Integration Definition and 1000 Threads

Grassmman variables are defined as anti-commuting, such that for two Grassmann variables ##\theta_1,\theta_2## such that ##\theta_1\theta_2=-\theta_2\theta_1##. Then for a Grassmann variable ##\theta##, the derivatives are such that $$\frac{\partial }{\partial \theta}\theta=1$$ as in usual...

I am trying to compute the following loop integral: $$ \require{cancel} \displaystyle \begin{align} I= \int \frac{d^4k}{(2\pi)^4}\bar u(p')\frac{[k^2k^\mu - (k\cdot p)\cancel{k}\gamma^\mu -(k\cdot p')\gamma^\mu\cancel{k}]}{[k^2-M^2+i\epsilon][(k-p)^2-m^2+i\epsilon][(k-p')^2-m^2+i\epsilon]}u(p)...

Example 1.2.24: Let C be a set in n-dimensional space, and let Q(C) = ∫...∫Cdx1dx2...dxn. If C= {(x1,x2,...,xn : 0 ≤ x1 ≤ x2 ≤ ... ≤ xn ≤ 1}, then Q(C) = ∫01 ∫0xn∫0xn-1...∫0x3∫0x2dx1dx2...dxn = 1/n!. (Apologies for the poor formatting of the integration bounds.) I haven't the foggiest idea how...

I came across an equation in a disconnected piece of my notes, and I remember that it was an instance of something, but I can't remember what. (I see how to do the integration, that is not what I am worried about.) Does anyone recognize either the function or the integration as an example of...

I am a highschool physics teacher from Brazil seeking ways to approach this subject within such themes, in a way that is integrated with other subjects such as history, geography, philosophy, biology, chemistry; all at once. I acknowledge that such an approach is both unrealistic and detrimental...

Can someone please explain the steps for the integrations in red circles on the attached page? This a page from Riley, Hobson, Bence - Mathematical Methods for Physics and Engineering 2006 edition. Thank you.

According to the author of this article, this integration formula is well-known but rarely taught. Do you know it? $$\int f(x)dx = xf(x)-\int_{x_{0}}^{f(x)}f^{-1}(t)dt $$ where x0 is a constant and f-1(x) is the inverse function of f(x).

I tried taking an element of length dx and tried calculating the force of gravity acting on it so that I could just integrate over the whole chain, but I couldn't figure out what to do of that displacement part since the dx element is not moving as is just at rest (hanging). So, how should I...

Integration in the order of dy then dx: $$\int_{-1}^{1} \int_{0}^{1-x^2} x^2 \sqrt{1-y} ~dy~dx$$ $$=\int_{-1}^{1} -x^2 \left[\frac{2}{3} (1-y)^{\frac{3}{2}}\right]_{0}^{1-x^2}dx$$ $$=\int_{-1}^{1}\left(-\frac{2}{3}x^5 + \frac{2}{3} x^2\right)dx$$ $$=\left. -\frac{1}{9} x^6 + \frac{2}{9}...

The problem, neater: Attempt at a solution:

Hi i have problems, to solve task a) Since I have to calculate the trace of the matrix ##Q##, I started as follows: $$\text{trace} (Q)=\sum\limits_{i=1}^{3}\int_{}^{}d^3x'(3x_i^{'2}-r^{'2}) \rho(x')$$ I then calculated further until I got the following form: $$\text{trace}...

I have an issue with (b). What I did was simply integrate ##dS##. It's a perfect gas, so, $$\left(\frac{\partial E}{\partial T}\right)_V=NC_V$$ and $$\left(\frac{\partial E}{\partial V}\right)_T=0$$ Next I used the relation that ##PV=NkT## to get ##\frac{P}{T}=\frac{Nk}{T}##, and after...

This really cracked me up! Unless there is something i am not seeing! part (a) is straightforward, using quotient rule: ##\dfrac{dy}{dx} = \dfrac{x⋅\dfrac{1}{x}- \ln x}{x^2}=\dfrac{1-\ln x}{x^2}## From here i was able to see that, ##\int \dfrac{\ln x}{x^2} dx= \int \dfrac{1}{x^2}- \dfrac{\ln...

Can someone please give as simple an example as possible to show what U substitution is about? I know basic integration rules but don't understand the point of u-substitution. I've read that it's used to "undo the chain rule", but I don't see how, and don't see how we can spot when we'd need to...

