0
votes
1answer
30 views

Rope question - integration

A 50-lb bucket is at the bottom of a 100-ft well. A 200 lb rope (also 100 ft long) is tied securely to the bucket. We will use rope to lift this bucket out of the wall, at a rate of 1 foot every ...
2
votes
1answer
46 views

Properties of the Double Layer Potential

Consider the double layer potential $$ W_{\nu}(x) = \int_{\partial\Omega} \nu(y) \frac{\partial}{\partial n_y}\left( \frac{1}{|x - y|} \right) d \sigma_y $$ for a bounded region $\Omega \in \mathbb ...
0
votes
0answers
28 views

Properties of functional integration

this question comes from theoretical Physics, the issue being the so called Path Integral. The measure of this thing is something written as $[d\phi]=\prod_x d\phi(x)$ And this should be the limit ...
2
votes
0answers
66 views

How to evaluate the integral $\int e^{ipx}e^{ipx} d^{3}x = 0$

I am embarrassed to ask this question. But I came across the following in a physics book: $$\int e^{ipx}e^{ipx} d^{3}x = 0$$ $d^{3}x = dydydz$, as @Semiclassical shows below. This came up in the ...
1
vote
3answers
53 views

integral of the sphere describing lambertian reflectance

A Lambertian surface reflects or emits radiation proportional to the cosine of the angle subtended between the exiting angle and the normal to that surface. The integral of surface of the hemisphere ...
1
vote
0answers
46 views

Integral calculation - Gravity - Free Fall

I have read this article http://physics.stackexchange.com/questions/3534/dont-heavier-objects-actually-fall-faster-because-they-exert-their-own-gravity. In the best answer by David Z - there are some ...
1
vote
1answer
46 views

An Integration Calculation

I'm just having a bit of difficulty understanding the last couple of steps made in the paper Horowitz & Hubeny - Quasinormal Modes of AdS Black Holes and the Approach to Thermal Equilibrium (p.8) ...
2
votes
1answer
84 views

The Gaussian Integral

Hi I am trying to calculate the expected value of $$ \mathbb{E}\big[x_i x_j...x_N\big]=\int_{-\infty}^\infty x_ix_jx_k...x_N \exp\bigg({-\sum_{i,j=1}^N\frac{1}{2}x^\top_i A_{ij}x_j}-\sum_{i=1}^Nh_i ...
3
votes
2answers
138 views

Integral $\int_0^\infty e^{imx^2}dx$

In evaluating an integral in path integrals in QFT, I am stuck with this integral (that came up from evaluating a functional integral), $$I = \bigg( \frac{m}{2\pi i\tau}\bigg) \int ...
0
votes
1answer
24 views

Deriving relative position from instanteous acceleration and time

I'm working on a mobile app that uses the accelerometer to move a cursor. Although it's technically a computer science problem, once you get past how you get the values, it's more of a math problem, ...
2
votes
2answers
140 views

Integral calculus question relating to particle motion

"A particle of mass m is attracted toward a fixed point 0 with a force inversely proportional to its instantaneous distance from 0. If the particle is released from rest, at distance L, from 0, find ...
0
votes
1answer
40 views

Integration involving multiple constants…

So I've been tackling the rather nasty integral of... $\int^R_s\frac{2r}{\sqrt{r^2-s^2}}.\frac{1}{2}(R-r)^2dr$ ...where R and s are constants. However, every method I try I seem to get stumped by ...
0
votes
0answers
94 views

Integrate the product of two exponential functions

I am trying to solve the following integral: $ \int^{\infty}_{-\infty} \exp{-3/2L [(r^{(0)}- \epsilon)^{2} + \sum_{a=1}^{n}(r^{(a)}-\Lambda \epsilon)^{2}]} d^{3}\epsilon$ I have try to use the the ...
0
votes
1answer
90 views

Center of mass in a straight rod

I got an assignment to prove that in a straight homogeneous rod, you can always choose a coordinate system in such a way that $$\int_S x_1 \, dx_1 \, dx_2=0 $$ $$\int_S x_2 \, dx_1 \, dx_2=0 $$ ...
3
votes
0answers
66 views

Two properties about Bessel function

Let $J_\nu(x)$ be the Bessel function of the first kind. $\int_0^\infty J_\nu(x)dx=1 , (Re(\nu)>-1)$. $\lim_{\nu\to+\infty}J_\nu(x)=0$ for any fixed $x$. I think the above two properties of ...
3
votes
0answers
96 views

How to perform this matrix integral?

Edit: some backgrouds added. In quiver matrix model which is reviewed DV or CKR, the path integral reduce to the matrix integral $$Z \sim \int \prod_{i=1}^r d\Phi_i \prod_{<a,b>} dQ_{ab} ...
7
votes
1answer
132 views

Positivity of the Coulomb energy in 2d

Let $$D(f,g):=\int_{\mathbb{R}^3\times\mathbb{R}^3}\frac{1}{|x-y|}\overline{f(x)}g(y)~dxdy$$ with $f,g$ real valued and sufficiently integrable be the usual Coulomb energy. Under the assumption ...
2
votes
1answer
62 views

Hankel trasformation of acoustic wave equation

We consider a simplified version of acoustic wave equation \begin{equation} \frac{\partial^2 p}{\partial r^2}+\frac{1}{r}\frac{\partial p}{\partial r}+\frac{\partial^2 p}{\partial z^2}+k^2 ...
0
votes
0answers
150 views

What is the Dirac mass on measure space?

