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5
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2answers
424 views

A four-dimensional integral in Peskin & Schroeder

The following identity is used in Peskin & Schroeder's book Eq.(19.43), page 660: ...
0
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2answers
2k views

Calculate the center of mass of a semicircle [closed]

How I determine the center of mass of a semicircle using the definition of center of mass? I only know solve this using the Pappus theorem. Consider that the semicircle is centered on the origin and a ...
-1
votes
3answers
217 views

What needs to be integrated to solve this problem?

An object is placed on a frictionless table with its one end attached to a cord which is connected to a pulley and the tension is maintained constant at 25 N. what is the change in kinetic energy ...
0
votes
1answer
182 views

Change of variables, Fermi Integral

This is a really basic question, but I'm kind of confused. I have this integral $$\int_{0}^{\infty}\frac{p^{2}dp}{e^{\alpha+\beta p^{2}/2m}+1}$$ where ...
1
vote
4answers
5k views

Deriving equations of motion using integration

Please refer to my school textbook pg48 (of the book, and not the pdf counter) here: http://ncertbooks.prashanthellina.com/class_11.Physics.PhysicsPartI/ch-3.pdf My doubt is in this context: (right ...
2
votes
3answers
974 views

Getting position from an accelerometer on an Android phone

I know that integrating acceleration twice will give me position (acceleration-->velocity-->position) but how can I do all this when I all I have are a set of data points (ex: 1 second = some # ...
1
vote
3answers
1k views

Integration by parts to derive relativistic kinetic energy

I have come across a weird integration during derivation of relativistic kinetic energy. Our professor states that i can get RHS out of LHS using integration by parts: $$ \int\limits_0^x \! ...
7
votes
1answer
238 views

Zeta regularization gone bad

This may sound as a mathematical question, but it should be very familiar to physicists. I am trying to perform an expansion of the function $$f(x) = \sum_{n=1}^{\infty} \frac{K_2(nx)}{n^2 x^2},$$ for ...
2
votes
2answers
179 views

Flux Over a Surface

I am teaching a multivariable calculus course and we are starting to go over surface integrals. I am a math professor with little knowledge of physics. At one point the book discusses fluid flow. ...
6
votes
1answer
339 views

Using $\frac{1}{A+i\epsilon} = PV\frac{1}{A}-i\pi\delta(A)$ in Feynman Integrals

Are the following operations O.K.? This is related to the Feynman parameter trick. $$F:= \int_0^1 \mathrm{d}x\int_0^{1-x}\mathrm{d}y \frac{1}{f(x,y)+\mathrm{i}\epsilon}.$$ Now using ...
3
votes
1answer
351 views

Crazy Dirac Deltas

I'm not expecting any rigor in the following and the answers...since we're dealing with Dirac deltas in the context of QFT. Consider the integral $$ \int d^4q\ \Theta(q_0)\Theta(p_{3,0}+q_0)\ ...
6
votes
1answer
1k views

Loop integral using Feynman's trick

I am trying to show for the one-loop integral with three propagators with different internal masses $m_1$, $m_2$, $m_3$, and all off-shell external momenta $p_1$, $p_2$, $p_3$ the following formula ...
8
votes
4answers
1k views

Possible ambiguity in using the Dirac Delta function

When doing integration over several variables with a constraint on the variables, one may (at least in some physics books) insert a $\delta\text{-function}$ term in the integral to account for this ...
1
vote
1answer
347 views

Shift operator (integral calculus involving Hermite polynomials) [closed]

I didn't know whether to pose this question on Physics.stackexchange or Math.stackexchange. But since this is the last step of a development involving the eigenfunctions of an Harmonic oscillator and ...
2
votes
1answer
418 views

Ashcroft Mermin Solid State Physics Eq. 2.60ff

I'm trying to follow the steps in Eq. 2.60 of said book. What I cant seem to figure out is how to change the integration variables from 'k' to 'E', as they state. The equation is $$\int ...
3
votes
2answers
406 views

Help with Greens function/Fourier transformation to solve screened Poisson equation

I am having trouble getting from one line to the next from this wiki page. I am referring to the text line Green's function in $r$ is therefore given by the inverse Fourier transform, where ...
2
votes
2answers
193 views

Are there general circuits that differentiate/integrate empirically?