This is part of the working from f(3x^2-1)^2xdx; I don't understand from when 6x becomes 1/6

I want to ask why the answer is not zero. If n approaches infinity, it means each term will approach zero so why the answer is not zero? Thanks

I'm hoping there's a reasonable answer to this. To summarize, data I acquired when imaging a particular target shows that I can retain 75% of my images for stacking at 10s exposure times, but only 50% of the images taken with 15s exposures. The difference is entirely due to tracking error and...

In, *An Introduction to Thermal Physics, page 235*, Schroder wants to evaluate the partition function $$Z_{tot}=\sum_0^\infty (2j+1)e^{-j(j+1)\epsilon/kT}$$ in the limit that $kT\gg\epsilon$, thus he writes $$Z_{tot}\approx\int_0^\infty (2j+1)e^{-j(j+1)\epsilon/kT}\,dj$$ But how is this...

I was recently very surprised when I had a looked up relativistic kinetic energy. All sources gave the kinetic energy as the difference between total energy and rest energy, in some or other variant of the formula ##E_k=(\gamma−1)mc^2##. I didn't really understand at first. It seemed overly...

Hello everyone, A few days ago I stumbled across the formula for the energy of a moving breather for the Sine-Gordon equation $$\Box^2 \phi = -Sin(\phi) $$ The energy in general is given by (c=1) $$ E = \int_{-\infty}^{\infty} \frac {1} {2} ((\frac {\partial \phi} {\partial x})^2+ (\frac...

Integral 7.2 is ok. I must employ the integration technique used in 7.2 to prove that integral equation 7.1 equals zero. For n<0 we have : $$\sum_{n=- \infty}^{-2} a_n \oint (z-z_0)^ndz$$For n>0 we have : $$\sum_{n=0}^{\infty} a_n \oint (z-z_0)^ndz$$ According to Cauchy's Integral Theorem...

I get that the bottom answer isn't a constant - but does this physically represent anything? When I set the two answers equal to each other, I get x = +- 1/sqrt(2) and I am wondering if this represents anything significant. I don't think (mathematically) there is anything wrong with the bottom...

Ok, so doing this one direction, with the range of x (0 to 2), I get the top minus the bottom equation of: ## y = 8 - x^3 ## Then, since it's squares, this would make it ##y^2##. So, integrating gives: ## \int_{0}^{2} (8-x^3)^2 = 82.3 ## That seems to be correct. However, I want to make...

So the solution is obviously given here, I'm just trying to understand it. I thought that integrating f(x) from -5 to 5 would give the area under the curve (including the areas below the "pond" at the edges of the image but above y=0. I don't really understand why we are subtracting the integral...

I am studying particle pair production using Parker and Toms book: Quantum Field Theory in Curved Spacetime. On page 48 they talk about converting the sum over momentum (k) into an integral. You assume boundary conditions so that k = 2*Pi*n/L, where n is an integer and L is the coordinate...

Could someone guide me on what change of variable was used to obtain equation 9.138 from equation 9.137? Book : Classical Dynamics of Particles and Systems 5th Edition by Stephen T. Thornton (Author), Jerry B. Marion (Author) They told us to check equation 8.38 and in that page they had...

Just went through this...steps pretty clear. I refreshed on Riemann integrals { sum of rectangles approximate area under curves}. My question is on the highlighted part in Red. The approximation of area under curve may be smaller or larger than the actual value. Thus the inequality may be ##<##...

i tried integrating the stuff but it didn't work what to do

I am interested specifically in solving this problem by factoring the quadratic term into complex linear factors. $$s^2+4=0$$ $$\implies s=\pm 2i$$ $$\frac{5s+6}{(s-2i)(s+2i)(s-2)}=\frac{A}{s-2i}+\frac{B}{s+2i}+\frac{C}{s-2}$$ We can solve for ##C## using the cover-up method with ##s=2## to...

$$h(t)=f(t)*g(t)=\int_0^t f(\tau)g(t-\tau)d\tau=\int_0^t g(\tau)f(t-\tau)d\tau\tag{1}$$ The Laplace transform is $$H(s)=\int_0^\infty h(t)e^{-st}dt=\int_0^\infty\left ( \int_0^t g(\tau)f(t-\tau)d\tau\right )e^{-st}dt\tag{2}$$ The Laplace transforms of $f$ and $g$ are $$F(s)=\int_0^\infty...

Hello everyone, If I have an integral ##\int_0^r \sqrt{(r^2 - x^2)}dx## and I'm integrating across the first quadrant to get the area of the first quater of a circle. And I change variables with ##x = r\cos{\theta}## and ##dx = -{r}\sin{\theta}{d\theta}## And I form a new integral that's...

