Divergence Definition and 746 Threads

Hi, Say you want to use a proof by contradiction to prove that a sequence diverges. So you assume that x(n)-----> L , and try to find a real number, call it M, such that |x(n) - L| can never get smaller than M, thus arriving at a contradiction. My question is: can M be of the form that...

I'm reading gravitation and having trouble with one of the exercises. aRab+bRgab is the general tensor the exercise asks to show that the divergence of this tensor vanishes if and only if b=-1/2a how do I go about solving this problem?

the divergence and the curl of a vector field "A" are specified everywhere in a volume V. The normal component of curl A is also specified on the surface S bounding V. Show that these data enable one to determine the vector field in the region

http://scienceblogs.com/principles/2010/02/physics_quiz_accelerated_twins.php The answer, they claim, is that Alice ages more than Bob. But say this were true, it would also mean that of two synchronized clocks placed on opposite sides of the earth, one at sunrise, and the other at sunset...

Divergence and Curl in cylindrical and spherical co are: \nabla \cdot \vec E \;=\; \frac 1 r \frac {\partial r E_r}{\partial r} + \frac 1 r \frac {\partial E_{\phi}}{\partial \phi} + \frac {\partial E_z}{\partial z} \;=\; \frac 1 {R^2} \frac {\partial R^2 E_R}{\partial R} + \frac 1 {R\;sin...

[b]1.The problem asks " use the divergence theorem to evaluate the surface integral \int\int F.ds for F(x,y,z) = <x3y,x2y2,−x2yz> where S is the solid bounded by the hyperboloid x^2 + y^2 - z^2 =1 and the planes z = -2 and z=2. i know that the \int\int F.ds = \int\int\int divFdv...

i need to prove that div(R/r^3) = 4πδ where R is a vector and r is the magnitude of the vector R. also δ is the dirac delta function. so div(R/r^3) is 0 everywhere except for the origin. i need to show that the volume integral of div(R/r^3) = 4π as well. using the divergence theorem we...

Not really a homework problem, just me wondering about this: why is there a problem here? Say you want to use the divergence theorem in conjunction with Stokes' theorem. So, from Stokes' you know: Line integral (F*T ds)= Surface integral (curl(F)*n)dS. And you know that Surface...

can anybody tell me the expansion for the divergence of tensor vector product \nabla.(\tilde{K}.\vec{b}) for the case of scalar and vector the expansion is given by \nabla.(a\vec{b})=a\nabla.\vec{b}+\vec{b}.\nabla a

In which geometry of physical system the \nabla.\overline{J} ie divergence of J is zero? How does the Maxwell equations turns out?

I am not able to find any good reference to answer my question, so I will post here how does divergence theorem translates to 4 dimensional curved spacetime. I understood how volume integral changes but I am not able to understand how surface integral changes. I will be glad if some one...

Homework Equations Hey guys I had a slight problem trying to find divergence of vector fields for the following equation: F(x,y,z)=(yzi-xzj-xyk)/(x^2 + y^2 + z^2) So I want to know if its possible of substitute (x^2 + y^2 + z^2) for 1 since that is the equation of a sphere? If not...

If F is a well defined vector field and divF=0 then does that mean the flux of F across any surface in 3D would also be 0? I know that in divergence theorem, divF=0 automatically implies that the integral will be 0 but what about across flat surfaces and planes?

Homework Statement Yeah I've been pondering over that, my book doesn't really do the justice of nailing it down for me. Does having 0 divergence means having "absolute convergence", like maybe at every point (or at a certain point) all the vectors are pointing towards a point? Like...

Absolute Convergence, Conditional Convergence or divergence... Homework Statement \sum_{n=1}^{\infty} \frac {(-2)^{n}}{n^{n}} Homework Equations \lim_{n \rightarrow \infty} | \frac {a_{n+1}}{a_n}| < 1 \;\; absolute\; convergence \lim_{n \rightarrow \infty} | \frac...

Evaluate http://webwork.latech.edu/webwork2_files/tmp/equations/93/91cfe28c766cad38444f0213c651281.png where http://webwork.latech.edu/webwork2_files/tmp/equations/59/a56001472f977192637ea927c607a61.png and is the surface of the sphere of radius 6 centered at the origin. Ok so I started by...

Homework Statement My professor warned us that a few proofs will be on the finals. This could be one of them. However, we did a proof in class where he listed out a bunch of terms and then did an inequality to say it is divergent. I personally hated that long proof. I don't want to...

