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0
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1answer
45 views

Where does this relativistic relation involving the delta function come from?

\begin{equation} \int\delta(E^2-\mathbf{p}^2-m^2)dE=\frac{1}{2E_\mathbf{p}} \end{equation} Shouldn't integrating the delta function like this just give 1?
-1
votes
0answers
45 views

Electric field outside a sphere

The sphere $K\subseteq R^3$ with radius $R$ has a homogeneous charge density $\rho$. Find the electric field E, produced by K outside of, meaning, find the integral ...
-1
votes
2answers
90 views

What does this equation mean? [on hold]

So I have just entered 11th grade and started limits on my own but my Physics textbook has an equation which I don't understand, I suspect it uses integration which I haven't learned yet. So can ...
0
votes
2answers
50 views

Derivation of $v=u+at$ [on hold]

I read the derivation of $v=u+at$ using integration. The steps are as follows - $$\frac{dv}{dt}=a dt$$ $$dv=adt$$ $$\int dv=\int a dt$$ $$\int dv=a\int dt$$ $$v=at+c$$ My questions are as follows - ...
0
votes
0answers
19 views

Integration for quantum amplitude of the coupling between two molecules

I am trying to solve following expression for quantum amplitude of coupling between two molecules, (arriving from the second order perubation) $$\frac{1}{p}\nabla_{j}\int e^{ipR\cos(\theta)} dT=i\int ...
5
votes
2answers
98 views

Elementary question about endpoint singularities

In George Sterman's book "An Introduction to Quantum Field Theory", on pages 413-414, there is a description of the endpoint singularity. One begins with the function $$ I(w) ~=~ ...
0
votes
1answer
49 views

Details of the radial Fourier transform pertaining to certain quantum integrals

Consider the integral $$U(t)=\int\frac{d^3p}{(2\pi)^3}e^{-ip^2t/2m}e^{i\vec p\cdot\Delta\vec x}$$ for the free non-relativistic propagator. I'm not quite sure about the gritty details of radial ...
-1
votes
0answers
29 views

Calculating the center of mass of a hemisphere (Solved) [duplicate]

Solved: $\theta$ goes from 0 to $\frac{\pi}{2}$ and not to $\pi$ EDIT: it was pointed out to me that this question was a duplicate of this post. In my opinion, the question asked on the other post, ...
0
votes
0answers
15 views

Integrating Charged Bodies

Why is it possible to integrate charged bodies by first taking a small charge and adding more small charges around it? Wouldn't the similarly charged particles exert an immense amount of force on ...
1
vote
2answers
96 views

Computing distance traveled from jerk

When dealing with higher time derivatives like jerk, how does one find the distance traveled? Can it be calculated by just knowing time?
2
votes
1answer
56 views

Ehrenfest's Theorem “contradiction”?

Ehrenfest tells us that for $\hat{p}$ $$\partial_t \langle p \rangle = \langle -\partial_x V \rangle$$ I also understand the basic steps in deriving this result directly by taking the time ...
0
votes
0answers
38 views

How to deal with this integral equation? [migrated]

While reading a paper I saw the following integral equation. $$\frac{1}{g} = \int_{-\pi}^{\pi} (\prod_{\sigma}\frac{\mathrm{d}p_{\sigma}}{2\pi}) \frac{\delta(\Sigma_{\sigma} ...
0
votes
1answer
43 views

Simplifying a Vector Integral

This question has (long) remained unanswered on MSE. While reading the book - Theory and Applications of Boltzmann Transport Equation by Cercignani, I found this integral which I am unable to ...
-4
votes
1answer
69 views

Computational physics using mathematica [closed]

So I was confused about this question on how to exactly begin to answer it. I am a novice in mathematica and I am teaching myself thus I require help in this question. From what I think I should do, ...
1
vote
2answers
29 views

Limits of Integration Trig, Mag Field Infinite Length Wire

I don't understand how the limits of integration should be defined when doing basic integrals of trig functions. It seems like it's an arbitrary decision, I don't understand it. Here's the set up: ...
0
votes
1answer
40 views

Confusion regarding area from graph

This might be a trivial question but is illustrated below. Why is the area 'below' the graph always taken for a velocity-time graph when finding the displacement? I mean why is the area with the time ...
1
vote
2answers
52 views

Why am I getting that work it's always the same in both directions?

I'm studying electrostatic and I'm getting pretty frustrated because with the definition of work I'm getting that it's always positive and it doesn't make any sense. So here I have 2 positive ...
0
votes
1answer
114 views

How to derive (the dimensionless coefficient in front of) the moment of inertia for common shapes?

