\pi\epsilon_0}\frac{Q_1Q_2}{a}$$ where $a$ is the distance between them. However, the total ...
2
votes
1answer
738 views

Trajectory of a photon around a Schwarzschild black hole?

Consider a photon coming from the infinity in a unbounded orbit to a Schwarzschild black hole (Schwarzschild radius $r_{s}$) (see this for illustration). Its impact parameter is $b$ and its distance ...
0
votes
1answer
720 views

Period of a pendulum [closed]

In the book 'Calculus the Early Transcendetals' at page 776 (7th edition) they give that the period of a pendulum with length $\text{L}$ that makes a maximum angle $\theta_0$ with the vertical is: $$\...
8
votes
2answers
1k views

Gaussian integral of a function with nonzero mean (generalizing Wick theorem)

From the wikipedia article, for a Gaussian integral of an analytic function we have that This is equivalent to the Wick theorem when f(x) is a polynomial. Now I'm trying to obtain a similar formula ...
1
vote
1answer
622 views

The computation of the propagator in two dimensions

I did the computation of the propagator in two dimensions at (19.26) in Peskin & Shroeder as follows. First I performed a Wick rotation. \begin{alignat}{2} \int\frac{d^2 k}{(2\pi)^2}e^{-ik\cdot (...
1
vote
1answer
142 views

Supersymmetric Localization (Mirror Symmetry)

I'm reading Chapter 9 of Mirror Symmetry book. As you can see in eq. (9.30) his model for SUSY is $$\begin{align} \delta_\epsilon X &=\epsilon^1\psi_1 + \epsilon^2\psi_2\\ \delta\psi_1 &= \...
9
votes
3answers
465 views

The invariant measure on an energy surface of a Hamiltonian system

Consider a Hamiltonian system with a time-independent Hamiltonian $H (p, q )$. By the Liouville theorem, the measure $d^np d^nq $ is conserved. However, one should also notice that the energy is ...
8
votes
5answers
490 views

Moment of inertia: why $\mathrm{dI}=r^2\mathrm{dm}$ instead of $\mathrm{dI}=m\mathrm{dr^2}$?

When computing the moment of inertia, I observed that people usually use the following logic: $$d I=r^2 dm,\\ \therefore I=\int r^2 dm$$ My question here is, why not use $dI=m ~d(r^2)$? I ...
3
votes
1answer
872 views

Fourier transform of the free propagator squared - $\int d^{4}p\ \frac{e^{-i p\cdot x}}{p^{2}+m^{2}-i\epsilon}$

The point of the question is to ask what is the function given by the following integral: $$ H(x,y) \ \equiv \ \int \frac{d^{4}p}{(2\pi)^{4}} \frac{e^{-i p \cdot (x-y)}}{(p^{2}+m^{2}-i\epsilon)^{2}} $$...
2
votes
1answer
123 views

Why does the integral symbol disappear when applying a functional derivative?

it is known that variation is defined by following: but could anyone tell me why the integral symbol disappears after following functional derivative?
1
vote
4answers
2k views

Area under and slope of the motion graphs

I wanted to ask in general what area under the graph means. Also which physical quantity is highlighted by area under distance vs time graph. I'm confused that area is a 2 dimensional concept and it ...
0
votes
2answers
343 views

Calculating the electric field of an infinite flat 2D sheet of charge [closed]

I was trying to calculate the electric field of an infinite flat sheet of charge. I considered the sheet to be the plane $z=0$ and the position where the electric field is calculated to be $(0,0,z_0)$,...
14
votes
4answers
3k views

Possible ambiguity in using the Dirac Delta function

When doing integration over several variables with a constraint on the variables, one may (at least in some physics books) insert a $\delta\text{-function}$ term in the integral to account for this ...
9
votes
1answer
989 views

A confusion from Weinberg's QFT text (a vanishing term in Lippmann-Schwinger equation)

I was reviewing the first few chapters of Weinberg Vol I and found a hole in my understanding in page 112, where he tried to show in the asymptotic past $t=−∞$, the in states coincide with a free ...
8
votes
1answer
601 views

Physical intuition/interpretation of fractional derivatives/integrals?

Oftentimes, when the derivative and integral operations are introduced within the realm of physics, we are taught some physical interpretation of them: Velocity is the derivative of position ...
4
votes
5answers
13k views

How area under Velocity-Time graph represents magnitude of displacement?

I understand that if the velocity is constant (acceleration=$0$) throughout the course of motion (where graph shows a rectangle) then it would simply be like playing with equation: (1). velocity=...
8
votes
2answers
5k views

Gravitational potential energy of any spherical distribution

The general formula to get the potential energy of any spherical distribution is this : \begin{equation}\tag{1} U = - \int_0^R \frac{GM(r)}{r} \, \rho(r) \, 4 \pi r^2 \, dr, \end{equation} where $M(r)$...
7
votes
1answer
1k views

Number of unique 2-electron integrals

Consider 2-electron integrals over real basis functions of the form $$(\mu\nu|\lambda\sigma) = \int d\vec{r}_{1}d\vec{r}_{2} \phi_{\mu}(\vec{r}_{1}) \phi_{\nu}(\vec{r}_{1}) r_{12}^{-1} \phi_{\lambda}(\...
6
votes
2answers
600 views

Derivation of $f(R)$ field equations, problem with integration by parts

I am following the derivation of the field equations on the the Wikipedia page for $f(R)$ gravity. But I do not understand the following step: $$ \delta S = \int \frac{1}{2\kappa} \sqrt{-g} \left(\...
4
votes
3answers
190 views

Why doesn't $\vec{E} =\frac{1}{4\pi\epsilon_0} \int\frac{\rho \hat{r}\;dxdydz}{r^2}$ blow up at $r=0$, when $\rho$ is finite?

Electric field at $(x,y,z)$ produced by a continuous distribution of charges is given by:$$\mathbf{E}(x,y,z) =\dfrac{1}{4\pi\epsilon_0} \int\dfrac{\rho(x',y',z') \mathbf{\hat{r}} \;\mathrm{d}x'\mathrm{...
2
votes
0answers
879 views

Sokhotski–Plemelj theorem [closed]

In quantum field theory we always deal with complex integrals. They arise, e.g. when we calculate Feynman diagrams in either $T=0$, $T \neq 0$ and non-equilibrium formalism (or just Fourier transforms ...
5
votes
1answer
447 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,\...