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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Gaps in derivation of thermodynamic property equations

If $h=h(T, P)$. Does $ dh = c_pdT + \left[v - T\left(\frac{\partial v}{\partial T}\right)_P \right]dP \Rightarrow h_2 - h_1 = \int_{T_1}^{T_2} c_pdT + \int_{P_1}^{P_2}\left[v - T\left(\frac{\partial v}...
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1answer
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Integration used in Derivations

I've seen many derivations in which Integration is used. But I don't understand the fact that why after going to a distance like $y$ or $x$, we take an element $dy$ or $dx$? Instead can't we take any ...
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Integral involving Dirac delta function over a finite interval

During the course of a textbook problem, I obtain the following (simplified to keep only important elements) : $$\int^{b}_{-b}dy\int^{b}_{-b}dy' \space exp\{A(y^{2}-y'^{2})\} \space \delta(y-y')$$ ...
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Use of Symmetry Arguments [migrated]

I am trying to solve the problem $$\int d^3x\,e^{i{\bf a}\cdot{\bf x}}e^{-br^2}$$ using symmetry arguments. Could someone direct me to a similar question or guide me through a similar problem so I can ...
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Kinematic formulas for constant $n$th derivative of position

I was wondering how to solve for $x(t)$ in the general case of constant $n$th derivative of $x$. This means to solve the equation $$\frac{\mathrm{d}^n x}{\mathrm{d} t^n}=q,$$ where $q$ is a constant. ...
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32 views

Why is normal force here not depended on angular frequency? [closed]

Q) A uniform cylinder of radius R is spinned about its axis to the angular velocity $\omega _0$ and then placed into a corner (figure shown above).The coefficient of friction between the corner walls ...
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1answer
35 views

Hamiltonian density from Klein-Gordon field

In the solution for Peskin & Shroeder 2.2 where the Hamiltonian density obtained from the Klein-Gordon Lagrangian is given by: $$ H = \pi^* \pi + \nabla \phi \cdot \nabla \phi^* + m^2 \phi^* \phi ...
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How to apply Leibniz's Rule to a Metal Pipe Temperature's Partial Derivative in this Example

Refer to this image showing the temperature of a metal pipe at the inlet and the outlet: The temperature $T(z,t)$ is a function of the length $z$ and time $t$. Let the average temperature be $$T_\...
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25 views

Superposition of electromagnetic waves in polarisation

Let's imagine an electromagnetic wave that points every direction (i.e., from $\theta = 0$ to $\theta = 2\pi$). For simplicity let's consider only the electric field vectors. The wave goes through a ...
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111 views

The Fourier transformation of $\cos(x)$ [migrated]

the Fourier transformation of $\cos(x)$, \begin{equation} f(k)=\int_{-\infty}^{+\infty}\cos(x)e^{ikx}dx \end{equation} $1$. on one hand we have \begin{equation} \begin{aligned} f(k)=&\int_{-\infty}...
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Integration of delta function of sum of square [migrated]

Does anyone know how to calculate $\int_{-\infty}^{\infty} \left(\prod_{j=1}^N dJ_j\right) \delta(-N + \sum_{j=1}^N(J_j)^2)$ or $\int_{-\infty}^{\infty} \frac{d\epsilon}{2\pi} \left(\int_{-\infty}^{\...
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Fetter and Walecka's derivation of perturbation theory of imperfect fermi gas

I've been learning the imperfect fermi gas, in Chapter 4 of Fetter & Walecka's book on Many-Body Physics. I have a hard time with one integral, equation (11.62) in P145. From this integral we can ...
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1answer
31 views

Closed form solution of the normal density of a superfluid for the Bogoliubov spectrum

I've been trying to solve the following definite integral $$ \int_0^\infty dx\, x^4\, \frac{e^{\sqrt{x^4+2 x^2}/Tp}}{\left(e^{\sqrt{x^4+2 x^2}/Tp}-1\right)^2}\, ,\quad Tp = \frac{T}{Un} $$ This is the ...
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Closed form solution to normal fluid density integral in the two fluid model [migrated]

I've been trying to solve the following definite integral $$ \int_0^\infty dx\, x^4\, \frac{e^{x^2+a}}{\left(e^{x^2+a}-1\right)^2}\quad , \qquad a>0\, . $$ This is the density of the normal part of ...
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Contour Integration in Mathematica [migrated]

I need to solve an integral similar to following for a project: \begin{equation} F[p^2\_,q^2\_]\equiv\int_0^{\infty}dk\, e^{-k/\lambda}\, \frac{k}{k-b[p^2,q^2]- I\epsilon}\ln(k-b[p^2,q^2]) \end{...
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Alternate way to calculate an integral

There's an infinite wire carrying current I on the origin along the z-axis. The question was to calculate $\int$B$\cdot$dl along the path PQ. I managed to get the correct answer by direct integration, ...
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How to calculate center of mass of a hollow hemi-sphere with some thickness?