Is it possible to construct simple circuits, that given a time-varying input, produce an output that represents the derivative or integral of the input with respect to time?
2
votes
1answer
323 views

Integral in Peskin and Schroeder

I'm having a bit of a slow day, and can't see how to do the following integral in Peskin and Schroeder (page 107 for anyone with the book). We've derived in the centre of mass frame the integral over ...
0
votes
0answers
122 views

Nicholas Kollerstrom article on the history of Calculus

Today, Newton´s birthday, I read an article posted in the arXiv by Nicholas Kollerstrom http://www.arxiv.org/abs/1212.2666 That basically claims that Newton did not invent Calculus. The article does ...
3
votes
2answers
304 views

Gaussian type integral with negative power of variable in integrand

How can we compute the integral $\int_{-\infty}^\infty t^n e^{-t^2/2} dt$ when $n=-1$ or $-2$? It is a problem (1.11) in Prof James Nearing's course Mathematical Tools for Physics. Can a situation ...
5
votes
1answer
484 views

Does the universe obey the holographic principle due to Stokes' theorem?

Does the universe obey the holographic principle due to Stokes' theorem? \begin{equation} \int\limits_{\partial\Omega}\omega = \int\limits_{\Omega}\mathrm{d}\omega. \end{equation} Can this ...
1
vote
2answers
198 views

Is air drag equation in term of momentum still valid?

This is the known equation of air drag: $$m{\bf a}=mg-\mathcal D=mg-b{\bf v}.$$ Considering this, is air drag equation in term of momentum still valid? $$m{\bf v}=mv_g-b{\bf r}.$$
-1
votes
1answer
186 views

What will happen when measuring unmeasurable object?

There is a set called Vitali Set which is not Lebesgue measurable. Analogously, there also exists a Vitali set $Y$ in $\mathbb R^3$ which is a subset of $[0,1]^3$ and $|Y\cap q|=1$ for all $q\in ...
5
votes
0answers
408 views

Gaussian Integrals : Functional determinant expressed as a trace

Be $A_{ij}$ a symmetric matrix. Then I can easily write $$ \int \exp\left(-\frac{1}{2}\sum_{i,j}x_i A_{ij} x_j+\sum_{i} B_i x_i\right)\; d^nx= \sqrt{(2\pi)^n}\exp\left\{-\frac{1}{2}\mathrm{Tr}\log ...
1
vote
1answer
185 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 ...
1
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1answer
302 views

What is the definition of density as a function?

(Before I start, I don't know which tag is suitable for this post. Please retag my post if it bothers you.) Let's say there is a string on $[0,1]$ with a mass given by $m(x)$. ($m(x)$ means the mass ...
1
vote
1answer
677 views

Derivation of the self gravitational potential energy of a sphere

I have been searching on the Internet but have not found a derivation of the formula for the self gravitational potential energy of a sphere. Can someone show how to do this? I assume it involved 6 ...
1
vote
1answer
232 views

Potential for charge distribution, finiteness

Consider a potential for charge distribution: $$v_H(\mathbf{r}) ~=~ \int \frac{\rho(\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|}d\mathbf{r'}$$ where $\rho(\mathbf{r'})$ is the charge density. This ...
1
vote
1answer
161 views

Integration of constant: $\int dp = \Delta p$ in impulse formula

In University Physics, it has something like: $$\int \sum F dt = \int \frac{dp}{dt} dt = \int dp = \underbrace{p_2 - p_1}_{\Delta p?}$$ But I thought $\int dp = p$? Though my maths is really rusty ...
3
votes
3answers
517 views

How to compute the expectation value $\langle x^2 \rangle$ in quantum mechanics?

$$\langle x^2 \rangle = \int_{-\infty}^\infty x^2 |\psi(x)|^2 \text d x$$ What is the meaning of $|\psi(x)|^2$? Does that just mean one has to multiply the wave function with itself?
8
votes
1answer
592 views

A confusion from Weinberg's QFT text (a vanishing term in Lippmann-Schwinger equation)

I was reviewing the first few chapters of Weinberg Vol I and found a hole in my understanding in page 112, where he tried to show in the asymptotic past $t=−∞$, the in states coincide with a free ...
2
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0answers
106 views

What is the correct way of integrating in astronomy simulations? [closed]

I'm creating a simple astronomy simulator that should use Newtonian physics to simulate movement of plants in a system (or any objects, for that matter). All the bodies are circles in an Euclidean ...
5
votes
3answers
471 views

Why we use $L_2$ Space In QM?

