Questions tagged [integration]

For questions about problems related to physics that involve evaluating integrals. Purely mathematical questions should be asked at math.SE.

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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 ...
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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
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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}}} ...
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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 ...
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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 ...
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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?
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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 ...
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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 ...
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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 ...
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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
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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?
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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 ...
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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 ...
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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,...
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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
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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 ...
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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 ...
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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 ...
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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
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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 ...
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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\...
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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{\...
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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 ...
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1answer
73 views

Bessel function of first kind [closed]

Can someone tell me how $$\frac{1}{T}\int_0^T e^{i(m-n)\omega t} e^{-ix\sin(\omega t)} e^{iy\sin(\omega t +\phi)}\, dt = J_{m-n}\left(\sqrt{x^2 +y^2 -2xy\cos(\phi)}\right)?$$
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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?
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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 ...
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2answers
163 views

QED integral is zero in dimensional regularization [closed]

Why is this integral zero in dimensional regularization? $$ \int\frac{d^Dk}{(2\pi)^D}\frac{1}{(k^2)^n} $$
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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
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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 \...
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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 ...
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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{\...
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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 ...
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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 =...
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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}\...
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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 ...
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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 ...
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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 ...
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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 ...
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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) =& \...
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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 $d\tau=rdF$...
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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 $B$. ...
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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
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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
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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 ...
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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 ...

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