Is there a way to derive (the dimensionless coefficient in front of) the moment of inertia for common shapes? I assume it has to do with the density of the shape, but I'm having trouble seeing it. ...
0
votes
1answer
47 views

What's my $dM$? Gravitational Potential inside a circle of mass

I'm trying to find the gravitational potential for an arbitrary point within a ring of uniform mass density. The point is constrained to be in the same plane as the ring. So we start with: ...
0
votes
1answer
41 views

How to find an equation for $x$ in terms of $t$ for a particle falling under gravity with resistance given by $mkv^2$? [closed]

Okay so I have determine the velocity $v$ and displacement $x$ as functions of $t$ for a particle falling under gravity with resistance given by $mkv^2$. I have set up the equation of motion divided ...
5
votes
1answer
213 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 ...
0
votes
0answers
39 views

Fourier Transforming a $n$-dimensional ket (QM)

I would like to evaluate the Fourier Transform of $n$ functions. I am aware from the derivation of the convolution how this is done for the case of $n=2$. How could this be generalised for $n=3$? ...
0
votes
1answer
34 views

Buckling of a slender column - total energy

I'm following Goldbart's Mathematics for Physics book, and I ran into a problem with exercise 1.4 (page 43). We have a formula for the energy stored in a slightly bent rod aligned on the $z$ axis: $ ...
0
votes
1answer
35 views

Integral limits when calculating the work

If I integrate $$dW= \vec{ F} \cdot d\vec{\ell}$$ which are the limits? In $$\int\limits_{W_{inf}}^{W_{sup}}dW= \int\limits_{\vec{\ell}_{1}}^{\vec{\ell}_{2}} \vec{ F} \cdot d\vec{\ell}$$ it is ...
1
vote
0answers
55 views

Shifting the integration variable in loop integrals

We know that, in four dimensions, shifting the integration variables is valid only for convergent and logarithmically divergent integrals. If we employ a hard cutoff $\Lambda$, is it permissible to ...
0
votes
2answers
59 views

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

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 ...
1
vote
0answers
47 views

Equivalence of integrals in Classical Electrodynamics

I have a technical question about a section from Jackson's Classical Electrodynamics 3rd ed. In chapter 14, Jackson derives an expression for $ \frac{d^2I}{d\omega d\Omega} $, the frequency spectrum ...
0
votes
1answer
30 views

Volume of highdimensional Sphere vs volume of spheres shell

When calculating the phase space volume $\Omega$ in the microcanoncial ensemble with fixed energy $E$, one integrates over a shell that includes all energies in between $E$ and $E+\delta E$: ...
0
votes
1answer
80 views

Struggling with an integral [closed]

I'm struggling with the following integral: $$ \int \int (r_1^2 + r_2^2) \exp \left( -\frac{b (r_1 + r_2)}{a} \right) \, \mathrm{d}V_1 \, \mathrm{d}V_2 $$ I tried to expand near $r_1 = 0 ;\; r_2 = ...
3
votes
1answer
55 views

$v^2 = 2ax$ or $v^2 = ax$?

As far as I am aware, $v^2 = 2ax$ is the formula to find the velocity in various questions. If kinetic energy = work, $$\frac{1}{2}mv^2=Fx$$ $$mv^2=2max$$ $$v^2=2ax$$ We use this formula to solve ...
1
vote
2answers
89 views

Calculation of co-moving coordinate separation for a moving object in a time-varying spacetime metric

My calculus has 30+ years of rust on it and I am stuck on the integration of the interval in General Relativity... I wish to calculate the spatial coordinate at time t of an object moving with ...
1
vote
1answer
63 views

Where do limits of integration come from in the equation of heat transfer by conduction?

I was watching the third lecture of Diffrential equations on OCW. As an application, the model of heat transfer by conduction is provided. We derived this equation which models the system where $T$ is ...
1
vote
0answers
97 views

Klein-Gordon field commutator integral identity [closed]

Consider a Klein-Gordon field $\phi$ on points $x,y$ of $\mathbb R^4$ Minkowski-spacetime. Here I'm writing $x=(x^0, \stackrel \rightarrow x)$ so that $\stackrel \rightarrow x$ gives the spatial ...
1
vote
1answer
41 views

Derivation of Fermi level for T>0

I am working through the derivation of the Fermi level $ \mu_0$ for T>0. However, at one point in the notes I have, it states without any explanation that: $$ \int_0^\infty F'(\epsilon) ...
-1
votes
3answers
124 views

What is physical interpretation gives integration?