When we calculate Center of mass (COM) of a hollow sphere, we assume that it's thickness is infinitesimally small, but in real world, we do not have any object with zero thickness, so how can we ...
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3answers
101 views

What is the linear momentum of a rotating body?

Let's say that a three dimensional object with continuous mass distribution is undergoing rotational motion about an axis that lies on the centre of mass. The translational velocity of the centre of ...
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90 views

What is wrong in my method of calculation of center of mass of a hollow right circular cone [closed]

In the figure above, it must be noted that $R-r$ is Infinitesimal small, but it is shown bigger in the figure. here $y_{cm}$ is $y$-coordinate of center of mass (COM), $m$ is the total mass of the ...
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72 views

Quantum Mechanics (Griffiths); Am I Missing A Subtle Argument In A Proof?

I'm working through a proof in Griffith's Quantum Mechanics book (Chapter 1.4 - Normalization) and feel like a subtle detail is being omitted. If anyone can supply clarity that would help. We have $\...
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2answers
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Integration of Torque for a Circular Current Loop in Magnetic Field [closed]

I am trying to derive the formula for Torque on a circular current loop inside a magnetic field. I know the formula is: $\tau = IAB\sin{\theta}$ Where I is the current, B is the magnetic field and A ...
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Discrepancy in two-point correlator integral of Harmonic Oscillator

The two point correlator of Quantum H.O. of natural frequency $\omega$, calculated using path integrals, is $$C_{2}=D\left(t_{2}-t_{1}\right) \propto \int \frac{d w^{\prime}}{2 \pi} \frac{e^{-i w^{\...
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Average value of alternating current for a long period

I came across the following in my physics text-book while reading alternating currents: Average value of current is given by: $$I_{av}=\frac{\int_{t_1}^{t_2}{I(t)dt}}{t_2-t_1}$$ Over a long period of ...
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Question about the VEGAS-algorithm for numerical integration

Disclaimer: I am not quite sure if this question belongs to Physics SE, if not feel free to move it. Question: I am currently using the VEGAS-algorithm (See e.g. here and here) and i am trying to ...
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Question About Radiative Transfer Equation Bounds of Integration

I'm unsure of what bounds to use for the integral involved in the formal solution to the radiative transfer equation. In some sources, the integral shown below goes from a general $s_0$ to $s_1$. In ...
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If we divide the second equation of motion by time $t$, why don't we get the first equation of motion where has $1/2$ come from? [duplicate]

The first equation of motion is $v = u + at$. The second equation of motion is $s = ut + \frac{at^2}{2}$. If we divide the second equation of motion by time $t$, why don't we get the first equation of ...
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What is this integration?

This is from the book introduction to mechanics by Kleppner D. Is it some typo, or something i don’t understand? Does F dt comes down in front of integration?
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Direction of Integration of Electrostatic force over a circular arc, direction of the resultant force changing with interchange in limits?

Problem: A uniformly charged circular arc with linear charge density $m$ subtends angle $\theta$, find the net force acting on a charge placed at its center if total charge of the arc is $Q$ and the ...
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How to make sense of this integral?

On page 10 of this paper on quantum field theory. There is the integral: $$\ln(N(T)) \propto \iiint\limits_{-\infty}^{\infty} \ln (\sinh(T \omega_k)) d^3k$$ where $\omega_k$ is the energy $\sqrt{|k|^2+...
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5answers
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Why the integral of a force gives Energy? [duplicate]

In the 10th grade, meaning few months back, I've studied the potential and Kinetic energy (I was ignorant about the importance of calculus), but When I learned calculus, and the Constant of ...
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1answer
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Where does the minus sign go when deriving the Stefan-Boltzmann Constant?

When deriving the stefan boltzmann law from planks law. You may make a substitution (http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/stefan2.html). This substitution will lead to a stray minus sign ...
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Surface element - spherical shell [closed]

Say there is a spherical shell with radius $R$ and width $h$. For a certain purpose, I wanted to divide this object into many "rings with holes". Signifying by $\theta$ the polar angle, we ...
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5answers
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Why are the number of magnetic field lines finite in a particular area?

