Theorem Definition and 1000 Threads

Hello! (Wave)We have $G(x,y)=-\frac{1}{4 \pi} \frac{1}{||\overline{x}-\overline{y}||}$ for $x, y \in \mathbb{R}^3$.I want to show that $\int_{\mathbb{R}^3} \Delta{G(x,y)} dx= 1$. It suffices to show that $\int_{\mathbb{R}^3} \Delta{G(x,0)} dx= 1$, since setting $\overline{x}=x-y$ we have...

I won't post the whole rigorous statement of the theorem, but basically the theorem states that If ##F(x,y) = 0## on a neighborhood of the form ##[x-\delta ,x+\delta ]\times [y- \epsilon ,y+\epsilon ]## and if ##\frac{\partial F(x,y)}{\partial y} \neq 0##, then there exists a function ##y=\phi...

Incorrect Theorem: Suppose x and y are real numbers and x + y = 10, then x != 3 and y != 8. (a) What’s wrong with the following proof of the theorem? Proof. Suppose the conclusion of the theorem is false. Then x = 3 and y = 8. But then x + y = 11, which contradicts the given information...

I've read the postulate that there could be a huge spherical dark matter halo extending far beyond the edges of the Milky Way. However, according to Newton's shell theorem, there is no net gravitational pull within a shell. How do they arrive at the conclusion of a halo so huge?

Simplify (find the sum) of $${30 \choose 0} + \frac{1}{2}{30 \choose 1}+ \frac{1}{3}{30 \choose 2} + ... + \frac{1}{31}{30 \choose 30}$$. Do this is two ways: 1. Write $$\frac{1}{i+1}{30 \choose i}$$ in a different way then add 2. Integrate the binomial thorem (don't forget the constant of...

Homework Statement Verify that a) ##\frac{1}{2\pi} \int_0^{2\pi} f(e^{it})\frac{e^{it}\bar z}{1-\bar z e^{it}}dt = 0##, if ##f(w)## is analytic for ##|w|<1+\epsilon##, and that b) ##\frac{1}{2\pi} \int_0^{2\pi} f(e^{it})\frac{e^{it}}{e^{it}-z}dt = f(z).## for ##z = re^{i\theta}## with ##r <...

At the bottom of the picture, I couldn't understand why differentiating with respect to x gives the first integral at the right-hand side 0. Thanks for reading.

Homework Statement why the f(t-1) isn't = 1-1 = 0 ? since f(t) = 1 , a=1 Homework EquationsThe Attempt at a Solution

I would like to post a comment for offtopic conversation in another thread. This is the point of no-communication theorem that measuring one particle does not change anything measurable about the other particle. But conclusions of no-communication theorem are limited by it's assumptions (as for...

Homework Statement We have an ordinary LRC circuit with inductance L, capacitance C and resistance R with an oscillating voltage with low frequency (U^e). Using the energy conservation law and Poynting's theorem, find the differential equation: $$L \frac{\partial ^2}{\partial t^2}I + R...

According to Castigliano's theorem, the local compliance of an elastic structure, e.g. a cantilever, can be determined by integrating the products of stress intensity factor weight functions over the length of said structure. However, if I do that for a double cantilever beam using simple beam...

Homework Statement 2. Homework Equations [/B] De Moivres Theorem/ Eulers formula The Attempt at a Solution Honestly don't know where to go with this now. I already applied De Moivres theorem at the very end. It feels like I have to do something more with either De Moivres theorem or...

Homework Statement I and J are open subsets of the real line. The function f takes I to J, and the function g take J to R. The functions are in C1. Use the mean value theorem to prove the chain rule. Homework Equations (g o f)' (x) = g'(f (x)) f'(x) MVT The Attempt at a Solution [/B] I know...

Just got confused that while applying the Work - Energy Theorem in a vertical Spring-Block system performing SHM (considering no other external forces other than gravity), when I apply the theorem from equilibrium position, do I consider the work done by gravity?

Homework Statement for the alternative form of second shift property (4.8) , why he integral of (e^-sp) g(p+a) dp isn't equal to integral of (e^-sp) g(t) dp ? why it will become L{ g(t+a) } ? Homework EquationsThe Attempt at a Solution

How do I prove the following theorem? If $A \subset B$ then $P(A) \le P(B)$ and $P(B-A) = P(B)-P(A)$ $A$ and $B$ are events and $P$ is the probability function. What I tried (but not sure if it's right or not): $P(B) = P((B\setminus A) \cup (B \cap A)) = P(B\setminus A)+P(B \cap A) \ge...

I need help with proving the theorem below Axiom 1: Each game is played by two distinct teams. Axiom 2: There are at least four teams. Axiom 3: There are at least one game played by each team Axiom 4: Each distinct team plays each of the other teams at most one time Theorem 2: At minimum...

