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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43 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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27 views

How do I evaluate the integral of the square root of a quartic equation? [migrated]

I'm currently trying to evaluate the integral $$\int^1_0 \text{d}t\sqrt{(1-t^2)(1-k^2t^2)}$$ where $k\in(0,1)$. Is it the case that this can be expressed in terms of elliptic integrals? I'm ...
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19 views

How did the equation 1.12 reduced in the given image below? I can't get how the 1/2 factor came into the picture? [closed]

I can't get the solution of W= m/2 integration of d/dt v^2 dt.
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25 views

Finding the work of a non-conservative force [closed]

I need some help with the problem below Calculate the work of the force $\vec{F} = (x^2+yz)\hat{x}+(2y^3-3z)\hat{y}+(-4z+2xy^2)\hat{z}$ from point the point $A(0,0,0)$ to the point $B(1,1,1)$ through ...
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2answers
536 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} \left(\...
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29 views
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3answers
132 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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16 views

Is it possble to do this complex symbolic calculation with Matlab? [migrated]

Sorry it's bit abrupt, but recently I am caught up in some symbolic calcualtion which is tedious and almost impossible with mere human hands, so just wondering is it possible to solve the double ...
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1answer
86 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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52 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
68 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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64 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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2answers
531 views

Limits of Integration Trig, Mag Field Infinite Length Wire

I don't understand how the limits of integration should be defined when doing basic integrals of trig functions. It seems like it's an arbitrary decision, I don't understand it. Here's the set up: ...
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1answer
38 views

Literature result for generic one-loop triangle integral

I've never really asked a literature based question here before, but was wondering if anyone knows where I may find a reliable source for symbolic expressions for generic one-loop triangle diagrams ...
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1answer
44 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 ...
3
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1answer
166 views

Density of states (DOS) integral when surface is not closed

According to the density of states (DOS) formula $$\rho(\varepsilon)\propto \int_{\varepsilon=\text{const}}\frac{dS}{|\nabla_k \varepsilon_k|}.$$ Since there is an integral on the constant energy ...
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2answers
154 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
429 views

Understanding Heaviside and Dirac Delta for Quantum step function

Looking at the solution for from this site I'm a bit confused on how two quantities necessarily reduce. I'm given this wavefunction $$ \psi(x) = \begin{cases} Ax & 0<x<a/2 \\ ...
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2answers
456 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 ...
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1answer
127 views

Regularising the Green's function in 2D

The Green's function for the 2D Helmholtz equation satisfies the following equation: $$(\nabla^2+k_0^2+\mathrm{i}\eta)\,{\mathsf{G}}_{2\mathrm{D}}(\mathbf{r}-\mathbf{r}',k_o)=\delta^{(2)}(\mathbf{r}-\...
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1answer
418 views

Cutoff-Scheme Renormalization and Order of Integration in QFT

The following is the result of Fubini's Theorem, describing when you can replace a double integral with an iterated integral safely: For a set $X \times Y \subset \mathbb{R}^2$, if $\iint |f(x,y)| d(...
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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 ...
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1answer
1k views

Why isn't the Gear predictor-corrector algorithm for integration of the equations of motion symplectic?

Okumura et al., J. Chem. Phys. 2007 states that the Gear predictor-corrector integration scheme, used in particular in some molecular dynamics packages for the dynamics of rigid bodies using ...
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2answers
82 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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38 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
45 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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2answers
296 views

Hamiltonian for single-mode field in cavity

Given that the Hamiltonian for a single-mode field is $$ H = \frac{1}{2}\int dV \left[ \epsilon_0E_x^2(z,t) + \mu_0^{-1}B_y^2(z,t) \right],$$ with \begin{align} E_x(z,t) =& \sqrt{\frac{2\omega^2}{...
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2answers
618 views

Gaussian integral formula for matrix product

I am looking for a way to prove that $$ \det (M \cdot N) = \det(M)\det(N) \tag{0}$$ Where $M$ and $N$ are matrices with continuous indices, so that $\det$ is a functional determinant. A way to show ...
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1answer
350 views

Gaussian integration and dimension argument

I made a mistake recently regarding the Gaussian density, by putting the determinant of the variance to the power $\frac{d}{2}$. Would the following argumentation be valid to highlight it should be to ...
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3answers
789 views

Finding the illuminance from a triangular light source

Since most light sources in games are point-like, it's pretty difficult to approximate area light sources with point sources. As triangles are a universal form to represent 3D models (thus area light ...
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3answers
62 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
134 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
49 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
113 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
153 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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4k views

Integrals over Grassmann numbers

I want to prove an identity from Peskin&Schroeder, namely that $$\left(\prod\limits_i^{} \int d \theta^*_i d\theta_i\right) \theta_m \theta_l^* \exp(\theta_j^* B_{jk} \theta_k)=\det(B) B^{-1}_{ml}...
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946 views

Contour integral of Feynman propagator

I am now reading the David Tong's lecture note on Quantum Field theory. I have some questions about the contour used in the integral \begin{equation} \int \frac{d^{4}p}{(2\pi)^{4}} \frac{i}{(p^{0})^{...
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1answer
67 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
327 views

Commutator of canonical fields in Quantum Field Theory

Let $\phi(\vec{x},t)$ denote the canonical fields and $\pi(\vec{x},t)$ denote the canonical impulses where they're given by: \begin{equation} \phi(x)=\int\frac{d^3\vec{p}}{(2\pi)^3\sqrt{2\omega_{\...
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1answer
113 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
51 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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2answers
2k views

Check dimensions of the integral of a function

I and a colleague are arguing about the dimensions of: $$\int_0^x f(x) dx $$ in this particular case $[f(x)]=m^2/s^3$ and $[x]=m$. Does it follow that $[\int_0^x f(x) dx]=m^2/s^3$ or $[\int_0^x f(x)...
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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
2k views

Making a cut trough a center of mass, can the masses of the pieces be equal?

Let's say point $P$ is the center of mass of an irregularly shaped object. If I make a straight cut trough point $P$ and split the object in two, is it possible for the two pieces to have the same ...
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2answers
68 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 ...
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1answer
63 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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2answers
2k 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}...
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72 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 ...
6
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1answer
155 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....