Integral Definition and 1000 Threads

I've seen a proof that the path integral formulation of quantum mechanics is equivalent to solving Schrodinger's equation. However, it appears to me that the proof actually depended on the Hamiltonian having a particular form. I'm wondering how general is the equivalence. Let me sketch a...

Hello, can anyone show me if this integral can be evaluated? ##\frac{1}{a_0^2}\int_\Sigma\frac{dy'dz'}{\bigg(y'^2+z'^2+\tfrac{1}{(2a_0)^2}\bigg)^2}##

Hello, I have difficulty in evaluating this integral. Can anyone assists? $\frac{1}{a_0^2}\int_\Sigma\frac{dy'dz'}{\bigg(y'^2+z'^2+\tfrac{1}{(2a_0)^2}\bigg)^2}$

Homework Statement Integrate from 0 to 1 (outside) and y to sqrt(2-y^2) for the function 8(x+y) dx dy. I am having difficulty finding the bounds for theta and r. Homework Equations I understand that somewhere here, I should be changing to x = r cost y = r sin t I understand that I can solve...

I am new to the world of calculus and the first thing that I learned is how to calculate the area under the range of a polynomial function, like: $$\int_1^3 x^2 \,dx$$ when I take the intergal of ##x^2##, I get ##\frac{x^3}{3}##due to the power rule, but it doesn’t make sense to me,why would...

Homework Statement ∫∞-∞|∂Ψ/∂x|2dx Homework Equations Can this be simplified at all The Attempt at a Solution

In many texts I have seen, Gauss theorem has the form of$$\frac{q}{\epsilon_0}=\oint\vec{E}d\vec{A}$$ Why a line integral symbol was used for this surface integral everywhere? The more I see it the more I believe there is something wrong with my understanding about this. I didn't think too much...

Let f(x) be a bounded continuous function on [0,1]. Let g(x)=f(x) on all rational points in [0,1]. Let g(x) be Riemann integrable on [0,1]. Does g(x)=f(x) almost everywhere in the interval? If so - proof? If not -counterexample.

Find the closed form (or) analytic expression form for the following integral $$ \hspace{0.3cm} \large {\int_{0} ^{\infty} \frac{\frac{1}{x^4} \hspace{0.1cm} e^{- \frac{r}{x^2}}\hspace{0.1cm}e^{- \frac{r}{z^2}} }{ \frac{1}{x^2} \hspace{0.1cm} e^{- \frac{r}{x^2}}+ \frac{1}{y^2}...

The question goes like: find the SA of the portion S of the cone z^2 =x^2 +y^2 where z>=0 contained within the cylinder y^2+z^2<=49 this is my attempt using the formula for SA, I could switch to parametric eqns, but even then I'd have hard time setting up limits of integration.

Homework Statement Given the graph of f(x) shown below, find the value of the integral. Photo attached. Homework Equations [/B] ∫23 5x·f(x2)dx The Attempt at a Solution [/B] I tried integration by parts to simplify the problem, but finding the integral of the composite function (f(x2))...

Homework Statement [/B] Calculate, and plot along with (on the same plot) the voltage seen below, the current flowing in the following circuit using the integral relationship between the voltage across an inductor and the current through the inductor. Verify your hand calculations and plot...

$\textsf{Evaluate the integral}$ $$I=\displaystyle\int\frac{x^2}{\sqrt{9-x^2}}$$ $\textit{from the common Integrals Table we have}$ $$\displaystyle I=\int\frac{u^2}{\sqrt{u^2-a^2}} \, du =\frac{u}{2}\sqrt{u^2-a^2}+\frac{a^2}{2} \ln\left|u+\sqrt{u^2-a^2}\right|+C$$...

I have the first and second orders that I use in a magnetic simulator, but i need the thirth also to do also with magnetic cylinders accordingly paper: Do anybody have it in any code? I should pass to C++

Homework Statement Use the integral test to show that the sum of the series gif.latex ##\sum_{n=1}^\infty \dfrac{1}{1+n^2}## is smaller than pi/2. Homework EquationsThe Attempt at a Solution I know that the series converges, and the integral converges to pi/4. As far as I´ve understood...

Homework Statement ##\int_{0}^{2\pi} cos^2(\frac{pi}{6}+2e^{i\theta})d\theta##. I am not sure if I am doing this write. Help me out. Thanks! Homework Equations Cauchy-Goursat's Theorem The Attempt at a Solution Let ##z(\theta)=2e^{i\theta}##, ##\theta \in [0,2\pi]##. Then the complex integral...

Homework Statement Acceleration is defined as the second derivative of position with respect to time: a = d2x/dt2. Integrate this equation with respect to time to show that position can be expressed as x(t) = 0.5at2+v0t+x0, where v0 and x0 are the initial position and velocity (i.e., the...