Homework Statement [/B] Use the Bounded Monotonic Sequence Theorem to prove that the sequence: \{a_{i} \} = \Big\{ i - \sqrt{i^{2}+1} \Big\} Is convergent.Homework EquationsThe Attempt at a Solution [/B] I've shown that it has an upper bound and is monotonic increasing, however it is to...

Suppose I have an element of [itex]SE(3)[\itex]. I know this can be thought of as a translation along an axis and rotation about that axis due to Chasles theorem. My question is simple: How do I go about computing the axis, length of the translation, angle of the rotation and radius of the...

Homework Statement in the third photo attached , why do we need to raise the whole pi _3 to power of -1/2 ? can we do so ? if we do so , the original pi_3 will be changed , right ? Homework EquationsThe Attempt at a Solution

Homework Statement So I've read a lot about this but still can't figure what's going on. I understand that to find the error of approximation all we have to do is: |E(x)| = |f(x)-Tn(x)| But what about M*(xn+1/(n+1)!) What's the point of this? and why does it have to be greater than or equal...

Okay, the task is really not that hard but I am getting strange numbers. 1. Homework Statement Find the current going through the resistor of 18 ohms. Circuit is shown in the first picture. Homework EquationsThe Attempt at a Solution I used norton theorem and superposition to find current In

Homework Statement Let f is differentiable function on [0,1] and f^{'}(0)=1,f^{'}(1)=0. Prove that \exists c\in(0,1) : f^{'}(c)=f(c). Homework Equations -Mean Value Theorem The Attempt at a Solution The given statement is not true. Counter-example is f(x)=\frac{2}{\pi}\sin\frac{\pi}{2}x+10...

Homework Statement http://www-mdp.eng.cam.ac.uk/web/library/enginfo/aerothermal_dvd_only/aero/fprops/dimension/node9.html can somoene expalin about step 4 in the first photo attached ? What does it mean by each group has all the repeating variables and non-repeating variable ? Homework...

I think this be Analysis, I Need some kind of convergence theorem for integrals taken over sequences of sets, know one? Example, a double integral taken over sets such that x^(2n)+y^(2n)<=1 with some integrand. I'd be interested in when the limit of the integral over the sequence of sets is...

Cooperstein (in Advanced Linear Algebra) and Roman (also in a book called Advanced Linear Algebra) give versions of the Correspondence Theorem for Vector Spaces ... but these 'versions' do not look like the same theorem ... can someone please explain how/why these two versions are actually the...

I am reading Bruce Cooperstein's book: Advanced Linear Algebra ... ... I am focused on Section 2.3 The Correspondence and Isomorphism Theorems ... ... I need further help with understanding Theorem 2.15 ... Theorem 2.15 and its proof read as follows...

I am reading Bruce Cooperstein's book: Advanced Linear Algebra ... ... I am focused on Section 2.3 The Correspondence and Isomorphism Theorems ... ... I need help with understanding Theorem 2.15 ... Theorem 2.15 and its proof read as follows:It appears to me (and somewhat surprises me)...

Homework Statement Verify the divergence theorem for the function V = xy i − y^2 j + z k and the surface enclosed by the three parts (i) z = 0, s < 1, s^2 = x^2 + y^2, (ii) s = 1, 0 ≤ z ≤ 1 and (iii) z^2 = a^2 + (1 − a^2)s^2, 1 ≤ z ≤ a, a > 1. Homework Equations [/B]...

I'm familiar with Bell's Theorem.. have studied it over the years. I'd just like to confirm if my belief is correct. In short. It shows either particles don't exist before measurement or there are hidden variables.. you know all those non-counterfactual and locality arguments.. Specker theorem...

I am reading Bruce Cooperstein's book: Advanced Linear Algebra ... ... I am focused on Section 2.1 Introduction to Linear Transformations ... ... I need help with understanding Theorem 2.7 ... Theorem 2.7, its proof and some remarks read as follows:I am having considerable trouble...

Homework Statement A high diver(m = 62kg) walks off a platform 15 meters above the water below (assume velocity inital = 0). The diver reaches a depth of 2.2 metres in the pool before coming to a stop. 1. What is the diver's change in kinetic energy (Answer: -9114J) 2. What is the average force...

I am reading Bruce Cooperstein's book: Advanced Linear Algebra ... ... I am focused on Section 1.6 Bases and Finite-Dimensional Vector Spaces ... I need help with the proof of Theorem 1.16 ... Theorem 1.16 and its proof reads as follows: Question 1 In the second paragraph of above proof...