I am reading the book "Lectures on Stochastic Analysis." But I know seldom about measure space. I meet with a symbol which the author call Dirac mass(in 9.3 of this book). Let E be a measurable space, ...
0
votes
0answers
32 views

Viscosity of a ball with known deceleration

For a metal ball going through a liquid with initial horizontal speed u, mass m and radius r and viscosity resistance constant C1, I found: ...
-1
votes
2answers
1k views

can an electric field exist at a point where the electrical potential there is zero?

can an electric field exist at a point where the electrical potential there is zero? 0 v=integral of E.dl
1
vote
1answer
73 views

Computing The Fourier Sine Series.

Compute the Fourier Sine series of the odd function: $f(x) = x^3 - 4x, -2 \leq x \leq 2 $. (Periodically extended with period 4) I know how to compute this of course where: $b_n = ...
2
votes
1answer
192 views

Evaluating an integral

Gaussian-profile initial condition has the solution, $$\phi (r,t)=\frac{R^{3}}{2}\frac{A}{\sqrt{\pi }}\int_{0}^{\infty }ke^{-R^{2}k^{2}/4}\frac{\sin (kr)}{r}\cos (\omega t)\ dk,$$ where A is an ...
3
votes
1answer
513 views

integrals with error function

Can anyone help me to compute these integrals? \begin{equation} \int_0^t\frac{1}{x}\exp\left(-\frac{a^2}{x}\right) \operatorname{erf}\left(\frac{b}{\sqrt{x}}\right)\,dx \end{equation} here ...
10
votes
3answers
329 views

Meaning of $\int\mathop{}\!\mathrm{d}^4x$

What the following formula mean? $$\int\mathop{}\!\mathrm{d}^4x$$ I know that this $\int f(x)\mathop{}\!\mathrm{d}x$ is the integral of the function $f$ over the $x$ variable, but the following ...
2
votes
0answers
119 views

Integration by parts for line integrals

I asked a question on physics stack exchange a few days ago here: http://physics.stackexchange.com/questions/67181/where-can-i-find-the-full-derivation-of-helfrichs-shape-equation-for-closed-mem and ...
1
vote
1answer
71 views

Is the following differentiating under the integral sign correct?

Suppose $$\frac{\delta f[u]}{\delta u(x)}\equiv \frac{\partial f}{\partial u}-\frac{\partial }{\partial x}\frac{\partial f}{\partial u_x}+\left(\frac{\partial }{\partial x}\right)^2\frac{\partial ...
2
votes
1answer
147 views

Trouble understanding a common vector calculus example

I have difficulty understanding the following vector calculus example. Text can be found here. It is the 5th Q&A -- starting with equation (31.1035).It concerns finding the vector potential of a ...
0
votes
1answer
325 views

Straight Line Motion w/ Acceleration and Deceleration Rates?

PROBLEM: A subway train travels 400ft between two stations. It starts from rest and accelerates at the rate of 8ft/sec^2 until it's velocity reaches 20ft/sec. It then moves at this constant velocity ...
6
votes
2answers
147 views

Arnold's Trivium problem 51

Calculate $$ f(k) = \int_{-\infty}^{+\infty} e^{ikx}\frac{1 - e^x}{1+e^x}dx.$$ As far as I know, this is not a function but rather the Fourier transform in tempered distributions. 1) What is ...
3
votes
1answer
394 views

Change of variables for a Dirac delta function

I have often seen the following equality in Physics textbooks. $$\int_{\mathbb{R}}\delta\left(\alpha x\right)f\left(\alpha x\right)|\alpha|dx=\int_{\mathbb{R}}\delta(u)f(u)du$$ or ...
1
vote
1answer
163 views

Finding the integral $\int_0^\pi\dfrac{d\theta}{(2+\cos\theta)^2}$ by complex analysis

Trying to find the integral $\int_0^\pi\dfrac{d\theta}{(2+\cos\theta)^2}$ by complex analysis, I let $z = \exp(i\theta)$, $dz = i \exp(i\theta)d\theta$, so $ d\theta=\dfrac{dz}{iz}$. I am trying ...
4
votes
1answer
129 views

How to do this integral

Prove that $$\int \frac{d^{n}q}{(2\pi)^{n} }\frac{q^{2a}}{(q^{2}+D)^{b}}=D^{-(b-a-n/2)}\frac{\Gamma (b-a-n/2)\Gamma (a+n/2)}{(4\pi )^{n/2}\Gamma (n)\Gamma (n/2)}$$ The angular part is easy to do as ...
4
votes
1answer
137 views

Existence Energy of Wave Equation

I was just going trhough some properties of the wave equation, including the energy of the wave equation given by $E(t)=\int_{-\infty}^{\infty}u_t^2+c^2u_x^2 dx$, i.e the sum of kinetic and potential ...
11
votes
4answers
775 views

Is this a dirac delta function?

I had this on an exam yesterday, and I'm not entirely convinced that the statement is true. We were asked to show that the function $\delta (x) = \int_{-∞}^{∞} \frac{1}{t(t-x)} dt$ is a dirac delta ...
0
votes
1answer
134 views

Proving that an integral is zero in order to prove Newton's shell theorem

Prove that for any $a\in(-1,1)$ $$\int_{0}^{\pi}\frac{(\cos t-a)\sin t}{\left(1+a^2-2a\cos t\right)^{3/2}}\,dt = 0.$$