Homework Statement Let G=x^2i+xyj+zk And let S be the surface with points connecting (0,0,0) , (1,1,0) and (2,2,2) Find ∬GdS. (over S) Homework EquationsThe Attempt at a Solution I parametrised the surface and found 0=2x-2y. I’m not sure if this is correct. And I’m also uncertain about...

What did the teacher meant with this: $$\int_{a}^{b} f(t)i + g(t)k dt $$ The two functions, a and b are all given. What is it to integrate a vector? From analytical geometry I know that something in the form of i + j + k is a vector.

Homework Statement (FYI It's from an Real Analysis class.) Show that $$\int_{0}^{\infty} (sin^2(t) / t^2) dt $$ is convergent. Homework Equations I know that for an integral to be convergent, it means that : $$\lim_{x\to\infty} \int_{0}^{x} (sin^2(t) / t^2) dt$$ is finite.I can also use the...

Hi All! I've been looking at this Fourier Transform integral and I've realized that I'm not sure how to integrate the exponential term to infinity. I would expect the result to be infinity but that wouldn't give me a very useful function. So I've taken it to be zero but I have no idea if you can...

I’m having trouble understanding the relationship between how work is both a dot product and integral. I know that work equals F • D and also the integral of F(x): the area under the curve of F and D. However, let’s say that I have a force vector <3,4> and a displacement vector of <3,0>. The...

Homework Statement Homework Equations E=KQ/R^2 The Attempt at a Solution I'm kinda confused at what the question is asked. It is in terms of x, but I thought the integral for potential is V=int(Edr)? Also, should it be integration starting from infinity? Why is the integration from -2 to 3...

Heya, So, I know this is a pretty simple problem, but I seem stuck on it nevertheless. Here's the question Calculate the upper and lower sums , on a regular partition of the intervals, for the following integrals \begin{align*} \int_{1}^{3}(1-7x)dx \end{align*} Please correct me if I'm doing...

Hello everyone! I am currently stuck at the two type of questions below, because I am not really sure what method should be used to calculate these question... Could you give me a hint how to do these questions? :(

Hello! I am reading from Schwarz book on QFT the Path Integral chapter and I am confused about something. I attached a SS of that part. So we have $$<\Phi_{j+1}|e^{-i\delta H(t_j)}|\Phi_{j}>=N exp(i\delta t \int d^3x L[\Phi_j,\partial_t \Phi_j])$$ What happens when we have the left and right...

Homework Statement Calculate \int_{S} \vec{F} \cdot d\vec{S} where \vec{F} = z \hat{z} - \frac{x\hat{x} + y \hat{y} }{ x^2 + y^2 } And S is part of the Ellipsoid x^2 + y^2 + 2z^2 = 4 , z > 0 and the normal directed such that \vec{n} \cdot \hat{z} > 0 Homework Equations All the...

Question $$\int_{-1}^{1} cos(x) P_{n}(x)\,dx$$ ____________________________________________________________________________________________ my think (maybe incorrect) $$\int_{-1}^{1} cos(x) P_{n}(x)\,dx$$ $$\frac{1}{2^nn!}\int_{-1}^{1} cos(x) \frac{d^n}{dx^n}(x^2-1)^n\,dx$$ This is rodrigues...

Homework Statement Find a > 0 so the integral int(exp(-ax)*cosx)dx from 0 to inf get as high value as possible. The Attempt at a Solution My way of solving this is to plot the integrand, i.e. exp(-ax)*cosx and check for different values of a. The larger a is, the smaller the area under the...

Hey, I've got this problem I've been doing, but I'm not sure if my approach is right. My textbook has pretty much less than a paragraph on this sort of stuff. My thinking was that since an integral is a sum, in order to get the range from 0 to 8, we should just be able to add or subtract the...

Hello everyone, I am stuck on this homework problem. I got up to (ln (b / (b+1) - ln 1 / (1+1) ) but I'm not sure how to go to the red boxed step where they have (1 - 1 / (b+1) ) if anyone can figure it out Id really appreciate it. thank you very much.

Homework Statement You are given the function f(x)=3x^2-4x-8 a) Find the values of a. Explain the answers using the function. Homework EquationsThe Attempt at a Solution a^3-2*a^2-8*a=0 a=-2 v a=0 v a=4 I found the answers, but I don't know how to explain my answers by using the function...

http://web.mit.edu/sahughes/www/8.022/lec01.pdf So I'm trying to understand how to get from F = ∫[(Q*λ)*dL*r]/(r^2) to F=∫q*λ*[(xx+ay)/(a^2+x^2)^(3/2)]*dx Like I don't understand why the x and y components of r are negative, or why "The horizontal r component is obviously zero: for every...