Homework Statement Here is a link to the paper I am working through: http://www.ams.org/journals/proc/1970-025-01/S0002-9939-1970-0262849-9/S0002-9939-1970-0262849-9.pdf Homework EquationsThe Attempt at a Solution [/B] I am working on the first line of the proof. This is what I thus far...

Homework Statement Hi everyone, I have a problem that has me stumped and would appreciate some pointers as to where I am going wrong and maybe point me in the right direction for solving the problem. The problem is in essence to use the "Work-Energy Theorem" to find the co-efficient of kinetic...

From the Ehrenfest theorem, we know that the equation below is correct for any state ## \psi ##. ##m\frac{d^2}{dt^2}\langle x \rangle_{\psi} =-\langle \frac{\partial V(x)}{\partial x} \rangle_{\psi} ## But then one of the definitions of coherent states is states for which the expected value of...

Hi. I've only ever seen Noether's theorem formulated ond proven in the framework of Lagrangian mechanics. Is it possible to do the same in Newtonian mechanics, essentially only using F=dp/dt ? The "symmetries" in the usual formulation of the theorem are symmetries of the action with respect to...

Can you suggest any book for learning multinomial theorem and its application in permutation and combinations problems? I am also looking for a book for learning Permutations and Combinations. (Right now, I am using a problem oriented book by Marcus. But I want a book for learning the basics as...

Hello all, Guys in my textbook they state that on a point mass at point outside spherical shell of uniform density, the gravitational force is just as if the entire mass of the shell is concentrated at the Centre of shell. The text also states, the force of attraction due to a hollow...

Why are the solutions satisfying ##\psi(x+l)=\lambda\,\psi(x)## (4.191) the only physically admissible solutions? (##l## is the period of the periodic potential.) We may argue that the probability of finding an electron at ##x##, ##|\psi(x)|^2##, must be the same at any indistinguishable...

I don't understand how the author get to these point. Please help me as i have been spending so much time trying to figure this out but to no avail. Thanks for your help Source: http://phys.columbia.edu/~nicolis/NewFiles/Noether_theorem.pdf

Homework Statement Prove that \sum\limits_{k=0}^l{n \choose k}{m \choose l-k} = {n+m \choose k}Homework Equations Binomial theorem The Attempt at a Solution [/B] We know that (1+x)^n(1+x)^m = (1+x)^{n+m} which, by the binomial theorem, is equivalent to: {\sum\limits_{k=0}^n{n \choose...

Usually its said that the violation of Bell's inequality means that any theory that contains the assumptions of locality and realism doesn't agree with QM and observations. But sometimes I hear people talk about counter-factual definiteness instead of realism(or maybe the presence of both!) as...

There's a theorem in Euclidean Geometry that says: "Let $\Delta$ and $\Delta'$ be two right triangles. If the hypotenuse and a leg of $\Delta$ has the same measure as the hypotenuse and a leg of $\Delta'$, then $\Delta\cong\Delta'$." Wikipedia says this is only a sufficient condition, by I don't...

Homework Statement Using Parseval's theorem, $$\int^\infty_{-\infty} h(\tau) r(\tau) d\tau = \int^\infty_{-\infty} H(s)R(-s) ds$$ and the properties of the Fourier transform, show that the Fourier transform of ##f(t)g(t)## is $$\int^\infty_{-\infty} F(s)G(\nu-s)ds$$ Homework Equations...

Assuming you are lifting a block up 1 meter from rest to rest with constant work. You know that the work is -deltaU or 10. However, you also know W=deltaKE which is 0. You finally know that W=Fx=10*F. How do you explain why the numbers are different? Thanks!

Hey! I want to discretize a fluctuation dissipation theorem for the white noise ζ of a stochastic differential equation on a 2D domain (sphere). For that I integrate over "Finite Volume" elements with area A and A' (see below). \begin{eqnarray*} \int_{A} d A \int_{A'} d A'...

I tried my best but I wasn't able to solve this can someone please provide me with a detailed solution. Here 's the question : Establish the expression of Vs/Ve (complex) using Thevnin's theorem Here is the circuit : I spent 4 hours trying to solve this but I had no clue how. I'am having...

I am a little confused over part 1 of the fundamental theorem of calculus. Part 2 makes perfect sense to me. I guess my confusion is if we have an integral g(x) defined from [a, b], and we are looking at point x, how do we know that g'(x) = f(x)? It makes sense in the idea that they are...

Homework Statement Calculate the moment of inertia of a uniform rigid rod of length L and mass M, about an axis perpendicular to the rod through one end. Homework Equations Parallel axis theorem: I = Icm + MD2 Long thin rod with rotation axis through centre: Icm = 1/12 ML2 Long thin rod with...