Questions tagged [integration]
For questions about problems related to physics that involve evaluating integrals. Purely mathematical questions should be asked at math.SE.
0
votes
1answer
18 views
Meaning of flux 2-integral
Can someone please explain the meaning of flux 2-integral in this sentence:
Mass is evaluated as a flux 2-integral at the asymptotic infinity.
For asymptotic infinity, I believe it is as ...
0
votes
2answers
48 views
Calculating power with force as a function of time
While doing an AP question in Physics C today I answered the question differently from the professor but I'm not sure what part of my reasoning is incorrect. The question gives the position of a block ...
4
votes
1answer
32 views
What are the scalar equations for velocity and displacement if acceleration obeys the inverse-square law?
In basic high school physics/calculus you learn that you can formulate equations for velocity and displacement under constant acceleration as:
$a(t) = a_0$
$v(t) = a_0t + v_0$
$x(t) = \frac{1}{2}...
1
vote
0answers
35 views
Deriving moment of inertia of a solid sphere [closed]
I have been trying to calculate it on my own, but the answer I get is different to the one I can find everywhere else, so I have to be wrong.
My attempt was a very straightforward one. I used ...
-2
votes
1answer
45 views
Integral over an area of spacetime [on hold]
Is it possible to evaluate this integral in spacetime?
$$\int_{\Sigma} \frac{dydz}{[a_{o}(y^{2}+z^{2})+2f_{o}y+2g_{o}z+c_{o}]^{2}}$$ where $$a_{o}c_{o}-f_{o}^{2}-g_{0}^{2}=\frac{1}{4}.$$
If it is ...
-1
votes
0answers
17 views
Action angle variables and Hamilton jacobi theory
Action P= Closed integral of Pdq , Why would we choose a closed integral, we know that hamiltonian flows preserve volume and area , so can we replace this closed integral with a definite integral ...
-2
votes
1answer
53 views
Integration Using Spherical Coordinates [closed]
So I had to find the moment of inertia of a hollow sphere of mass $M$, radius $R$, and negligible thickness.
$dI=R^2 \cdot dm$
where $dm = \dfrac{M}{4\pi R^2}\cdot R^2\sin(\theta)\cdot d\theta\cdot ...
3
votes
1answer
109 views
The acceleration of the particles by finite difference [closed]
I would like to approximate the acceleration of a molecular dynamics system.
I'm following an online tutorial to solve a set of equations for molecular dynamics. I can use $F=ma$ to calculate the ...
-2
votes
1answer
51 views
Given the parameters of the electrostatics problem, is this integral possible to evaluate analytically? [closed]
A cone with apex at the origin has a height $h$ and a top radius $h$, a uniform charge density with no charge on the top face.
I need to find the potential $V$ at a position $z$ on the cone's axis ...
4
votes
2answers
213 views
Sum to an integral in deriving equipartition theorem
I'm reading this derivation of the equipartition theorem for ideal gases. On the second page, it is mentioned that the partition function as a simple sum,
$${\displaystyle Z=\sum _{i}e^{-\varepsilon ...
0
votes
0answers
65 views
Integration by parts in the time derivative of a functional [migrated]
I have a question regarding some calculations in Torres del Castillo's paper "Hamiltonian structures for classical fields". Let $\phi_{(a)}$ ($a=1,2,\dots,n$) be the variables that determine the state ...
1
vote
1answer
49 views
Perpendicular weight force on an object that is tipping over [closed]
I'm currently working on a problem I can't seem to find an answer to. I have an object that is hanging over a cliff. This object is exactly 12m in length, and it starts off in equilibrium (6m over the ...
-4
votes
1answer
55 views
No clue about a term [closed]
$\int_S\int \vec{A}\cdot\hat{n}dS= \int_S\int A cos(\theta)dS= \int_S\int \left(A_xdS_x+ A_ydS_y+ A_zdS_z\right)$
I have no clue about the term $$\int_S\int \left(A_xdS_x+ A_ydS_y+ A_zdS_z\right)$$
...
2
votes
1answer
103 views
Integration of the splitting function
I have a problem performing the following integration provided in the paper by Catani and Seymour (arXiv: hep-ph/9605323) page 27. Given is the integral
$$
\mathcal{V}=\int_0^1 (z(1-z))^{-\epsilon} \...
1
vote
2answers
100 views
How to determine the units of my integral? [closed]
Given that I have some coefficient(i.e. a number) which is to be determined from a radial integral:
$$b_{n00} = \frac{(2\pi)^{1/4}}{\sigma^{3/2}} \frac{1}{\sqrt{3}} [C(000|000)]^2 \int^{\infty}_{r = ...
1
vote
0answers
33 views
Integration by parts in dimensional regularisation
I have a question concerning integration by parts identities in dimensional regularisation. Appearently, almost every textbook about dimensional regularisation claims that
$$\int d^Dl_1...d^Dl_L \...
1
vote
0answers
33 views
Explicit integration of the time-dependent Schrodinger equation when eigenvalues are unknown
Let's consider the hydrogen atom Hamiltonian
$$H = - \frac{1}{2}\Delta - \frac{1}{r}$$
The solution for the corresponding time-dependent Schrodinger equation is the following:
$$\psi = \psi (t = 0){...