Homework Statement F(x,y,z) = (2x-z) i + x2y j + xz2 k and the volume is defined by [0,0,0] and [1,1,1]. Homework Equations flux integral = \int\int\int div F dV The Attempt at a Solution \int\int\int div F dV = \int\int\int (2+x2-2xz)dxdydz = 2 + 1/3 - 1/2 = 11/12 But I...

Homework Statement Evaluate the double integral over M (F \circ dS) where M is the surface of the sphere of radius 3 centered around the origin. (Sorry! I couldn't figure out how to use math symbols!) Homework Equations double integral(F\bulletdS)=triple integral (\nabla\bullet F)dV due...

Hello Friends, I am at a loss to understand a proof concerning the proof of divergence of (-1) ^n sequence. According to the book: "To prove analytically that the sequence is convergent, it must satisfy both of the following conditions: A: |-1-L| < epsilon B: |+1 - L| < epsilon " (+1...

i don't really understand the difference :( ∇2V versus ∇ (∇ . V) ? can anyone give me a simple example to showcase the application difference? thanks!

Let W be the solid bounded by the paraboloid x = y^2 + z^2 and the plane x = 16. Let = 3xi + yj + zk a. Let S1 be the paraboloid surface oriented in the negative x direction. Find the flux of the vector field through the surface S1. b. Let S be the closed boundary of W. Use the Divergence...

Homework Statement Determine whether the series is absolutely convergent, conditionally convergent, or divergent. \sum (-1)^n\frac{e^{1/n}}{n^4} Homework Equations The Attempt at a Solution I used the root test so \sqrt[n]{\frac{e^{1/n}}{n^4}} --> \lim_{n\to \infty...

Homework Statement Determine if the series the summation form n=2 to infinity of n/((n2+1)ln(n2+1)) is convergent or divergent. Homework Equations The Attempt at a Solution I applied the integral test and got positive infinity, so I it diverges. But I want to know if I'm right...

Homework Statement Is \sum(-1)^(n-1)*arcsin(1/n) absolutely convergent, conditionally convergent, or divergent? 2. The attempt at a solution The original function is alternating, so by the alternating series test, the function is convergent, because 0 < arcsin(1/(n+1)) <arcsin(1/n)...

Homework Statement what is the divergence of <y,z,x>? Homework Equations The Attempt at a Solution is the answer 0? seems too easy, lol, because the actual question is "compute the surface integral for F dot prod dS over domain T where T is the unit sphere and F = <y,z,x>"...

Please teach me this: Why the naively divergence of a diagram is ln(lambda) (where lambda is ultraviolet cutoff) when the superficial degree of divergence D=0(the divergence of lambda^D when D=0)).I am reading the renormalization theory in Schroeder&Peskin and I do not understand this.I do know...

Homework Statement For example in electromagnetism and I think it's true for any vector field, the relation \vec \nabla \cdot (\vec \nabla \times \vec E)=0. As far as I know, the curl of a vector field is a vector. So basically the above expression takes the divergence of a vector? It...

Homework Statement This problem I have been set is to find real life applications of divergence theorem. I have to show the equivalence between the integral and differential forms of conservation laws using it. 2. The attempt at a solution I have used div theorem to show the equivalence...

Homework Statement \Sigma from n=0 to infinity (-10)n/n! Determine the absolute convergence, convergence, or divergence of the series. Homework Equations In this section, it's suggested that we use the following to determine a solution: A series is called absolutely convergent if the series...

I am just curious how you find the divergence of the following vector field Heres my example u = xz^(2)i +y(x^(2)-1)j+zx^(2) y^(3)k Am I right in thinking U take the derivative with respect to x for first term derivative with respect to y for second term... giving me...

Hi, I am reviewing some vector calculus and have a problem on the derivation of the divergence in the spherical coordinate. Assume there is a small volume located at r_0, \theta_0, \phi_0 with a volume of r_0^2\sin\theta_0 \Delta r \Delta \theta \Delta \phi. My question is that why...

Hi. I've been reading PF for quite a while and have decided to ask my first question. Please be gentle. (I'm a retired computer programmer, not a student)... I've been learning Gauss' divergence theorem and I understand what "flux density" is when considering things like fluid transport or...

Why is this true? \vec \bigtriangledown \cdot \vec f ( \vec r ) = \frac {\partial}{\partial r} (r^2 | \vec f ( \vec r ) | )

I've tried to make sense of this conjecture but I can't wrap my head around it. We've been learning about the divergence theorem and the Neumann problem. I came across this question. Use the divergence theorem and the partial differential equation to show that...