One can draw/imagine as many unique (curved/straight) lines as he/she wants in some specified finite area (assuming that each line is unique if it doesn't overlap with another line). Then how can the ...
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3answers
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Does $\int{\frac{1}{dx}}$ have any meaning in physics?

Recently I came across this problem : There are two identical parallel plates of length $L$ and breadth $B$ on the XZ plane . One plate passes through $Y = 0$ and the other passes through $Y = d$. ...
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Path independence of a conservative force

My book Halliday et al. gives a proof of the path independence (conservative force). It is said that the net work to move a particle from a to b and then from b to a is zero. Thus the work done from a ...
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1answer
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Approximation of an integral on a certain limit

In Peskin & Schroeder - Chapter 6 - the authors make the following approximation when $-q^2\rightarrow\infty$ $$\int_0^1 \!\!d\xi\, \frac{-q^2/2}{-q^2\xi(1-\xi)+m^2} \simeq \frac{1}{2}\int_0d\xi\, ...
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What is the difference between zero and an infinitesimal number?

In a standard Atwood machine physics problem, the string going over the pulley is considered massless. So does that imply mass = 0 or mass = dm? General question: what is the difference between 0 and ...
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Torque on an arbitrary current carrying line placed in a 2d plane due to non uniform magnetic field [duplicate]

Given any arbitrary current carrying Line in a plane whose shape is defined by the function y = f(x) which is kept under a non uniform magnetic field vector B now calculate the net torque about the ...
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3answers
129 views

Integral of a dot product

If I integrate a dot product (e.g. $\vec E \cdot \mathrm d \vec s$), then the dot product itself becomes $|\vec E| |\mathrm d \vec s| \cos\theta $. But when I try to integrate the dot product itself, ...
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How to change the integration measure between vectors and complex numbers?

When using $CP(1)$ representation, we will write the unit vector $\boldsymbol{n}$ as the the spinor form, i.e. complex number: $$\boldsymbol{n}=\left(\begin{array}{cc}z_{1}^{\dagger} & z_{2}^{\...
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Using integrals in electric field in ring [closed]

When finding the electric field due to a ring. We take on the cos component as the sin component gets cancelled out. Now if we integrate the sin component of the electric field of the sin component ...
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1answer
53 views

How to integrate radial distribution function? [closed]

i have this equation in my lecture notes, where $g(r)$ is the radial distribution function, $n(r)$ is the average number of particles within r to r + dr, and professor said this g(r) can be integrated ...
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2answers
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Why can we equate these two integrals related to blackbody radiation?

I was reading this Wikipedia article which describes how Planck’s Law of blackbody radiation is derived. Letting $B(v,T)$ represent the energy emitted at frequency $v$ and temperature $T$, the article ...
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Why does Nambu-Goto action need parametrization but Einstein-Hilbert action does not?

In the following book (page 19 / equation 2.7 --- part of the free preview at google-books), we have the parametrization general volume element, defined as $$ d\mu_p=\sqrt{-\det G_{\alpha\beta}}d^{p+1}...
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1answer
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Proving 3D Hamiltonian operator is Hermitian

A Hermitian operator $H$ is defined as $$\int f^*(Hg) d^3\vec{r} =\int (Hf)^*gd^3\vec{r}$$ where $f$, $g$ are 3D square integrable functions and the integrals are taken over all coordinates. I am ...
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1answer
53 views

How to decide which way are the limits of an integral?

So, I'm trying to calculate the electric field at a point $r$ distance away on the perpendicular bisector of a finite line charge having uniform charge density $\lambda$ I arrive at the following ...
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If charges is quantised, how can we use integrals in electrostatics?

In electrostatics we say that charge is quantised. Then my question is how can we integrate them?
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45 views

Gravitational potential energy and integration

Suppose a chain hangs a bit off a table. The work done to put the hanging part on the table is $mgh$ where $m$ is mass of hanging part, $h$ is height of centre of mass from table. NO integration ...
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One-loop Feynman integral over Euclidean momenta

I am trying to perform the following one-loop computation $$ \int \frac{d^Dq}{(2\pi)^D} \frac{(k+q)^2 q^2}{((k+q)^2+m^2)(q^2+m^2)} $$ where $k$ is fixed and everything is on the Euclidean setting, so ...
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Bloch Component under Lorentz Condition

Suppose we have Gaussian state in the momentum representation $$ a(\textbf p) = (2\pi)^{-3/4}w^{3/2}exp(-\textbf{p}^2/2w^2) $$ and a state $$b(\textbf p) = K\text{sinh}(\frac{\alpha}{2}) q_z a(\...

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