Questions tagged [integration]
For questions about problems related to physics that involve evaluating integrals. Purely mathematical questions should be asked at math.SE.
753
questions
0
votes
0answers
35 views
Ridiculously difficult Newtonian physics problem [closed]
I have been trying out with this one for a long time with no luck getting the exact expression. I am pretty sure this is somehow related to Kepler's third law but unable to establish the connection ...
3
votes
2answers
85 views
Is there a name for the integral of 4-force with respect to spacetime distance?
I understand that the integral of force with respect to distance is known as Energy. As I understand it even in special relativity the force is the same as the derivative of momentum with respect to ...
3
votes
1answer
83 views
Analytical continuation in QFT
My question is quite basic and generic. It is known that scaleless integrals that appear in QFT such as
$\int \frac{d^dk}{(k^2)^2} = \frac{1}{\epsilon_{\mathrm{UV}}} -
\frac{1}{\epsilon_{\mathrm{IR}}} ...
0
votes
1answer
21 views
Clarification on the notation of a paper about hybrid
Here is a screenshot from this paper by J. P. Foster and F. Weinhold. This paper focuses on a model of hybridization. It therefore considers movement of electrons in three dimensions.
The author does ...
0
votes
0answers
10 views
Numerical integration methods: Explicit vs Semi-Implicit vs Newton-Euler 1, 2 and use in cyclone physics engine [migrated]
I am trying to understand the difference between explicit Euler and semi-implicit Euler integration, where in explicit Euler the current position is calculated as
...
0
votes
2answers
39 views
For the kinematics of a particle with constant acceleration, what does this assumption mean exactly? [closed]
I don't understand what the assumption is referring to how could it be possible that the velocity isn't equal to the the initial velocity at time zero, isn't that the definition of initial velocity?
-1
votes
0answers
29 views
Center of mass of hemisphere
While calculating the center of mass for hemisphere,
In solid hemisphere, we can calculate without taking the trignometric limit,
Elements can be taken in terms of $y$ (vertical axis) and I got the ...
1
vote
1answer
55 views
Dirac delta, Fourier transform & exponentials
Consider the following equation/identity:
$$
\int d^3x e^{i(\vec{p}+\vec{q})\cdot\vec{x}}=(2\pi)^3\delta^{(3)}(\vec{p}+\vec{q}).
$$
I am trying to calculate some commuters I'm encountering in my ...
11
votes
4answers
1k views
Integrating acceleration - wrong choice of bounds in textbooks?
I've noticed in my physics textbook (and in a lot of other popular sources), that the process of integrating non-constant acceleration to get to a velocity formula, the integrating bounds imposed on ...
0
votes
0answers
24 views
Modelling the velocity of a bouncing ball [closed]
I have to write a paper under the topic: mathematically modelling the velocity of a bouncing ball. I have attached an image which contains my handwritten working as of now. So far, I have attempted to ...
1
vote
1answer
64 views
Integral divergent in Peskin and Schroeder eq. (7.90)
I'm working on the Eq. (7.90) in Peskin (page 252).
However, I don't understand why it diverges logarithmically. Does $\Gamma(0)$ mean logarithmically divergence?
0
votes
0answers
46 views
Regularization of an integral
I really need to properly evaluate the following integral.
\begin{equation}\label{1}
\frac{1}{2}\int dx \, dy \,\frac{\rho(x)\rho(y)}{(x-y)^2}
\end{equation}
Where $\rho$ is some density function ...
0
votes
0answers
24 views
Vector field surface integral in spherical coordinates [migrated]
I am trying to show the divergence theorem holds for
$$\textbf{v}=r^2cos\theta \hat{r}+r^2cos\phi\hat{\theta}-r^2cos\theta sin\phi \hat{\phi}$$
over a spherical volume centred at the origin. So I ...
2
votes
0answers
51 views
What does it mean to integrate with respect to matrices?
In Random matrix theory, the following definition of a partition function for an ensemble is common.
$$Z=\int dM \exp [-N Tr(M^2)]$$
where $M$ is a Random matrix of dimension $N \times N$.
In general,...
1
vote
0answers
30 views
Normalization constant of the collision wave function [closed]
The problem
The quantum system under consideration is the 1-dimensional system with the step potential
\begin{equation}
\mathcal{V} =
\begin{cases}
0 \quad & x<0\\
U_0 \quad & 0&...
3
votes
1answer
72 views
Is this box integral divergent or finite when “pinched” at one point?
Let us define the following conformal integral:
$$X_{1234} = \int \frac{d^4 x_5}{(2\pi)^8} \frac{1}{x_{15}^2 x_{25}^2 x_{35}^2 x_{45}^2}\tag{1}$$
This is the box integral in position space, and it ...
