Divergence Definition and 775 Threads

Homework Statement If the sum of a sub n to infinity (n=1) converges then the limit of n as n tends to infinity of an = 0 Homework EquationsThe Attempt at a Solution an =(a1+a2+...an)-(a1+...+an-1) = limit of an (n tends to infinity) = sn -s(n-1) =0 The area I'm confused is why do we assume...

1. Problem: Consider vector field A##\left( \vec r \right) = \frac {\vec n} {(r^2+a^2)}## representing the electric field of a point charge, however, regularized by adding a in the denominator. Here ##\vec n = \frac {\vec r} r##. Calculate the divergence of this vector field. Show that in the...

Prove that the sequence :(-1)^n diverges by using the ε-definition of the limit of a sequence

Consider the following integral $$\int \frac{d^4k}{k^2}$$ It is UV divergent but is it IR finite or IR divergent? The integrand is singular as ##k \rightarrow 0## so this suggest an IR divergence but this is no longer the case if I make a shift of the loop momenta by say ##p_1## and write the...

Homework Statement I attempted to solve the problem. I would like to know if my work/thought process or even answer is correct, and if not, what I can do to fix it. I am given: Calculate the divergence of the vector field : A=0.2R^(3)∅ sin^2(θ) (R hat+θ hat+ ∅ hat)Homework Equations [/B] The...

Homework Statement Find the divergence of the function ##\vec{v} = (rcos\theta)\hat{r}+(rsin\theta)\hat{\theta}+(rsin\theta cos\phi)\hat{\phi}## Homework Equations ##\nabla\cdot\vec{v}=\frac{1}{r^2}\frac{\partial}{\partial r}(r^2v_r)+\frac{1}{r sin\theta}\frac{\partial}{\partial...

Homework Statement Given ##b_n = 1 / n## if ##n## odd and ##b_n = 1 / n^2## if ##n## even, show that the series $$\sum_{n=1}^{\infty} (-1)^n b_n$$ diverges. Homework Equations Did'nt find any for this problem The Attempt at a Solution I assumed that ##\sum_{n=1}^{\infty} (-1)^n b_n =...

Consider a scenario where in one frame R, I have a magnet at rest and a solid slab of charges with an arbitrarily large mass moving at velocity v. The overall acceleration of the slab is trivial, however, the v x B exerted on the slab is divergent, thus compressive/tensile stresses are exerted...

Homework Statement By Gauss' law, how is it able to obtain ## \nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0} ## ? By Coulomb's law, ##\vec{E} = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} \hat{r}## I calculate the divergence of ##\frac{1}{r^2} \hat{r}## and get the result is zero That means the...

{Moderator note: Member advised to retain and use the formatting template when starting a thread in the homework sections} Hey guys Question: Calculate the divergence as a function of radius for each of the following radially symmetrical fields in which the magnitude of the field vector: (a)...

It is my understanding that the task of enumerating all of the divergent diagrams in a quantum field theory can be reduced to analyzing a hand full of diagrams (well, at the moment I know that this is at least true for QED and phi^4 theory), and that all other divergent diagrams are divergent...

Hello! I have been doing a previous exam task involving the divergence theorem, but there is a minor detail in the answer which i can't fully understand. I have a figur given by ${x}^{2} +{y}^{2} -{z}^{2} = 1$ , $z= 0$ and $z=\sqrt{3}$ As i have understood this is a hyperboloid going from...

I'd like to use 2-d for simplification. Divergence is the rate of change of a component of a field F as you travel along that component's direction. So Fx represents the part of F flowing the X direction and same with Fy along the Y direction, and so divergence is calculated by dFx/dx +...

Homework Statement F(x,y,z)=4x i - 2y^2 j +z^2 k S is the cylinder x^2+y^2<=4, The plane 0<=z<=6-x-y Find the flux of F Homework Equations The Attempt at a Solution What is the difference after if I change the equation to inequality? For example : x^2+y^2<=4, z=0 x^2+y^2<=4 , z=6-x-y...

Homework Statement For the vector field F(r) = Ar3e-ar2rˆ+Br-3θ^ calculate the volume integral of the divergence over a sphere of radius R, centered at the origin. Homework Equations Volume of sphere V= ∫∫∫dV = ∫∫∫r2sinθdrdθdφ Force F(r) = Ar3e-ar2rˆ+Br-3θ^ where ^ denote basis (unit vectors)...

I’m currently evaluating the "realism" of two survival models in R by comparing the respective Kullback-Leibler divergence between their simulated survival time dataset (`dat.s1` and `dat.s2`) and a “true”, observed survival time dataset (`dat.obs`). Initially, directed KLD functions show that...

Homework Statement [/B] Let f and g be scalar functions of position. Show that: \nabla f \cdot \nabla(\nabla ^2 g)-\nabla g \cdot \nabla(\nabla ^2f) Can be written as the divergence of some vector function given in terms of f and g. Homework Equations [/B] All the identities given at...

