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2
votes
1answer
80 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
134 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 ...
0
votes
2answers
80 views

Electric Field and Calculus: What is the physical significance of infinitesimal $dA$ in the equation of Gauss's Theorem?

In many equations we see infinitesimals $dA$, $dS$, $dx$ and so on. What is is the physical significance of these? Someone told me it signifies a small entity. For example,in case of $dA$ it signifies ...
2
votes
1answer
108 views

Integral over a product of two Green's functions

Need some help here on a frequently encountered integral in Green's function formalism. Forgive me since I am a junior student. I have an integral/summation as a product of a retarded and advanced ...
1
vote
1answer
112 views

Difficulty with the usage of Cauchy's integral formula in Griffiths QM book

On page 410 of Griffiths QM 2nd Ed. book, he begins an analysis to evaluate the integral: $$\frac{1}{2i}\int_{-\infty}^\infty \frac{s \sin{(sr)}}{(s-k)(s+k)}\mathrm{d}s.$$ To exploit Cauchy's formula, ...
1
vote
3answers
107 views

Integral ambiguity

I'm a bit confused with some notation I encounter in physics calculus. Consider this: Taken from here. Integration operates on functions, correct? What does it mean to integrate $\frac{d{\bf ...
4
votes
2answers
256 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 ...
0
votes
0answers
45 views

simplification of Green's intergral for diffusion

\begin{eqnarray} I_1 = \int^{\infty}_{0} \frac{1}{\tau}d\tau \int^{+\infty}_{-\infty} \exp\left[-\frac{p}{2\tau}(x-x')^2 + (z-z' + \tau)^2\right]dx' \end{eqnarray} where, $p$, $x$ $z$ and $\tau$ are ...
2
votes
1answer
58 views

Mellin-Barnes (MB) integrals and hypergeometic functions

I'm trying to understand a step in arXiv:1104.2661. Equation 3.4 reads, \begin{equation} ...
2
votes
0answers
67 views

Finding the moments of the Boltzmann/Gibbs Distribution

I am trying to compute the moments of the Boltzmann distribution using a moment generating function, by taking the Fourier transform of the distribution and then taking derivatives to find the ...
0
votes
2answers
142 views

Problem deriving displacement from accelerations

I have a problem deriving displacement from an accelerometer; I want a time series of displacement so I used numerical integration twice; I based my code on the trapezium rule and so did something ...
0
votes
1answer
157 views

help with absolute pressure to gauge pressure derivation steps

I would like some help with the explicit math steps to go from equation 2 to 3. These equations are presented in a paper that I am reading. I will show where these equations came from and my attempt ...
2
votes
1answer
57 views

Torque on wire summarized with magnetic moment

The magnetic moment of a current-carrying wire loop $L$ is $$ \boldsymbol\mu = \frac I2\oint_L\mathbf{r} \times \mathrm{d}\mathbf{r} $$ so the torque it experiences under a uniform magnetic field ...
0
votes
0answers
61 views

Compute energy from displacement via time integration (using ODE)

How to compute energy from displacement given by solving an ODE via, e.g., ODE45. Say one has: $$ \ddot{z}+\zeta \dot{z}+z=\kappa\cos\left(\Omega t\right),\quad z=z\left(t\right) $$ and wishes to ...
2
votes
1answer
77 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} ...
4
votes
4answers
2k 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 ...
1
vote
3answers
453 views

Calculate vertical distance using acceleration

I have the following scenario: an iPhone is connected to a vertical rail (it's motion is restricted to the Y axis). Through mechanical means the phone is forced upwards and then allowed to return to ...
0
votes
0answers
25 views

Verlet timestep

I am using velocity Verlet in molecular dynamics. Is just a simple question, if a simulation using time-step femtosecond, in velocity Verlet is just necessary $dt = 10^{-15}$ to use femtosecond?
3
votes
1answer
165 views

Hamiltonian with Dirac Delta function

I've to compute this expression $$ \hat{H} ~=~\frac{1}{4}g_2\int d^3R\int d^3r\ \bar{\Psi}(\vec{R}+\frac{\vec{r}}{2})\bar{\Psi}(\vec{R}-\frac{\vec{r}}{2}) $$$$ \times \left[ ...
1
vote
1answer
474 views

Inverse Fourier transform of Yukawa potential (troubles with Mathematica)

It can be proved that the potential $\frac{e^{-u|r|}}{|r|}$ has Fourier transform $\frac{4\pi}{u^2+q^2}$. Now, I'm trying to go backwards and do the inverse Fourier transform but I'm running into ...
0
votes
0answers
40 views

Geometric Interpretation of these equations of motion?