0
votes
1answer
43 views
Confusion about volume component in Gauss's Law for a cylinder
I am currently working on a problem in which we use Gauss's Law to find the electric field within an infinitely long cylindrical shell (inner radius r, outer radius R) of charge density $\rho = \rho_0 ...
-1
votes
2answers
150 views
Evaluating the Coulomb's law integral $\int \frac{\vec{r}-\vec{r}^{\prime}}{\left|\vec{r}-\vec{r}'\right|^{3}} d \tau^{\prime}$ [closed]
I am trying to evaluate the integral $$\int \frac{\vec{r}-\vec{r}^{\prime}}{\left|\vec{r}-\vec{r}'\right|^{3}} d \tau^{\prime}$$
over the spherical volume $r'<R$ with $\vec{r}$ inside the ...
0
votes
1answer
54 views
What does the area under an acceleration-displacement curve represent?
Considering the equation where,
$$
\frac {1}{2} \left (v^2_f - v^2_i \right) = \int_0^s ads\,
$$
What does the left-hand side of the equation actually represent? Is there an intuitive explanation ...
1
vote
1answer
54 views
Lebesgue Integral in physics [duplicate]
I study physics and in this year I have to formule and write my bachelor thesis. I have a lot of ideas but some of them looks more interesting for me.
A few days ago I thought about situations in ...
2
votes
3answers
139 views
Struggle in understanding the definition of voltage
I ask some help in understanding better the concept of voltage. The voltage is a difference in electric potential between two points $a$ and $b$. It is defined as
$$V_{ab}=-\int^a_b\mathbf{E}\cdot d\...
1
vote
0answers
53 views
Deriving potential from central force [closed]
I've read in a book that for a central force of the form $$ f(r)= \frac{{-ke^ {-r/a}}}{r^2} $$ the adequate potential is $$ V(r)= \frac{{-ke^ {-r/a}}}{r} $$
I'm trying to understand why $$ -\frac{\...
-1
votes
1answer
92 views
Integration and average in physics? [closed]
Many applications of physics theory involve computations of integrals. Examples are voltage, force due to liquid pressure, surfaces...
In some cases, when there is linear dependence between two ...
-3
votes
1answer
73 views
Bessel function of first kind [closed]
Can someone tell me how
$$\frac{1}{T}\int_0^T e^{i(m-n)\omega t} e^{-ix\sin(\omega t)} e^{iy\sin(\omega t +\phi)}\, dt = J_{m-n}\left(\sqrt{x^2 +y^2 -2xy\cos(\phi)}\right)?$$
0
votes
1answer
70 views
$\phi^4$-theory: Feynman diagrams loop integral calculation [closed]
I am studying quantum field theory by myself, could anyone help me with this integral? How can I get this result? Be more specific?
0
votes
1answer
48 views
Equations of motion for an object with non-constant acceleration related to its velocity [duplicate]
If I have an object flying through space with an initial velocity $v_0$ and undergoing constant acceleration $a$, then I can easily compute its velocity or displacement at any point in time $t$ using ...
0
votes
2answers
163 views
QED integral is zero in dimensional regularization [closed]
Why is this integral zero in dimensional regularization?
$$
\int\frac{d^Dk}{(2\pi)^D}\frac{1}{(k^2)^n}
$$
0
votes
1answer
23 views
Converting density=mass/volume to relative rate equation for integration
In order to get the mass of an object from density, we might use
\begin{equation}
m = \int\rho(x)dx \tag{1}
\end{equation}
I understand why this works on a conceptual level, but I would like to be ...
2
votes
2answers
83 views
Riemann sum of completeness relation in continuous basis
Suppose I have a wave function $\psi $ we express it in a continous states as
$$\psi= \int_{-\infty}^{\infty} dxC (x)\rvert x\rangle = \int_{-\infty}^{\infty} dx\rvert x\rangle \langle x \rvert \...
0
votes
0answers
39 views
Hypersurface four-vector, or a familly of four 3-forms?
While reading my old personal notes on forms in relativity, I got confused about some aspects of the mathematical formalism (integration on tensors and p-forms). The energy-momentum flux across some ...
2
votes
1answer
46 views
Is the output of a line integral over a scalar field a vector?
In my physics book of "mathematical methods for physics", the author writes that line integral of a scalar function $\phi$ over a curve $C$ can be written as the following:
$$\int_C\phi\,\text d{\...
1
vote
0answers
38 views
Why does integrability imply compatibility?
In mechanics, we have the so called compatibility conditions, which quarantee that when a body deforms, the strains are "compatible" in such a way to no discontinuities or gaps for inside the body as ...
-1
votes
3answers
69 views
Derive gravitational potential energy for this system [closed]
This is on a study guide for my Physics 221 final. I feel like I almost got it but I am off by a sign error. Here is the question:
Here is what I got so far:
Known:
$$F_g = \frac{GMm}{r^2}$$
$$U_g =...