It is clear that a server and a client are programs communicative iwth each other using one or more protocols (HTTP, TCP, etc.) I conceptually understand what an API is: it is like an intermediary between two programs that makes integration easy. For example, we build app A and want to connect...

I am calculating the temperature distribution and utilizing the obtained results to calculate the current distribution. In order to do this , I employ a table in which I stock all the current distribution for each value of radius . Subsequently, I aim to identify the radius corresponding to a...

Hi I struggle with integration generally. Could you be able please to talk me through the stages of this one? thanks martyn

I proceeded as follows $$\int\frac{2(\sqrt3-1)(cosx-sinx)}{2(\sqrt3+2sin2x)}dx$$ $$\int\frac{(cos(\pi/6)-sin(\pi/6))(cosx-sinx)}{(sin(\pi/3)+sin2x)}dx$$ $$\frac{1}{2}\int\frac{cos(\pi/6-x)-sin(\pi/6+x)}{sin(\pi/6+x)cos(\pi/6-x)}dx$$ $$\frac{1}{2}\int cosec(\pi/6+x)-sec(\pi/6-x)dx$$ [FONT=times...

Some sources state a similar format of the following $$\int_a^{a+da}f(x)dx=f(a)da$$ Which had me thinking whether the following integration can exist $$\int_a^{a+dx}f(x)dx=f(a)dx$$ I have difficulty grasping some aspects about these integrations 1. Regarding the 1st integration, shouldn't ##a##...

This is a bit confusing...conflicting report from attached wolfram and symbolab. Which approach is correct?

I tried to prove this but I fall into a loop when I try to apply integration by factors, that is I prove that the integral is equal to itself. Any helpfull tips?

Idea: Given a system of two coupled oscillators in which 2 masses are connected to a spring in the middle. Each of the two masses is coupled to another spring on the left and right, which have fixed ends but are not connected to each other. So we have 3 springs, two masses and the springs also...

In the book, I see the following: ##\int_{x_1}^{x_1 + \epsilon X_1} F(x, \hat y , \hat y') dx = \epsilon X_1 F(x, y, y')\Bigr|_{x_1} + O(\epsilon^2)##. My goal is to show why they are equal. Note that ##\hat y(x) = y(x) + \epsilon \eta(x)## and ##\hat y'(x) = y'(x) + \epsilon \eta'(x)## and...

Picture of question: Part (a) : ##\nabla \times \vec F = 0## so a Potensial exists. I don't have problem with this part. Part (b) : what I've done : First experssion is 0 because ##\theta = \dfrac {\pi} {2}##. I don't know how to integrate over ##\theta ## when it is a constant.

Hi. What exactly is happening mathematically when you integrate ##\frac{1}{x}## $$\int_a ^b \frac{1}{x} dx=\ln{b}-\ln{a}=\ln{\frac{b}{a}}$$ if there's units? Sure, they cancel if you write the result as ##\ln{\frac{b}{a}}##, but the intermediate step is not well-defined, so why should log rules...

My attempt: (a) I don't think I completely understand the question. By "evaluate ##\lim_{n\to \infty f_n (x)}##", does the question ask in numerical value or in terms of ##x##? As ##x## approaches 1 or -1, the value of ##f_n (x)## approaches zero. As ##x## approaches zero, the value of ##f_n...

using the equation mentioned under Relevant Equations I can get, $$\mathbb{P}(2X > Y |1 < 4Z < 3) = \frac{\mathbb{P}(2X>Y, 1<4z<3)}{\mathbb{P}(1<4z<3)}$$ I can find the denominator by finding the marginal probability distribution, ##f_{Z}(z)## and then integrating that with bounds 0 to 1. But I...

The characteristic equation has a zero discriminant and the sole root of ##-1##. The general solution to the associated homogeneous equation is thus $$y_h(x)=e^{-x}(c_1+c_2x)\tag{1}$$ Now we only need to find one particular solution of the non-homogeneous equation. The righthand side of the...

Was solving a problem in mathematics and came across the following integration. Unable to move further. Can somebody provide answer for the following ( a and b are constants ).

I need to integrate time dilation with derivative, how could I do that?

I am reading the Horatiu Nastase's Introduction to quantum field theory (https://professores.ift.unesp.br/ricardo.matheus/files/courses/2014tqc1/QFT1notes.pdf ) ( Attached file ) or Peskin, Schroeder's quantum field theory book, p.105, (4.77). Through p.176 ~ p. 177 in the Nastase's Note, he...