Homework Statement Verify the Divergence Theorem for F=(2xz,y,−z^2) and D is the wedge cut from the first octant by the plane z =y and the elliptical cylinder x^2+4y^2=16 Homework Equations \int \int F\cdot n dS=\int \int \int divF dv The Attempt at a Solution For the RHS...

Homework Statement I have a couple of series where I need to find out if they are convergent (absolute/conditional) or divergent. Σ(n3/3n Σk(2/3)k Σ√n/1+n2 Σ(-1)n+1*n/n^2+9 Homework Equations Comparison Test Ratio Test Alternating Series Test Divergence Test, etc The Attempt at a...

I would like to prove that the following integral is logarithmically divergent. $$\int d^{4}k \frac{k^{4}}{(k^{2}-a)((k-b)^{2}-x)((k-y)^{2}-a)((k-z)^{2}-a)}$$ This is 'obvious' because the power of ##k## in the numerator is ##4##, but the highest power of ##k## in the denominator is ##8##...

Homework Statement Find te gradient of the following function f(r) = rcos(##\theta##) in spherical coordinates. Homework Equations \begin{equation} \nabla f = \frac{\partial f}{\partial r} \hat{r} + (\frac{1}{r}) \frac{\partial f}{\partial \theta} \hat{\theta} + \frac{1}{rsin\theta}...

Nabla operator is defined by \nabla = \sum^3_{i=1} \frac{1}{h_i}\frac{\partial}{\partial q_i}\vec{e}_{q_i} where ##q_i## are generalized coordinates (spherical polar, cylindrical...) and ##h_i## are Lame coefficients. Why then div(\vec{A})=\sum^3_{i=1} \frac{1}{h_i}\frac{\partial}{\partial...

Surface S and 3D space E both satisfy divergence theorem conditions. Function f is scalar with continuous partials. I must prove Double integral of f DS in normal direction = triple integral gradient f times dV Surface S is not defined by a picture nor with an equation. Help me. I don't...

Today when I ask a professor about maxwell eqation He tells me " it seems that the unknowns exceed the number of equations. What are the missing ingredients? The answer is the boundary condition .With appropriate boundary conditions, zero divergence and zero curl will nail down a unique solution...

Homework Statement The laser beam is not a point source. It is known that it has a rectangular shape with a divergence of 30 mrad x 1 mrad. I would like to know how large my laser lobe will be at a distance of 250 mm from the laser source. Homework Equations I think you can use trigonometri...

$\tiny{206.f3a.}$ $\textsf{Use the divergence Test to detemine whether the series is divergent}$ \begin{align} \displaystyle &\sum_{k=1}^{\infty}\frac{\arctan(2k)}{1+4k^2}\\ \textit{take limit}\\ =&\lim_{{k}\to{\infty}}\frac{\arctan(2k)}{1+4k^2}\\ \\ =&\frac{\arctan(\infty)}{\infty}...

$\tiny{206.8.8.17}$ $\textsf{Evaluate the following integral, or stat that it diverges.}\\$ \begin{align*} \displaystyle && I_{17}& =\int_{0}^{\infty}36x^8 e^{-x^9}\, dx& &(1)&\\ && &=36\int_{0}^{\infty}\frac{x^8}{e^{x^9}}\, dx & &(2)&\\ \end{align*} $\textit{first what be the recommended...

I have $$\sum_{n = 2}^{\infty} \frac{(lnn)^ {12}}{n^{\frac{9}{8}}}$$ I'm trying the limit comparison test, so I let $$ b = \frac{1}{n^{\frac{9}{8}}}$$ and $a = \sum_{n = 2}^{\infty} \frac{(lnn)^ {12}}{n^{\frac{9}{8}}}$ $\frac{a}{b} = (lnn)^ {12}$ therefore I know the limit of this as n...

Why is it possible that ∫∫∫ V f(r) dV ≠ 0 even if f(r) =0

Homework Statement ∇ . r = 3, ∇ x r = 0 Homework EquationsThe Attempt at a Solution So far I've gotten up to ∇(∇^2 r)

Why must steady currents be non-divergent in magnetostatics? Based on an article by Kirk T. McDonald (http://www.physics.princeton.edu/~mcdonald/examples/current.pdf), it appears that the answer is that by extrapolating the linear time dependence of the charge density from a constant divergence...

I'm using the book of Jerome Keisler: Elementary calculus an infinitesimal approach. I have trouble understanding the proof of the following theorem. I'm not sure what it means. Theorem: "An increasing sequence <Sn> either converges or diverges to infinity." Proof: Let T be the set of all real...

Hi, I'm looking at the following graph, but there are a few things I don't get. For instance: curl should always be zero in circles where the field lines are totally straight (right-most figure) curl should always be non-zero in circles where the field lines are rotating (center figure in 2nd...

Homework Statement The problem states that it is required to prove that ∇f(r)=f'(r)R/r where r is the vector field Homework Equations ∇=(∂/∂x)i + (∂/∂y)j + (∂/∂z)k R= xi + yj +zk r = √(x^2+y^2+z^2) The Attempt at a Solution The trouble that I am having with this problem is the inability to...