2
votes
1answer
142 views
Peskin QFT Contour Integral — Chapter 6
On page 178 of Peskin's QFT, they have the vector potential
$$A^\mu(x)=\int\frac{d^4k}{(2\pi)^4}e^{-ik\cdot x}\frac{-ie}{k^2}\left(\frac{p'^\mu}{k\cdot p'+i\epsilon}-\frac{p^\mu}{k\cdot p-i\epsilon}\...
-1
votes
1answer
50 views
Solving a two variable integration [closed]
I was going through the solid state book by Philip Phillips. I came across an integral similar to:
$$\int_{0}^{\beta}d\tau d\tau^{'}e^{-E_c|\tau-\tau^{'}|}$$
where $\beta E_c >> 1$.
I am not ...
2
votes
2answers
120 views
Integral form of work during an irreversible process?
Question
Why can't the work during an irreversible process be integrated? Where is my understanding amiss?
Motivation (for this question)
A lot of my physics background seems to say this is a math ...
0
votes
3answers
154 views
Why does the solenoidal term vanishes in a barotropic fluid?
In fluid dynamics, and in particular in atmospheric dynamics, the so-called solenoidal term is the line integral:
$$\oint \frac{\vec{\nabla p}}{\rho}\cdot d\vec l$$
where $p$ and $\rho$ are the ...
3
votes
1answer
118 views
Why is $(-\frac{e^2}{4\pi \epsilon_0}) = (-\frac{\hbar ^2}{ma})$?
Note: No, this is NOT a homework question. I am struggling to understand how two physical concepts are related and truly think this could be helpful to a broader audience. Also, I already have the ...
1
vote
1answer
53 views
How can I relate this integral to dimensional regularization?
In the paper "Scattering into the Fifth Dimension of $\mathcal{N}=4$ Super Yang-Mills", the authors give the following result for an integral:
$$\begin{align} I^{(1)}(x_{13}^2,x_{24}^2,m) =& \...
3
votes
1answer
77 views
Torque due to continuous force distribution / pressure [closed]
In my fluid mechanics course, I was exposed to some cases where I need to calculate the torque due to the pressure and all solutions manuals or online tutorials take it as a known fact that $d\tau=rdF$...
1
vote
1answer
27 views
Torque experienced by a coplanar loop of current in a uniform magnetic field
There are a lot of posts on this already, but apparent all of them just consider some special case. I am now struggling with this more general case.
Let there be a magnetic field with strength $B$. ...
2
votes
2answers
71 views
Electric fields in continuous charge distribution
My question may be very basic, but I can't think of a reasonable explanation for this.
Consider a solid charged sphere. Now, we have an electric field inside the solid sphere, but at any particular ...
3
votes
1answer
66 views
Regulating a divergent integral in QED
When we try to regulate a divergent integral, we introduce another parameter, say $\lambda$ and then compute the integral. We finally take a limit (either $\lambda \rightarrow 0, \infty $) to restore ...
1
vote
0answers
78 views
Can I use dimensional regularization with this integral?
I would like to extract the divergence of this integral in 4d Euclidean space:
$$\int d^4z \frac{1}{(x-z)^4}\tag{1}$$
This divergence is expected to cancel with other divergences, which I got using ...
1
vote
1answer
119 views
When can I set $d=4$ in dimensional regularization?
I am using dimensional regularization to extract the divergence of some complicated integral. I work in $d=2\omega$ dimensions, with $\omega\approx 2$. After I extract the divergence, I have an ...
0
votes
1answer
76 views
Work done to compress a liquid in container [closed]
Into a compressed container containing water with pressure p and volume V we want to pump additional water. What is the work done? Unlike in the ideal gas, the work cannot be simply found out using ...
0
votes
0answers
42 views
Integrating a product of Gaussian distributions
I'm stuck at this relatively easy looking integral where I have gaussian distributions
\begin{equation}
\sigma(x,y)=\frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{x^2+y^2}{4\sigma^2}}
\end{equation}
and the ...
0
votes
3answers
113 views
Confusion about this simple electrostatics line integral
Suppose we want to find the electrostatic potential $\phi_{0}$, with reference to infinity, at $r_{0}$ resulting from a positive charge $q$ located at the origin. For simplicity, let us assume we are ...
0
votes
2answers
70 views
Evaluation of contour integration [closed]
Consider the integral $\int^{\infty}_{-\infty}\frac{q\exp(iqR)}{q^2-k^2}dq$. This kind of equation appears in evaluation of Green's function in scattering theory.We use contour integration to evaluate ...
1
vote
3answers
74 views
Find velocity from acceleration equation [closed]
Suppose the acceleration of a particle is a function of $x$, where
$$a(x) = (2.2 s^{-2})x$$
(a) If the velocity is zero when x = 1.0 m, what is the speed when x = 3.4 m?
(b) How long does it take ...