4
votes
1answer
105 views
Time evolution operator in QM
I am reading a introduction to quantum mechanics right now. There is a part about the time evolution operator:
\begin{align*}
i\hbar \partial_t \,\psi(\vec r, t) = \hat H(t)\, \psi(\vec r,t)
\end{...
0
votes
1answer
73 views
Integrating rigid body equations for a game engine simulation
I'm a mechanical engineer who's trying to implement a physics engine for a 3D game simulation, so I apologize for being incorrect or simply ignorant of some aspects of computation. I'm implementing ...
0
votes
1answer
29 views
How can I calculate the total rotation a detuned pulse will apply a nuclear spin?
I'm trying to model the effect a radiofrequency pulse will have on a nuclear spin at different detunings. The pulse has a sech lineshape, a pulse area (time integral of the pulse envelope) of $\frac{π}...
0
votes
1answer
39 views
How to combine limits when integrating in the frequency domain
I want to combine the signal of two separate pulses in the frequency domain in order to calculate their overlap (by multiplying the two signals together and integrating). However, one of these signals ...
0
votes
1answer
25 views
Integrating Carnahan-Starling Pressure
Given the Carnahan-Starling equation of state for a solution of hard-spheres,
$$ Z = \frac{P}{\rho k_BT} = \frac{1 + \eta + \eta^2 - \eta^3}{(1-\eta)^3}$$
where $\rho = N/V$ is the number density and ...
1
vote
1answer
81 views
Computing integrals for divergent loop amplitudes?
I am trying to compute the cross-section for the diagram below with a divergent triangle loop:
$\qquad\qquad\qquad\qquad\qquad$
where $X^0$ and $X^-$ are some fermions with zero and negative charge ...
0
votes
1answer
23 views
How to find the net electric force exerted on a uniformly charged rod by another, same rod on the x-axis (they don't touch)? [duplicate]
How to find the net electric force exerted on rod 2 by rod 1, both being on the x-axis, both having the same length and constant linear charge density, being some distance apart?
More specifically, ...
1
vote
1answer
41 views
The average velocity of a particle
The Maxwell distribution of velocities is:
$$p (v) = (\frac{m}{2\pi K_b T})^{\frac{3}{2}} e^{\frac{-mv^2}{2 k_b T}}$$
I want to understand how to obtain the average value of the velocity. The ...
0
votes
1answer
48 views
About the quadratures method
in the Classical Mechanics (2nd. Ed.) book of Herbert Goldstein, p. 75
it says: "Equations 3-18 and 3-20 are the two remaining integrations,
and formally the problem has been reduced to quadratures..."...
1
vote
0answers
21 views
Hylleraas Multi-dimensional integral [closed]
Evaluate the hylleraas integral
$$\int \int \frac{\exp(-ar_{1}-br_{2}-cr_{12})}{r_{1}r_{2}r_{12}}d^{n}r_{1}d^{n}r_{2} $$
with $r_{1}=|\vec{r_{1}}|$, $r_{2}=|\vec{r_{2}}|$, $r_{12}=|\vec{r_{2}}-\...
0
votes
1answer
69 views
How do I interpret this summation-integral notation?
Reading a paper I came across this abomination of a notation, my question is, how do I interpret it?
For context I'll post the whole page + the notation as a separate.
The notation in mind is this ...
0
votes
1answer
35 views
Integration in order to derive formulas for self-inductance
When deriving formulas for Self Inductance, I stumbled upon the following integration:
$$\frac{V_o}{L}\int_0^tsinωtdt=\int_0^t\frac{dI}{dt}$$
The result logically is:
$$-\frac{V_o}{ωL}cosωt +...
1
vote
1answer
153 views
Wightman Function for complex scalar field - timelike separations?
For a complex scalar field $\Phi$, the field has the expansion
$$
\Phi(x^0,\mathbf{x}) = \int \frac{d^{3}\mathbf{p}}{\sqrt{ 2 E_{\mathbf{p}} (2\pi)^3 } }\ \bigg[ e^{- i E_{\mathbf{p}}x^0 + i \mathbf{p}...
0
votes
1answer
27 views
Application of Green's theorem, free electron model
I am just reading this paper about a free electron model in conjugated molecules and got a bit confused about equation (1.38).
There is an integral:
$$
\int_0^{a_B}
f_B(x_B) \left(\frac{d}{d x_B}\...
0
votes
2answers
39 views
Derivation of the formula for Electric Potential Energy
I just learnt the formula for calculating Electric Potential Energy
$W=\frac{1}{C}\int_0^Qqdq = \frac{1}{C}[\frac{1}{2}q^2]_0^Q=\frac{Q^2}{2C}$
I understand the methodology, but what I do not ...
3
votes
3answers
162 views
Peskin & Schroeder: Free particle propagation
In Peskin & Schroeder Ch. 2, p. 14, in proving that the NRQM propagation amplitude for a free particle is nonzero everywhere, they move from
\begin{equation}
U(t)~=~ \frac{1}{(2\pi)^3} \int d^3p \...