Homework Statement How to I explain that maxwell's equation has well defined divergence Homework Equations All four EM Maxwell's equation The Attempt at a Solution I discussed it by showing one of the property of Maxwell's equation that is the Divergence of a Gradient is always zero (With...

I have this series: $$\sum_{k = 1}^{\infty} {4}^{\frac{1}{k}}$$ To solve this, I am trying to compare it to this series $$\sum_{k = 1}^{\infty} {4}^{k}$$ So, I can let $a_k = {4}^{\frac{1}{k}} $ and $b_k = {4}^{k}$ These seem to be both positive series and $ 0 \le a_k \le b_k$ Therefore...

I have $$\sum_{n = 2}^{\infty} \frac{{(\ln\left({n}\right)})^{12}}{n^{\frac{9}{8}}}$$ We can compare it to $ \frac{1}{{n}^{\frac{1}{8}}}$. $ \sum_{n = 1}^{\infty} \frac{1}{{n}^{\frac{1}{8}}}$ diverges because $p < 1$ in this case. So, if I can prove that $...

##\nabla p = \rho \nabla \phi ## My textbook says that by taking the curl we get: ## 0=\nabla \rho X \nabla \phi ## ** I don't follow. I understand the LHS is zero, by taking the curl of a divergence. But I'm unsure as to how we get it into this form, from which it is clear that the gradients...

Hi, I have a question about identifying closed and open surfaces. Usually, when I see some exercises in the subject of the divergence theorem/flux integrals, I am not sure when the surface is open and needed to be closed or if it is already closed. I mean for example a cylinder that is...

Hi PF! I have a question on the dyadic product and the divergence of a tensor. I've never formally leaned this, although I'm sure it's published somewhere, but this is how I understand the operators. Can someone tell me if this is right or wrong? Let's say I have some vector ##\vec{V} = v_x i +...

Prove that \sum\frac{1}{2n+1} diverges. I understand that \sum\frac{1}{n} i.e. the harmonic series diverges (I say this because of the comparison test, that is, \frac{1}{2n+1}\leq\frac{1}{2n}\leq\frac{1}{n}). However, this doesn't correctly imply that 1/(2n + 1) diverges. Then I decided to use...

I have a few questions regarding the derivation of the degree of divergence for feynman diagrams. The result is $$D = [g_E] - \sum_{n=3}^{\infty} V_n [g_n]$$ (following notation in Srednicki, ##P118##) I am trying to understand what ##[g_E]## is here? Since in this set up we are summing over...

The Navier-Stokes equation is: (DUj/Dt) = v [(∂2Ui/∂xj∂xi) + (∂2Uj/∂xi∂xi)] – 1/ρ (∇p) where D/Dt is the material (substantial) derivative, v is the kinematic viscosity and ∇p is the modified pressure gradient (taking into account gravity and pressure). Note that the velocity field is...

I just took a calc 2 test and got 3/8 points on several problems that asked you to show convergence or divergence. The reason being that I didn't use the correct test of convergence? The answer was right, if you get to the point where you know the series converges, then why does it matter which...

Homework Statement \sum\limits_{n=1}^{\infty}\frac{n-1}{(n+2)(n+3)} Homework Equations S=\sum\limits_{n=1}^{\infty}a_n (1) \lim\limits_{n\rightarrow\infty}\frac{a_{n+1}}{a_n}\gt 1\rightarrow S\ is\ divergent (2) \lim\limits_{n\rightarrow\infty}\frac{a_{n+1}}{a_n}\lt 1\rightarrow S\ is\...

Hi. I am reading a paper about gaussian beams and the author says that gaussian beams have simultaneously minimal divergence and minimal transversal extension. In order to prove it, the author states that \mathrm{divergenece} \propto \int_{-\infty}^{+\infty} \frac{d\,k_{x}}{2\pi}...

Homework Statement I just want to focus on the divergence outside the cylinder (r >R) Homework Equations The Attempt at a Solution For r > R, I said ∇ * E = p/ε But that's wrong. The answer is ∇ * E = 0 I'm confused because there is definitely an electric field outside the cylinder (r...

Homework Statement Determine which of the sequences converge or diverge. Find the limit of the convergent sequences. 1) {asubn}= [((n^2) + (-1)^n)] / [(4n^2)] Homework Equations [/B] a1=first term, a2=second term...an= nth term The Attempt at a Solution a) So I found the first couple of...

Homework Statement Sorry- I've figured it out, but I am afraid I don't know how to delete the thread. Thank you though :) Homework Equations Below The Attempt at a Solution Photo below- I promise its coming! I've started by using cylindrical coordinates, but I wasn't sure if spherical...

Divergence theorem states that $\int \int\vec{E}\cdot\vec{ds}=\int\int\int div(\vec{E})dV$ And Gauss law states that $\int \int\vec{E}\cdot\vec{ds}=\int\int\int \rho(x,y,z)dV$ If $\vec{E}$ to be electric field vector then i could say that $div(\vec{E})=\rho(x,y,z)$ However i can't see any...