I asked this question for many people/professors without getting a sufficient answer, why in QM Lebesgue spaces of second degree are assumed to be the one that corresponds to the Hilbert vector space ...
1
vote
1answer
236 views

dimensional analysis of Grassmann integration/differentiation

There is another paradox that I need to resolve: The Berezin integration rules for Grassmann odd variables give the same result as differentiation: If $f=x+\theta\psi$ is a superfunction, the ...
1
vote
1answer
454 views

Questions regarding solving the Brachistochrone problem using Lagrangian

brachistochrone problem: Suppose that there is a rollercoaster. There is point 1 ($0,0$) and point 2 ($x_2, y_2)$. Point 1 is at the higher place when compared to the point 2, so the rollercoaster ...
3
votes
0answers
269 views

Integrals given by Landau [closed]

Discussion about Landau's "Theoretical Minimum" has already been posted here. Unfortunately I couldn't find much about some examples of questions he gave to students. There are three questions in the ...
16
votes
3answers
1k views

When is Lebesgue integration useful over Riemann integration in physics?

Riemann integration is fine for physics in general because the functions dealt with tend to be differentiable and well behaved. Despite this, it's possible that Lebesque integration can be more ...
2
votes
2answers
9k views

Determining the center of mass of a cone

I'm having some trouble with a simple classical mechanics problem, where I need to calculate the center of mass of a cone whose base radius is $a$ and height $h$..! I know the required equation. But, ...
6
votes
2answers
472 views

What the circled integral?

What the circled integral $$ \oint $$ means? I saw this symbol in a lot of books about advanced physics. How is his definition? What kind of integral it is? It is used only in physics or also in ...
6
votes
4answers
3k views

How do you do an integral involving the derivative of a delta function?

I got an integral in solving Schrodinger equation with delta function potential. It looks like $$\int \frac{y(x)}{x} \frac{\mathrm{d}\delta(x-x_0)}{\mathrm{d}x}$$ I'm trying to solve this by ...
4
votes
1answer
475 views

Number of unique 2-electron integrals

Consider 2-electron integrals over real basis functions of the form $$(\mu\nu|\lambda\sigma) = \int d\vec{r}_{1}d\vec{r}_{2} \phi_{\mu}(\vec{r}_{1}) \phi_{\nu}(\vec{r}_{1}) r_{12}^{-1} ...
1
vote
1answer
6k views

Calculation of Distance from measured Acceleration vs Time

I have an Accelerometer connected to a device that feeds the instant values of the acceleration in the 3 directions. I've tried to calculate the distance for a vertical movement using these values ...
0
votes
1answer
87 views

Vehicle acceleration

What I'm essentially doing is Kalman Filter. If anyone is familiar with (but it doesn't really matter in this case). Consider the following formulas: $$x_k=x_{k-1}+v_{k-1}dt+a_{k-1}\frac{dt^2}{2}$$ ...
2
votes
1answer
411 views

Meaning of $d\Omega$ in basic scattering theory?

In basic scattering theory, $d\Omega$ is supposed to be an element of solid angle in the direction $\Omega$. Therefore, I assume that $\Omega$ is an angle, but what is this angle measured with respect ...
3
votes
1answer
280 views

Symplectic integrators of the pendulum equation?

In particular, a symplectic integrator to solve: $$\ddot{\theta} + \dfrac{g}{l} \sin(\theta) = 0.$$ I'm currently using velocity verlet - by realizing that $$\ddot{\theta} = -\nabla (-cos(\theta)) ...
4
votes
2answers
441 views

About an electrostatics integral and a delta-function kernel

I'm having trouble with an integral and I would like some pointers on how to "take" it: $$ \int \limits_{-\infty}^{\infty}\frac{3\gamma a^{2}d^{3}\mathbf r}{4 \pi \left( r^{2} + ...
4
votes
1answer
462 views

Integration of partition-function over many momentum variables

My integral looks like $$Z(\beta) = \frac{1}{h^3}\int d^3p\ \exp{\left(-\frac{\beta}{2m}\sum^{3N}_{i=1}p_i^2\right)}.$$ I'm confused about how to integrate over seemingly 3N variables in only a ...
9
votes
3answers
1k views

Principal value integral

I am reading A. Zee, QFT in a nutshell, and in appendix 1 he has: Meanwhile the principal value integral is defined by: $$\int dx\,{\cal P}{1\over x}f(x)~=~ \lim_{\epsilon \rightarrow 0} \int ...
1
vote
0answers
126 views

How to integrate twice of this viscous term?

I am reading a paper, and I do not understand why the author said the following term when integrated twice will become, $\int\limits_\Omega {{\rm{d}}\Omega {{\bf{\psi }}^{\bf{u}}}\cdot\nabla ...
0
votes
2answers
234 views

How was transformed an integral below?

I know how transform an integral below, $$ \iint f(\mathbf v_{1})f(\mathbf v_{2})d^3\mathbf v_{1}d^3\mathbf v_{2}, $$ using relative speed coordinates: we just use $$ m_{1} \mathbf v_{1} + ...