I was reading my Engineering Mechanics book, and it derived some strange looking integrals I'll have to apply. I could memorize them, but I'd rather understand them - then I won't have to memorize. ...
0
votes
1answer
73 views

Change of variables in calculating the integral of multivariable differential entropy

I have already asked this question in math.SX but here might be more proper. So I decided to put a copy here and delete the one which is not the one that got an answer: I know that for one ...
1
vote
1answer
37 views

integration for upper ocean mixed layer equation

I am trying to work through the following paper specifically trying to get from equation (1) to equation (6). Equation 1 states that $$ Q = \beta S e^{-\beta z} + 2B \delta(z) $$ where $\delta z$ ...
1
vote
2answers
114 views

Planetary motion: integration of equation of motion

I was reading Planetary Motion (page 117) in Barry Spain's Tensor calculus, and stupidly enough, I didn't understand this. The equations are : $$\frac{d^2\psi}{d\sigma^2} + ...
0
votes
2answers
106 views

Gamma functions integral identity

In Srednicki's QFT book, eq. $14.27$ is a result used over and over again for computing loop correction. It is the following integral evaluated in terms of gamma functions: $$ \int d^dq ...
1
vote
3answers
62 views

Moment of inertia of a cylinder [closed]

When I tried to calculate the moment of inertia ($I_C$) of a cylinder (mass M, height H, radius R) around the rotating axis going symmetrically through its middle, I came up with a different result ...
4
votes
1answer
211 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 ...
0
votes
0answers
113 views

Langreth rules and Keldysh formalism

I am trying to confirm the proof of Langreth's theorem / rules as seen in http://www.iue.tuwien.ac.at/phd/pourfath/node52.html . My problem is equation 3.55. I would do it like this: $\int_{C_{1}} ...
0
votes
1answer
231 views

Calculating stiffness of a beam of non-constant cross section

I am considering a paraboloid shape which is fixed at its base and is being compressed downwards. I am trying to find its stiffness so I can calculate how much it deforms. When I attempt to do this I ...
1
vote
1answer
69 views

Calculating electrodynamic momentum of a dumbbell (consisting of two point charges) in longitudinal motion

I'm working through a paper on momentum in electrodynamics that requires the integration below and would greatly appreciate any help. I'm pretty sure it evaluates to $2/d$ but I can't quite figure ...
3
votes
0answers
169 views

Propagator in massless scalar field theory

Suppose we have the following Lagrangian: $\mathcal{L} = \frac{1}{2} \phi \Box \phi + V(\phi)$, where $\Box = \partial _ {\mu} \partial ^ {\mu}$ and $V$ is the interaction term. We use the $(-+++)$ ...
0
votes
0answers
90 views

Difference between normal mode methods and wavenumber integration?

Normal mode and wavenumber integration methods allow the evaluation of an integral transform solution. The text book that I am reading states that the normal mode method evaluates the field as a sum ...
0
votes
0answers
28 views

Order of Monte Carlo integration and frequency summation

I am currently trying to calculate an integration formula of a linear response function by Monte Carlo method. It is a multiple integration over three 3D vectors, i.e., nine dimensions in all. And ...
2
votes
2answers
83 views

How do you integrate an expression over a variable in the limit of an integral?

I am trying to follow the steps to solve the integro-differential equation that arises from a plasma sheath problem given in this paper. This is the step I can't follow: ...
3
votes
2answers
118 views

How do you know which way to choose the limits of an integral?

