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1
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0answers
37 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
56 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
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2answers
81 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
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1answer
668 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
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1answer
58 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 ...
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0answers
90 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
911 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 ...
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2answers
138 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
425 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
369 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
285 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
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0answers
47 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
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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
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2answers
148 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
114 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
578 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.
7
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4answers
410 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
90 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
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1answer
631 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
79 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
84 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
199 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
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2answers
123 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
147 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
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1answer
127 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
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3answers
108 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 ...
5
votes
2answers
305 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
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0answers
46 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
64 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
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0answers
75 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
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2answers
199 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
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1answer
171 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
63 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
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0answers
63 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
87 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
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4answers
3k 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
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3answers
515 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
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0answers
30 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
180 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
644 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
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0answers
42 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
88 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
39 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
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2answers
124 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
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2answers
120 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
73 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
319 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
1answer
308 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
72 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
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0answers
198 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 $(-+++)$ ...