4
votes
0answers
69 views
Integration of Differential Forms
I want to understand what it actually means to integrate a differential form on a manifold. Being a mathematician, the explanation I always get is that they simply follow the right transformation rule....
0
votes
0answers
49 views
Calculation of Fresnel Integral Table
Below is how Fresnel approximate the eponymously "Fresnel Integral". In his own words:
Let $i$ and $i+t$ be the narrow limits between which it is proposed to integrate $\cos(qv^2) \, dv$. ...
1
vote
1answer
57 views
Orthonormal Basis integration and Kronecker delta
Given that this integral I'm trying to solve is
$$\frac{2}{\pi}\sum^{\infty}_{l=0}\sum^{l}_{m=-l}\int_{r=0}^{\infty}\int_{k=0}^{\infty} R_{nl}(r)b_{lm}(k)j_{l}(kr)k^2 r^2 \int_{\theta = 0}^{\pi}\int_{\...
1
vote
1answer
36 views
What is the definition of the moment of inertia tensor?
I can find volume integrals for the moment of inertia in 2D and 3D, but is there a definition that works in an arbitrary number of (spatial!) dimensions?
1
vote
3answers
129 views
Electric field of an infinite sheet of charge [closed]
I am trying to derive the formula for E due to an infinite sheet of charge with a charge density of $ \rho C/m^2$. I assumed the sheet is on $yz$-plane. I used Coulomb's law to get an equation and ...
0
votes
1answer
202 views
Derivation of Rotational Motion Equations using Calculus
How are the equations for rotational motion derived using calculus and the following general equations ?
$$\mathbf{v}(t) = \mathbf{v}_0+\int_{t_0}^t \mathbf{a}(t')dt'$$
$$\mathbf{r}(t) = \mathbf{r}...
2
votes
1answer
86 views
Integrating $1/x$ in radioactive decay derivation
I have a question concerning, for example, the derivation of the equation for radioactive decay.
You start with the following differential equation
$$-\lambda \cdot N=\frac{\mathrm dN}{\mathrm dt}$$
...
0
votes
1answer
226 views
Why is displacement equal to the area of velocity-time graph? [duplicate]
why is the distance of a body equal to the area of its speed-time graph?
the general formula of speed(v) is
v=distance(s)÷time taken(t)
so the formula of distance(s) should be s=v×t
so if the speed-...
0
votes
0answers
50 views
What's the rotational inertia of a tetrahedron at the origin?
The Problem
What is the inertia matrix, about the origin, for a tetrahedron $OABC$, where O is the origin, and A, B, and C are arbitrary points in 3D space, with a density of $1 kg/m^3$?
What I ...
0
votes
0answers
55 views
Massive versus Massless $\phi^4$ Sunset Diagram - does $\frac{1}{\epsilon^2}$ term vanish for $m=0$?
In a real scalar massive $\phi^4$-interacting theory consider the amputated sunset diagram. This is the integral out of Kleinert and Schulte-Frohlinde Critical Properties of $\phi^4$-Theories:
The ...
4
votes
2answers
302 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 ...
1
vote
0answers
37 views
Closed form of Iterated integrals arising in Fredholm's integral equation solution in the context of Nonequilibrium Quantum field Theory?
While solving a Non-equilibrium quantum field theory problem I came across this class of $2n_{}^{\text{th}}$ order iterated integral :
$$F(T_{}^{},T_{0}^{},\epsilon)=\int_{T_{0}^{}}^{T_{}^{}}dt_{1}^{}...
19
votes
4answers
2k views
Is dimensional analysis valid for integrals
Can we apply dimensional analysis for variables inside integrals? Ex: if we have integral $$\int \frac{\text{d}x}{\sqrt{a^2 - x^2}} = \frac{1}{a} \sin^{-1} \left(\frac{a}{x}\right),$$ the LHS has no ...
2
votes
0answers
52 views
Vector potential of a partially-known magnetic field
let's consider a three-dimensional space permeated by a known magnetic field $\vec{B}$. Let's consider in this space a topologically spherical surface $\mathcal{S}$ centred in the origin. I put a ...
-1
votes
1answer
146 views
Find velocity using integration method or relative velocity approach [closed]
In the diagram shown below, jeep moves with a speed of 60kmph and the car's velocity as observed from the moving jeep is 20kmph. we need to find the velocity of the car. I used relative velocity ...
-2
votes
1answer
106 views
Help with proof in Griffiths QM book [closed]
So, I'm having trouble with this proof in Griffiths' QM text. I don't get how Griffiths exactly goes from the text(circled in read) on page 47 to the next step(also circled in red). He says that he ...
0
votes
0answers
50 views
Get rid of the derivatives and relativistic mass in Feynman lectures
i have a problem with get rid of the derivatives in Feynman lectures (chapter 15, Equivalence of mass and energy). The problem:
we have $\frac {d(mc^2)}{dt} = v\cdot \frac {d(mv)}{dt}$, then we ...