Homework Statement This is from a fluid mechanics text. There are no assumptions being made (i.e., no constants): Show that \frac{\partial{}}{\partial{t}}\int_V e\rho \,dV + \int_S e\rho\mathbf{v}\cdot\mathbf{n}\,dA = \rho\frac{De}{Dt}\,dV\qquad(1) where e and \rho are scalar quantities...

Apparently one can deduce the form of divergence in polar (and spherical) coördinates using the theorem of Gauss and Ostrogradsky, namely that the volume integral over the divergence is equal to the flux integral over the surface. I can't see a way to do that, do you?

So I know what they are and I've been given some really vague and weak interpretations, but I want to build up my intuition and know more about the specifics of curl and divergence. To my understanding now I know that curl is similar to a paddle wheel spinning in a direction dependent on the...

In the notes attached to this thread: https://www.physicsforums.com/showthread.php?t=457123 On page 110, how has he gone from equation (369) to eqn (370). He claims to have done it by "integration by parts using the divergence theorem to eliminate derivatives of \delta g_{ab} if present". (The...

Homework Statement A vector field h is described in cylindrical polar coordinates by ( h equation attached ) where i, j, and k are the unit vectors along the Cartesian axes and (er) is the unit vector (x/r) i+(y/r) j Calculate (1) by surface integral h through the closed surface...

http://arxiv.org/abs/1012.4216 4-dimensional Spin-foam Model with Quantum Lorentz Group Muxin Han 22 pages, 3 figures (Submitted on 19 Dec 2010) "We study the quantum group deformation of the Lorentzian EPRL spin-foam model. The construction uses the harmonic analysis on the quantum Lorentz...

Hi people, It is my first post here :) It is not a homework problem because i solved it (and at least i think i am right...). The question is as follows: In problem 1.16 (Intro. to eletrodynamics, Grifitths, 3rd edition) he asks to calculate the divergence of the function v=1/r2r (bold is...

Dear Experts, I started to look deeper into the electromagnetic fields. So I would like to write a simple code in C/C++, which is capable of calculating the divergence or rotation of the vector fields. Could someone helps me please, to get this started? How to illustrate partial...

Homework Statement I have stared at this too long and do not know which test to approach it with, even writing it out. The problem is State the Convergence or Divergence of the given series: Summation n=1 to Infinity of 1 / sqrt (n^3 + 2n) Homework Equations I narrowed it down to...

Homework Statement Evaluate div v at P = (0, 0, 0) by actually evaluating (\int_S\mathbf{\hat{n}}\cdot \mathbf{v}\,dA)/V and taking the limit as B-->0. Take B to be the cube |x|\le\epsilon,|y|\le\epsilon,|z|\le\epsilon. Let \mathbf{v} = x\mathbf{\hat{i}} + 2y\mathbf{\hat{j}} -...

Homework Statement Calculate the Divergence of a second-order tensor: \sigma _{ij}(x_{i})=\sigma_{0}x_{i}x_{j} Homework Equations \bigtriangledown \cdot \sigma_{ij}=\sigma_{ij'i} The Attempt at a Solution \sigma_{ij'i}=\frac{\partial }{\partial x_{i}}\cdot\sigma_{0}x_{i}x_{j}...

Homework Statement The equation is the summation from n=1 to infinity of [(-1)^n] / [sqrt(2n+3)]. Homework Equations If the series An is compared to a a series Bn that diverges and the series An is greater than the series Bn they both diverge. If the limit from n to infinity of...

Homework Statement http://i324.photobucket.com/albums/k327/ProtoGirlEXE/q8.jpg Attempt to the solution: ok I got that it only converges on (1,infinity) because I solved it and q>1 is where it only converges, so for the rest it diverges. But I'm having trouble with putting the divergency in...

Homework Statement Calculate the divergence of the vector function f = a/s^2 (s hat) where s is the radial distance from the z axis, expressed in cylindrical coordinates. Homework Equations The Attempt at a Solution Using the divergence theorem I relate the volume integral of...

Homework Statement Let D be the region x^2 + y^2 + z^2 <=4a^2, x^2 + y^2 >= a^2, and S its boundary (with outward orientation) which consists of the cylindrical part S1 and the spherical part S2. Evaluate the ux of F = (x + yz) i + (y - xz) j + (z -((e^x) sin y)) k through (a) the whole...