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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Cauchy's Integral with quadratic exponential term [migrated]

As I was studying the Cauchy's integral formula, I tried to do the integral: \begin{equation} I = \int_{-\infty}^{\infty} \frac{1}{x - a} e^{(i A x^2 + i B x)} dx \end{equation} with $A>0, B>0$ ...
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Kinetic theory: Computing the flux density

The problem is the following: Set up: A sealed box contains a collisionless classical gas of particles, each of mass $m$. The box is coupled to a thermostat such that interactions between particles ...
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2answers
102 views

Recasting integrals from Lagrangian to Eulerian frame

Working on a research problem in the continuum mechanics of fluids. For clarity, uppercase will be used for tensors in the reference configuration, and corresponding spatial items will be in lowercase....
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17 views

Maxwell model subject to reversing constant strain rate

I've been reading about viscoelastic models and using Excel to plot some of their characteristics. I'm particularly interested in their response to a constant strain rate which reverses periodically ...
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Does changing the direction of paths change the value of the path-integral?

When we use path-integral formalism to calculate some quantities, we often specify the starting and the ending points in the phase space, say $A$ and $B$. Let say $\gamma_A^B$ denote a path that ...
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1answer
79 views

What are some areas of physics, where the concept of “natural integral” may arise?

Natural integral (as we will define it) is a distinguished antiderivative of a function that can be understood as interpolation of the sequence of consecutive derivatives to the $-1$. It has a ...
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191 views

Is the average force calculated from $F(x)$ the same as that calculated from $F(t)$?

Say a force is doing work on an object in one dimension. I could calculate the average force over the distance with $$\frac{1}{\Delta{x}}\int_{x_1}^{x_2} F(x) \text dx$$ If I also formulated force ...
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1answer
41 views

Issues with Feynman parameters

As a sanity check, I have tried to evaluate a Feynman parameter integral, and have been unable to reproduce the textbook result. I wish to verify the identity $$\frac{1}{ABC} = \int\limits_0^1\int\...
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485 views

Moment of inertia: why $\mathrm{dI}=r^2\mathrm{dm}$ instead of $\mathrm{dI}=m\mathrm{dr^2}$?

When computing the moment of inertia, I observed that people usually use the following logic: $$d I=r^2 dm,\\ \therefore I=\int r^2 dm$$ My question here is, why not use $dI=m ~d(r^2)$? I ...
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32 views

Potential inside an insulating sphere integrating from $V_0 = 0$ vs. $V_{\infty} = 0$ [closed]

I'm having trouble finding the potential inside an insulated sphere by setting the potential at the center of the sphere to 0 and then integrating from there. I know that if I set the reference ...
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2answers
46 views

Radioactive decay differential equations [closed]

I am trying to form a differential equation between two different isotopes, Uranium-238 and Thorium-234. The rate of decay of an isotope is proportional to the amount present. So that: $$ \frac{dx}{...
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335 views

Understanding the differential in integrals

After years, I still find myself having trouble really internalizing the meaning of various differentials in integrals—specifically, when they come about via reasoning regarding physical phenomena. ...
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561 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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Product over integral representations and steepest descents

Suppose we have a product of Dirac or Heaviside functions in the context of a spin model and we use an integral representation to express these in order to do some manipulations, more specifically ...
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3answers
42 views

Work done by electric force when moving like charges together

Consider two positive charges. I would like to find the work done by the electric force on charge +q as it is brought closer to charge +Q from radius a to b. I first define electric force as a ...
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1answer
52 views

Moment of inertia and centre of masses of continuous bodies [duplicate]

I'm quite accustomed with integration and all those calculus involved in finding moment of inertia as well as center of mass. But a random thought is wiggling in my mind. Why do we take $dm$ instead ...
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1answer
206 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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4answers
63 views

Electric potential - a disc

I have a small question. The electric field exerted by a disc with radius $R$ (charge density $\sigma$) on the $xy$ plane on a point $z$ on the $z$-axis is: $$\vec{E}=\frac{\sigma}{2\epsilon_0}[1-\...
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Relative distance frame of reference partition function

