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0
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1answer
89 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
37 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
38 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
0answers
82 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
37 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
16 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
44 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
52 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
43 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
25 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
77 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
46 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
74 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
54 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 ...
2
votes
0answers
94 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
32 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
118 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
113 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
47 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
32 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
votes
0answers
12 views

Integration over a spherical surface in this case [duplicate]

I have asked this on this link http://math.stackexchange.com/q/1058307 But I will try to ask it here again. In this paper http://www.hcs.harvard.edu/~jus/0302/song.pdf Song explains the following ...
1
vote
1answer
81 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
79 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
105 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
34 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
252 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
54 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
35 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
50 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
66 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
288 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
52 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
57 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 ...
3
votes
2answers
441 views

Why and when do we differentiate or integrate equations in physics? [closed]

I'm an engineering student and none of my professors ever explained why do we use derivations and/or integrations in physics. So I have this task, it goes like: The object is moving in a positive ...
0
votes
2answers
108 views

Definition of torque for a continuous body

I am working on basic physics definitions. Given a particle at position $r$ (in some coorinate reference system) upon which acts a force $F$, the $torque$ $\tau$ is defined by \begin{equation} ...
5
votes
2answers
308 views

Electron's self-energy in QED in arbitrary gauge

Recently I've tried to evaluate electron's self-energy in QED in the second order of perturbation theory by using dimensional regularization. Corresponding 1PI-diagram leads to $$ \Sigma_{1loop} = ...
5
votes
1answer
342 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: $$\mathrm{Tr} \log (\partial^2+m^2) = \sum_k \log(-k^2+m^2)$$ ...
1
vote
2answers
177 views

Is a double integral required to find the moment of Inertia of a non-uniform sphere?

Consider some ball of given radius $R$, with a mass density function that depends on the radial variable, $\rho=\rho(r)$ where $r$ is the distance from the center of the sphere. What is the moment ...
1
vote
0answers
37 views

Change to relative coordinates [closed]

I am looking at a weakly interacting bose gas and I am stuck on this integral: $$\frac{1}{V^2} \int_V d^3r_1 \int_V d^3 r_2 W(|\vec{r_1}-\vec{r_1}|)\, \exp{\frac{i}{\hbar}r_1(\textbf{p}_1 - ...
2
votes
0answers
51 views

Set of orthotogonal complex functions [closed]

Show that the functions $e^{in\pi x/l}$, n = 0, ±1, ±2, ..., are a set of orthogonal functions on $(-l, l)$ using: $A(x)$ and $B(x)$ are orthogonal on $(a,b)$ if $$\int^b_a A^*(x)B(x)dx = 0$$ ...
0
votes
2answers
139 views

Difficulty evaluating a complex integral on Griffiths

This actually a question from Griffiths QM. (Q2.21) I have difficulty understanding integrals involving imaginary components. In this example, it looks like the first term (encircled in red) explodes ...
4
votes
1answer
107 views

How is taking the average of an integral over an interval justified?

I have been studying classical mechanics. Often when going through a worked problem, I see a step where there is an integral from 0 to 2$\pi$ of $\sin^{2} \theta \ d\theta$. Instead of using the ...
4
votes
2answers
568 views

A basic math identity often used in integrals [closed]

I'm just wondering about why $y_i=A_{ij}x_j$ implies $$d^Ny=|\det A|d^Nx.$$ I see that $\det A$ is the product of the eigenvalues of a diagonal matrix but still don't exactly see how. Please help.
6
votes
4answers
291 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 ...
2
votes
1answer
76 views

First variation of the action in relativistic notation - Landau & Lifshitz “Classical theory of fields”

In Landau & Lifshitz's book, Classical theory of fields, the action for a free particle is defined as: $$\tag{8.1} S= \int ^b _a {-mc \ \text d s}=0,$$ where $$\text d s=c\,\text d ...
1
vote
1answer
307 views

How to use accelaration data of moving object to calculate distance?

I read couple of similar question on this forum and few blogs on web, though I am still confused,I am determined to calculate object displacement using accelerometer data. So, I tried using ...
0
votes
1answer
72 views

Rotational symmetry in integration

Can someone please tell me why $$4\int d^4x \, x^\mu x^\nu ~=~\int d^4x \, g^{\mu\nu}x^2 $$ by some rotational symmetry argument?
2
votes
1answer
81 views

change of variable in a 2-loop integral

given the 2 loop integral $$ \int dq_{1} \int dq_{2}F(q1,q2) $$ (1) then in dimension D=4 our integral will be a 8-dimensional integral so why can not make a change of variable to 8-dimensional ...
0
votes
1answer
135 views

From Paris to … London [closed]

(Excuse the pun in the title, couldn't resist) Paris and London are connected by a straight underground tunnel, as shown in the diagram below. A train travels between the two cities powered only by ...