Homework Statement Evaluate ##\int\int_{R} (x+2)(y+1) \; dx \; dy## where ##R## is the pentagon with vertices ##(\pm 1,0)##, ##(\pm 2,1)## and ##(0,2)##. Homework EquationsThe Attempt at a Solution After drawing ##R## I split ##R## into two sections ##R_1## (left half) and ##R_2## (right half)...

Homework Statement Calculate the integral \int_{S} (\frac{A}{r^2}\hat{r} + B\hat{z}) \cdot d\vec{S} Where S is the sphere with r = a. 2. The attempt at a solution I have no clue how to solve this problem. I have thought of introducing spherical coordinates and somehow finding a connection...

The question provides the vector field (xy, 2yz, 3zx) and asks me to confirm Stokes' theorem (the vector calc version) but I am trying to use the generalized differential forms version. So, I am trying to integrate \omega = xy\,dx + 2yz\,dy + 3zx\,dz along the following triangular boundary...

I would like to learn (self-study) the theory behind the n-dimensional Riemann integral (multiple Riemann integrals, not Lebesgue integral). I am from Croatia and found lecture notes which Croatian students use but they are not suitable for self-study. The notes seem to be based on the book: J...

1. At pg.212, Hartle book (2003) writes equation 9.81 as an approximation of 9.80, directly. 2. $$ΔΦ=\int_0^{w_1}\frac{(1+\frac{M}{b}w)}{(1+\frac{2M}{b}w-w^2)^\frac{1}{2}}dw$$ equation(9.80) $$ΔΦ≈\pi+4M/b$$...

Hi. I am working my way through some complex analysis notes(from a physics course). I have just covered Cauchy's theorem which basically states that the integral over a closed contour of an analytic function is zero. this is then used to show that contours of analytic functions can be deformed...

So folks, I'm learning complex analysis right now and I've come across one thing that simply fails to enter my mind: the Cauchy Integral Theorem, or the Cauchy-Goursat Theorem. It says that, if a function is analytic in a certain (simply connected) domain, then the contour integral over a simple...

Hello A simple question. I have a linear integral operator (self-adjoint) $$(Kx)(t)=\int_{a}^{b} \, k(t,s)\,x(s)\,ds$$ where $k$ is the kernel. Can I say that its norm (I believe in $L^2$) equals the spectral radius of $K?$ Thanks! Sarah

Hi! I would like solve this kind of relation: \phi = \int_0^r \phi (r') 4 \pi r'dr' But I don't know how to proceed... Can you advise me ? Thank's in advance !

I like use OpenOffice to do mathematics documents (yes, not as good as LaTex, but it's not as much of a hassle), but I find the fonts that I currently have don't have a good set of good-looking integral signs, so I looking for a FREE font that has a good set of such signs. A small search for...

Homework Statement If ##a \neq 0##, evaluate the integral $$\int \frac {dx} {a~\sin^2~x + b~\sin~x~\cos~x + c~\cos^2~x}$$ (Hint: Make the substitution ##u = \tan x## and consider separately the cases where ##b^2 - 4ac## is positive, zero, or negative.) The Attempt at a Solution $$\int \frac...

Is the definite integral $$\int_{1}^{\infty}\left(\arcsin \left(\frac{1}{x}\right)-\frac{1}{x} \right)\,dx$$ of indeterminate form or not? Prove your statement.

Hi, I have the following integral that I want to evaluate: \int_0^{\infty}y\,e^{-y\left[(z+1)(K-1)+1\right]}Ei\left(y_2(K-1)\right)\,dy In the table of integrals there is a similar integral in the form \int_0^{\infty}x^{v-1}\,e^{-\mu...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.3: Modules Over Principal Ideal Domains ... and I need some help in order to formulate a proof of Proposition 4.3.5 Part (iii)... ... Proposition 4.3.5 reads as...

While deriving the volume of sphere formula, I noticed that almost everyone substitute the limits 0 to 360 for the angle (theta) i.e the angle between the positive x-axis and the projection of the radius on the xy plane.Why not 0to 360 for the angle fi (angle between the positive z axis and...

Hello all, I need to evaluate the following 3-dimensional integral in closed-form (if possible) \int_{y_1=0}^{\infty}\int_{y_2=0}^{\infty}\int_{x_2=0}^{zy_2}\exp\left(-\min(x_2,\,y_1(z-\frac{x_2}{y_2}))\right)e^{-(K-1)x_2}e^{-y_1}e^{-y_2}\,dx_2dy_2dy_1 where ##z## is real positive number, and...

Hello all, Is there a closed form solution for the following integral \int_0^z\frac{1}{1+z-x}\frac{1}{(1+x)^K}\,dx for a positive integer ##K\geq 1##, and ##z\geq 0##? I searched the table of integrals, but couldn't find something similar. Thanks in advance for any hint