Questions tagged [integration]
For questions about problems related to physics that involve evaluating integrals. Purely mathematical questions should be asked at math.SE.
853
questions
0
votes
2answers
45 views
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}...
-1
votes
1answer
35 views
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 ...
0
votes
2answers
46 views
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) :
$$\int^{b}_{-b}dy\int^{b}_{-b}dy' \space exp\{A(y^{2}-y'^{2})\} \space \delta(y-y')$$
...
1
vote
0answers
63 views
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 ...
0
votes
0answers
42 views
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. ...
0
votes
1answer
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 ...
0
votes
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:
$$
H = \pi^* \pi + \nabla \phi \cdot \nabla \phi^* + m^2 \phi^* \phi ...
0
votes
1answer
32 views
How to apply Leibniz's Rule to a Metal Pipe Temperature's Partial Derivative in this Example
Refer to this image showing the temperature of a metal pipe at the inlet and the outlet:
The temperature $T(z,t)$ is a function of the length $z$ and time $t$. Let the average temperature be
$$T_\...
0
votes
1answer
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 ...
1
vote
0answers
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}...
0
votes
0answers
46 views
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}^{\...
0
votes
0answers
16 views
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 ...
2
votes
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 ...
1
vote
0answers
67 views
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 ...
-2
votes
0answers
24 views
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{...
0
votes
1answer
63 views
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, ...
0
votes
2answers
74 views
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 ...
2
votes
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 ...
1
vote
0answers
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 ...
0
votes
1answer
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 $\...
1
vote
2answers
46 views
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 ...
0
votes
0answers
89 views
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
$$C_{2}=D\left(t_{2}-t_{1}\right) \propto \int \frac{d w^{\prime}}{2 \pi} \frac{e^{-i w^{\...
0
votes
2answers
31 views
Average value of alternating current for a long period
I came across the following in my physics text-book while reading alternating currents:
Average value of current is given by:
$$I_{av}=\frac{\int_{t_1}^{t_2}{I(t)dt}}{t_2-t_1}$$
Over a long period of ...
1
vote
0answers
43 views
Question about the VEGAS-algorithm for numerical integration
Disclaimer: I am not quite sure if this question belongs to Physics SE, if not feel free to move it.
Question:
I am currently using the VEGAS-algorithm (See e.g. here and here) and i am trying to ...
0
votes
0answers
20 views
Question About Radiative Transfer Equation Bounds of Integration
I'm unsure of what bounds to use for the integral involved in the formal solution to the radiative transfer equation. In some sources, the integral shown below goes from a general $s_0$ to $s_1$. In ...
-3
votes
2answers
44 views
If we divide the second equation of motion by time $t$, why don't we get the first equation of motion where has $1/2$ come from? [duplicate]
The first equation of motion is $v = u + at$.
The second equation of motion is $s = ut + \frac{at^2}{2}$.
If we divide the second equation of motion by time $t$, why don't we get the first equation of ...
0
votes
1answer
72 views
What is this integration?
This is from the book introduction to mechanics by Kleppner D.
Is it some typo, or something i don’t understand?
Does F dt comes down in front of integration?
0
votes
0answers
31 views
Direction of Integration of Electrostatic force over a circular arc, direction of the resultant force changing with interchange in limits?
Problem:
A uniformly charged circular arc with linear charge density $m$ subtends angle $\theta$, find the net force acting on a charge placed at its center if total charge of the arc is $Q$ and the ...
1
vote
0answers
68 views
How to make sense of this integral?
On page 10 of this paper on quantum field theory. There is the integral:
$$\ln(N(T)) \propto \iiint\limits_{-\infty}^{\infty} \ln (\sinh(T \omega_k)) d^3k$$
where $\omega_k$ is the energy $\sqrt{|k|^2+...
0
votes
5answers
997 views
Why the integral of a force gives Energy? [duplicate]
In the 10th grade, meaning few months back, I've studied the potential and Kinetic energy (I was ignorant about the importance of calculus), but When I learned calculus, and the Constant of ...
0
votes
1answer
27 views
Where does the minus sign go when deriving the Stefan-Boltzmann Constant?
When deriving the stefan boltzmann law from planks law. You may make a substitution (http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/stefan2.html). This substitution will lead to a stray minus sign ...
-1
votes
2answers
56 views
Surface element - spherical shell [closed]
Say there is a spherical shell with radius $R$ and width $h$. For a certain purpose, I wanted to divide this object into many "rings with holes". Signifying by $\theta$ the polar angle, we ...
30
votes
5answers
4k views
Why are the number of magnetic field lines finite in a particular area?
One can draw/imagine as many unique (curved/straight) lines as he/she wants in some specified finite area (assuming that each line is unique if it doesn't overlap with another line). Then how can the ...
2
votes
3answers
176 views
Does $\int{\frac{1}{dx}}$ have any meaning in physics?
Recently I came across this problem :
There are two identical parallel plates of length $L$ and breadth $B$ on the XZ plane . One plate passes through $Y = 0$ and the other passes through $Y = d$. ...
0
votes
2answers
63 views
Path independence of a conservative force
My book Halliday et al. gives a proof of the path independence (conservative force). It is said that the net work to move a particle from a to b and then from b to a is zero. Thus the work done from a ...
2
votes
1answer
73 views
Approximation of an integral on a certain limit
In Peskin & Schroeder - Chapter 6 - the authors make the following approximation when $-q^2\rightarrow\infty$
$$\int_0^1 \!\!d\xi\, \frac{-q^2/2}{-q^2\xi(1-\xi)+m^2} \simeq \frac{1}{2}\int_0d\xi\, ...
1
vote
1answer
36 views
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 ...
0
votes
0answers
26 views
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 ...
2
votes
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, ...
1
vote
0answers
27 views
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:
$$\boldsymbol{n}=\left(\begin{array}{cc}z_{1}^{\dagger} & z_{2}^{\...
-4
votes
1answer
55 views
Using integrals in electric field in ring [closed]
When finding the electric field due to a ring. We take on the cos component as the sin component gets cancelled out. Now if we integrate the sin component of the electric field of the sin component ...
0
votes
1answer
53 views
How to integrate radial distribution function? [closed]
i have this equation in my lecture notes, where $g(r)$ is the radial distribution function, $n(r)$ is the average number of particles within r to r + dr, and professor said this g(r) can be integrated ...
0
votes
2answers
36 views
Why can we equate these two integrals related to blackbody radiation?
I was reading this Wikipedia article which describes how Planck’s Law of blackbody radiation is derived. Letting $B(v,T)$ represent the energy emitted at frequency $v$ and temperature $T$, the article ...
1
vote
0answers
41 views
Why does Nambu-Goto action need parametrization but Einstein-Hilbert action does not?
In the following book (page 19 / equation 2.7 --- part of the free preview at google-books), we have the parametrization general volume element, defined as
$$
d\mu_p=\sqrt{-\det G_{\alpha\beta}}d^{p+1}...
0
votes
1answer
49 views
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 ...
1
vote
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 ...
8
votes
2answers
1k views
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?
0
votes
0answers
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 ...
1
vote
0answers
45 views
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 ...
0
votes
0answers
16 views
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(\...