It is my understanding that the integration is the inverse process of differentiation and its meaning is a fine sum (in fact, so is its symbol) but what physical interpretation do we get from this? At ...
2
votes
1answer
118 views

From acceleration to displacement

Hi I am a major in Computer science and this question should be really easy for all the physics geniuses here: I have a set of data points from an accelerometer on a moving object that basically ...
1
vote
0answers
53 views

Partial Integration of outer product of del and position vector

I am trying to understand the solution I have been given to prove the following relation for a current density $\vec{j}(\vec{r})$ that is concentrated around the origin: $$ \int_V dV \, ...
0
votes
0answers
33 views

How can one approximate integral def. of Z by the max value of the integrand?

I am taking a course in statistical physics, and while reviewing my notes from the lectures I came across something that I cannot get my head around. We arrive at an integral expression for the ...
1
vote
1answer
84 views

Problem evaluating a holomorphic path integral [closed]

Equation (4.11) on page 146 of Sidney Coleman's book Aspects of Symmetry is the holomorphic path integral, \begin{equation} I=\int \exp(-z^{*}Az)\Pi dzdz^{*}=\frac{1}{\det(A)}, \end{equation} where ...
2
votes
0answers
95 views

Klein-Gordon propagator integral in the light-like case

In Kerson Huang's Quantum Field Theory From Operators to Path Integrals (Amazon link), pages 28 and 29, he calculates the propagator in the following case: time-like, space-like and light-like. First ...
3
votes
1answer
109 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 ...
1
vote
0answers
36 views

Normalization constant of the Vacuum polarization

In the article "On gauge invariance and vacuum polarization" by Schwinger, at some point the equation $$\frac{C}{s^2}\int e^{i\frac{x^2}{4s}} \, dx =1$$ is said to have the solution ...
1
vote
2answers
274 views

Triple integral $\iiint_{\mathbb{R}^3} d^{3}q ~\delta^{3}(\vec{q})\frac{(\vec{p}\cdot\vec{q})^2}{q^{2}} $ involving Dirac Delta function

I am trying find $$\iiint_{\mathbb{R}^3} d^{3}q ~\delta^{3}(\vec{q})\frac{(\vec{p}\cdot\vec{q})^2}{q^{2}},$$ where $\vec{p}$ is some fixed vector. The answer should be $\frac{p^2}{3}$. Below is ...
2
votes
1answer
58 views

Integration of $e^{-it\sqrt{\mathbf{p}^2 + m^2}}$ for QM amplitude

My question might be more about maths than physics, but it originated in a Physics context. Take $\hbar$ = $c$ = 1. I was looking at the amplitude for a free particle to propagate from an initial ...
1
vote
0answers
36 views

Why we don't integrate intital velocity in body cast equation?

On this site I've found a formula for calculating the $x, y$ coordinates for a body throwed by an angle to a horizon. It looks like this: $$x(t) = V_0 t \cos(\alpha); $$ $$y(t) = V_0 t ...
0
votes
1answer
54 views

Do logarithms appear inside the divergent UV integrals? If so why? [closed]

Do logarithms appear inside the UV divergent integrals of $q\cdot f\cdot t$? I mean expressions of the form of $ \int_{V}d^{r}f(p)log(p^{2}+m^{2}) $ In this case, can we approximate it by $ log(p)= ...
0
votes
2answers
73 views

Relation between electric field and dipole moment

I want to show the following equality $$\int_{\left|\vec{r}\right|<R}d^3r\vec{E}\left(\vec{r}\right)=-\frac{\vec{p}}{3\epsilon_0}$$ where $\vec{p}$ is the dipole moment of a charge distribution ...
0
votes
1answer
549 views

Electric Field of a circular arc at a point

Given that the circular arc wire with radius 'r' has a linear charge density λ. What is the Electric field at the origin? I took a small segment dy, which is 'θ' above the x-axis with charge ...
0
votes
1answer
55 views

In statistical mechanics, what does integrating with respect to the position of a molecule mean?

So, this is probably a dumb question, but I cannot visualize or make sense of integrating over the position of a molecule in space. Okay, so an example in my thermodynamics textbook: we have N = 5 ...
0
votes
0answers
77 views

Magnitude of Electric field vector on the axis of a conducting half sphere

I need to calculate the electric field of a conducting half-sphere shell ( just the outer shell, it has no base ) of a radius $a$ and surface charge density $\sigma$ on the arbitrary point on axis ...