I am preparing for an exam on Statistical Mechanics and came across the following integral: $$\int d^2q_1d^2q_2e^{-\frac{\beta}{2k}|q_1-q_2|^2} $$ where $q_i\in\mathbb{R}^2$ and $0\leq q_{i,x}\leq L$...
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What is wrong with this calculation of work done by an agent bringing a unit mass from infinity into a gravitational field? [duplicate]

Let us assume that a gravitational field is created by a mass $M$. An agent is bringing a unit mass from $\infty$ to distance $r < \infty$, both measured from mass $M$. The agent is always forcing ...
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1answer
175 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
37 views

Is the off-diagonal part of this rank-2 tensor integrand odd?

Peskin and Schroeder in Introduction to Quantum Field Theory consider the following tensor integral (Eq. 6.46): $$\int \frac{\mathrm{d}^4l}{(2\pi)^4} \frac{l^\mu l^\nu}{D^n} = \int \frac{\mathrm{d}^4 ...
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3answers
102 views

What type of integrals are these?

Gauss's law $\rightarrow$ $$\oint\vec E\cdot d\vec A=\frac {Q_{encl}}{\epsilon_0}$$ Gauss's law for magnetism$\rightarrow$ $$\oint\vec B\cdot d\vec A=0$$ Faraday's law$\rightarrow$ $$\oint\vec E\...
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28 views

$dE$ stands in my way to know the density of states in bulk crystal, how to get rid of it?

In a book about semiconductors, I found the following formula for the density of states: $$D(E)dE=\frac{(2m)^{3/2}E^{1/2}}{2\pi^2\hbar^2}dE. \tag{1}$$ In that book, the important lesson from this ...
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142 views

Electric field of given charge density

Given the charge density $$\rho(\vec{x})=\rho_0\delta(x_1)\delta(x_2),$$ where $\delta$ denotes the Dirac-distribution and $\vec{x}=(x_1,x_2,x_3)$, I am asked to calculate the electric field which is ...
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98 views

What should I do if I want to put the “zero potential value” on another point instead of one at infinite distance from the source?

When calculating electric potential using : $$\ V(r) = \frac{q}{4\pi\epsilon_0 r}$$ (r pointing any point). It's implied we are referring to this full formula : $$\ V(r) - V(\infty)=\frac{q}{4\pi\...
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1answer
478 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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90k views

What does an integral symbol with a circle mean?

I have frequently seen this symbol used in advanced books in physics: $$\oint$$ What does the circle over the integral symbol mean? What kind of integral does it denote?
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How to get the solution of the QCD renormalization group equation at higher orders

In Peskin & Schroeder Exercise 17.1, it is asked to find the solution at higher order of the renormalization group equation with $\beta$-function at $$ \beta (g) = -\frac{b_0}{(4\pi)^2} g^3 - \...
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Galactic Dynamics Binney and Tremaine

I was reading up on galactic dynamics, especially rotation curves and came across this equation in Galactic Dynamics by Binney and Tremaine(2.208 pg 111): $$ \psi(m)= \rho_{b0}\int^m_0 dm^2 \left(\...
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4answers
1k views

Non-unique zero function in the function space (Hilbert space)

I have just started studying about quantum mechanics, and I was studying the definition of the inner product for functions; I am also quite new to linear algebra. While studying I think I encountered ...
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1answer
48 views

Sign of work against an electrostatic field

I think I'm slightly confused with the signs. The work needed in order to bring point charges $q_i$ from infinity to a distance $r$ from some shape of charge $Q(r)$ with a varying charge (because we ...
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1answer
71 views

Why integration? [closed]

This is the expression of an energy stored in an inductor, i know it came from integrating inductance (as a constant) and current with respect to time, but my question is why it was integrated? What ...
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91 views

Saddle point approximation and finite action configurations forming a set of zero measure