I am reading http://www.feynmanlectures.caltech.edu/I_13.html#Ch13-S4 In the beginning of equation 13.18, in which Mr. Feynman calculates the potential energy of an object outside a spherical shell, ...
1
vote
0answers
53 views

Changing Coordinate Systems in Two-Loop Integrals

Suppose we have the two-loop integral $\int \mathrm{d} ^ 4 k _ {2} \int \mathrm{d} ^ 4 k _ {1} \, f(k _ {1}, k _ {2})$, where $k _ {1}$ and $k _ {2}$ are four-dimensional vectors in Euclidean space. ...
3
votes
0answers
60 views

Coordinate Systems in Loop Integrals

Let us consider a two-point two-loop integral $\int \mathrm{d} ^ 4 k _ {2} \int \mathrm{d} ^ 4 k _ {1} \, f(k _ {1}, k _ {2}, p)$, where $k _ {1}$, $k _ {2}$ and $p$ are four-dimensional vectors in ...
0
votes
0answers
55 views

Jacobian in Loop Integrals

Let us consider a two-loop integral $\int \mathrm{d} ^ 4 k _ {2} \int \mathrm{d} ^ 4 k _ {1} \, f (k _ {1}, k _ {2})$, where $k _ {1}$ and $k _ {2}$ are four-dimensional vectors in Euclidean space. We ...
2
votes
1answer
212 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 ...
4
votes
4answers
298 views

Evaluation of expectation values

I will denote operators with hats. Suppose we got an operator of the form $i[\hat p, \tan^{-1}(e^{\hat x})]$ and we want to calculate the amplitude for a transition from a state $|p_i\rangle$ to the ...
0
votes
1answer
71 views

How to integrate two-electron interaction in He? (variational method)

A probe wavefunction in the variational method is $$\psi(r_1, r_2) =\frac{\alpha^6}{\pi^2}e^{-\alpha(r_1+r_2)}$$. In $\left<\psi \right|H\left|\psi\right>$ with $$H = \frac{p_1^2+p_2^2}{2m} - ...
3
votes
1answer
78 views

What does $\int_C V \,d\mathbf{l}$ mean?

What does $$\int_C V \,d\mathbf{l}$$ mean? I initially thought it was simply a line integral around $C$, that is, if $\mathbf{r}: [0,1] \longrightarrow \mathbb{R}^3$ is a paremetrization of $C$, then ...
4
votes
0answers
130 views

Lebesgue integration [closed]

I know this question is probably not adequate to this SE either, but let me explain my situation: I'm civil engineering's college, so, there isn't a SE for civil engineering, and my doubts about ...
0
votes
2answers
99 views

Non-uniform circular motion, computing the angle

There is an object moving in circle, with this law: $$ \alpha = -k^2 \theta $$ With $ \alpha = \frac{ d \omega }{dt} $, and $ \omega= \frac{d \theta}{dt} $, $\theta$ = angle, $k$ positive ...
0
votes
1answer
73 views

Movement with non-constant acceleration [duplicate]

Suppose we have a material point. If it is moving from position $X_0$ with initial velocity $V_0$ and constant acceleration $A$, then from elementary physics course I remember that its movement is ...
1
vote
2answers
220 views

Position dependent speed, how to compute position

I can't solve a problem: $ A= 0.5 (ms)^{-1}$, $ x_0 = 0.5 m $, $v(t)= A \cdot x^2 $, I have to compute the position at $t=3$ ($x_0$ is the initial position). So my guess is that I should be able ...
7
votes
4answers
660 views

How are electric flux calculations not double integrals?

A disk of radius 0.10 m is oriented with its normal unit vector $\hat{n}$ at 30$^{\circ}$ to a uniform electric field $\vec{E}$ of magnitude 2000 N/C. What is the electric flux through the disk? ...
1
vote
1answer
92 views

Moment of inertia of a sphere

I'm looking at sample calculations of moment of inertia of a sphere here. In the first example (disc method), it has the integral as $dI = \frac{1}{2}r^2 \,dm$, while in the second example (shell ...
0
votes
0answers
65 views

Self-inductance of a toroidal inductor

I am trying to determine the self-inductance of a toroidal coil of mean radius $R$ with $N$ loops of radius $a$ with a current $I$ flowing within them. I have calculated the magnetic field by noting ...
5
votes
3answers
386 views

Integral form of Gauss's law for magnetism from Stokes' theorem?

How can the integral form of Gauss's law for magnetism be described as a version of general Stokes' theorem? How does it follow?