In Coleman's "Aspects of Symmetry", chapter 7, section 3.2, he makes a claim that configurations of finite action form a set of zero measure and are therefore unimportant. Further, he goes on to prove ...
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802 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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90 views

Integral in direct calculation of anomalies

I am trying to follow Weinberg's triangle diagram calculations in section 22.3 of volume II of The Quantum Theory of Fields. He reduces the calculation to evaluating the integral \begin{equation} \...
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3answers
37 views

Failing to understand use of one-dimensional upper limit in area variable

I am trying to learn capacitance of a parallel plate capacitor, while learning i noticed the portion where they are calculating potential using line integral. Now, there is a term "ds" which is the ...
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1answer
61 views

Confusion about angular integration in spherical polar coordinates

I'm having to perform an integral of the following form, $$\int\frac{d^3\mathbf{p}}{(2\pi)^3}f(|\mathbf{p}|)\mathbf{\hat{p}}\cdot\mathbf{A}\exp\left(i\mathbf{p}\cdot\mathbf{B}\right)$$ Here, $\mathbf{...
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1answer
47 views

Using Integration To Find Impulse from Force-Time Graph [closed]

Good day all, This question is asking me to find impulse, J, for each cart by integrating force. I don't know how to pull the information from this force-time graph to set up the integral. Just to ...
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1answer
64 views

Scalar potential vector

I am trying to find the scalar potential, $\phi(\vec r)$, of a conservative vector field $\vec a(\vec r)$. I am integrating along a straight line from $\vec r_0$ to $\vec r$ which is parametrised by $\...
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3answers
109 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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2answers
92 views

Why does the range of this integral work out this way?

I have a bit of trouble in finding the same limits for the integral in Eq. (17.111) from Peskin & Schroeder. We have something like $$ \int_0^1 dx' \int_0^1 dz f(x',z) \delta(x-zx').$$ Posing $y=...
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86 views

Why is integral of product of a test function and derivative of Dirac-delta function seems to diverge? [closed]

Suppose,we have to evaluate the integral $\int_{-\infty}^{\infty}f(x)\delta'(x)dx$ Traditionally to solve this,we integrate by parts so that the integral is equal to$-f'(0)$,which is finite if $0$ is ...
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81 views

Is heat $\delta Q$ an exact differential for an isochoric process (ideal gases)?

Generally speaking, heat and work are path-dependent, thus $\delta Q$ and $\delta W$ are not exact differentials. By first law of thermodynamics, we know that $dU=\delta Q - \delta W$ but $\delta W=0$...
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2answers
62 views

Indefinite integral of a density function

Suppose that $\rho(x)=\frac{dm}{dx}$ is the linear density of a rod. Can we find the mass at each point of the rod by integrating $\rho(x)$, so that:$$m(x)=\int\rho(x)dx.$$ Can we do the same with ...
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1answer
43 views

Approximating and simplifying an electric field

I calculated the electric field exerted by a square lying on the $xy$ plane with side length $a$ on a random point in space $(x,y,z)$, and got this expression: $$\vec{E}=\frac{\sigma}{4\pi\epsilon_{0}}...
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1answer
81 views

$\oint{A}=0\implies$ A is a State function?

If $A$ is a thermodynamic variable (ex:Pressure, volume, entropy). then If $\oint{A}=0$, then does it imply that $A$ has to be a state function? I'm trying to prove that Entropy is a state function. ...
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0answers
16 views

Finding rate of change of momentum in a fire extinguisher [closed]

I am trying to find the rate of change of momentum as asked by part ii. I have begun the question by saying "dp/dt= -dN/dt x momentum of a particle". To cancel out the sqrt(pi) factor in dN/dt it ...
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47 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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30 views

How to calculate the total gravitational potential energy of a vertical object (do we use integration?)

Hello I was reading another question asked by zach466920, and when he was trying to calculate the total GPE of a water 'tower', he used this explanation: He basically used integration to calculate ...

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