Divergence Definition and 746 Threads

I have followed a derivation of the continuity equation, and it uses the divergence theorem at some point, but looking at the actual meaning of the equation, it almost seems like it is saying the same thing as the theorem. That is, local conservation. So my question is... Can the continuity...

Homework Statement Use the divergence theorem in three dimensions \int\int\int\nabla\bullet V d\tau= \int \int V \bullet n d \sigma to evaluate the flux of the vector field V= (3x-2y)i + x4zj + (1-2z)k through the hemisphere bounded by the spherical surface x2+y2+z2=a2 (for z>0)...

I don't know whether it was proved or can be prove. I don't know whether it is useful. maybe it can be used in string theory or some other things. any comment or address will be appreciated.

My notes say that if we know the divergence and curl of a field then that uniquely determines the field. Can somebody give me an example of how, given only the div and curl of a field, we can deduce the field? I considered the electric field where we have, \nabla \cdot...

1. I was just trying to understand what divergence means so I hope someone can help me out. Well from what I have read if I take a vector field and use an infinitesimal region, if the vector going in is smaller than the vector going out there is positive divergence. Does this mean if i...

I need to show that, using Gauss' Theorem (Divergence Theorem), i.e. integration by parts, that: \int_V dV e^{-r} \nabla \cdot (\frac{\vec{\hat{r}}}{r^2}) = \int_V dV \frac{e^{-r}}{r^2} any ideas on where to start?

Homework Statement Is the series \Sigma\stackrel{\infty}{k=1} (\sqrt{k+1} - \sqrt{k})/k convergent or divergent? Homework Equations The Comparison Test: 0<=ak<=bk 1.The series \Sigma\stackrel{\infty}{k=1} ak converges if the series \Sigma\stackrel{\infty}{k=1} bk converges. 2. The...

This may well be the wrong place to post this so apologies for that if it's the case. Anyway, I'm stuck on this question, any help appreciated Use Gauss' Theorem to show that: (i) If \psi($\mathbf{r}$) ~ \frac{1}{r} as r \rightarrow \infty , then, \int_V {\psi \nabla^{2} \psi}...

Homework Statement Check the Divergence Theorem \int_V(\nabla\cdot\bold{v})\,d\tau=\oint_S\bold{v}\cdot d\bold{a} using the function \bold{v}=<y^2, 2xy+z^2, 2yz> and the unit cube below. Now when I calculate the divergence I get (\nabla\cdot\bold{v})=2y+2x+2y but Griffith's...

I need some help understanding a definition: This is supposed to be an explanation of what the author did on the page before. He had just described how to construct a (complex) Hilbert space from a (real) smooth manifold with a smooth nowhere vanishing volume element, and then moved on to...

Homework Statement Determine whether the following converges or diverges: \sum\frac{n^{n}}{(n+1)^{(n+1)}} with the sum going from n=1 to n=infinity. Homework Equations Comparison Test, Ratio Test. The Attempt at a Solution This should be do-able with the above two tests...

I was told this problem could be done with divergence theorem, instead of as a surface integral, by adding the unit disc on the bottom, doing the calculation, then subtracting it again. Homework Statement Homework Equations The Attempt at a Solution for del . f I get i + j =...

Homework Statement Evaluate the surface Integral I=\int\int_S\vec{F}\cdot\vec{n}\,dS where \vec{F}=<z^2+xy^2,x^100e^x, y+x^2z> and S is the surface bounded by the paraboloid z=x^2+y^2 and the plane z=1; oriented by the outward normal.The Attempt at a Solution...

Homework Statement evaluate https://instruct.math.lsa.umich.edu/webwork2_files/tmp/equations/71/7816ab9562fbe29a133b96799ed5521.png if https://instruct.math.lsa.umich.edu/webwork2_files/tmp/equations/65/11ed69ea372626e9c4cee674c8dc6f1.png and S is the surface of the region in the first...

Homework Statement This is just a general question. My fundamentals aren't very solid because I'm studying on my own at the moment. \int_V (\triangledown \cdot \bold{v}) dV = \int_S \bold{v} d\bold{a} I am trying to find out the sign of the area integral on a surface defined by spherical...

Hello! I got one issue with proving divergence of series. I start covering this part of mathematics and don't understand how to prove it. Here is the issue: I got one harmonic series: \sum_{n=1}^{\infty}{\frac{1}{n}}=1 + \frac{1}{2} + \frac{1}{3} +... We need to show that the series of...

Homework Statement Find the flux of the vector field out of the closed surface bounding the solid region x^2 + y^2 ≤ 16, 0 ≤ z ≤ 9, oriented outward. F = x^3 + y^3 + z^3 Homework Equations The Attempt at a Solution I found the divergence which is 3x^2+3y^2+3z^2. And...

Homework Statement \sum (2n^{2}+3n)/\sqrt{5+n^{5}} index n=1 to infinity Homework Equations The Attempt at a Solution I tried both the Ratio Test (limit as n goes to infinity of a_{n+1}/a_{n}) and the Limit comparison test (limit as n goes to infinity of a_{n}/ b_{n}) but wasn't...

Homework Statement If the current density is time independent and divergence free, show that the Maxwell Equations separate into independent equations for \vec{E} and \vec{B}. Homework Equations Maxwell's equations The Attempt at a Solution The only Maxwell equation with \vec{j}...

Homework Statement Show that following statement is true: If Σa_n diverges, then Σ|a_n| diverges as well. Homework Equations Comparison Test: If 0 ≤ a_n ≤ b_n for all n ≥ 1, and if Σa_n diverges, then Σb_n diverges as well. The Attempt at a Solution I tried to prove the...

Homework Statement 2. Verify the divergence theorem for the vector field: F =(r2cosθ) r +(r2cosφ) θ −(r2cosθsinφ) φ using the upper hemisphere of radius R.Homework Equations Is this any close to be correct? The question marks indicate parts I am not sure about please help. Anyone know what are...

\sum\frac{1*3*5 ... (2k-1)}{1*4*7 ... (3k-2)} from k=1 to infinity Does this series converge or diverge?? I have no idea where to begin, I don't understand it's format. Aren't series usually A+B+C ... but this is just multiplication ?! ? So ?? HELP!

Just for reference, i got this question from reading an online ebook: http://www.plasma.uu.se/CED/Book/EMFT_Book.pdf The bottom equation on page 24 is where i these equations came up. I have been reading some stuff and i keep coming across an annotation which looks exactly like a...

Hello I am trying to get my head around what the divergence actually represents physically. If you have some vector field v, and the components of v, vx, vy, vz have dimensions of kg/s ("flow" - mass of material per second) the divergence will have units of kg/(s*m) (mass per time distance)...

Homework Statement A vector field is defined by A=f(r)r a) show that f(r) = constant/r^3 if \nabla. A = 0 b) show that \nabla. A is always equal to zeroHomework Equations divergence and curl relationsThe Attempt at a Solution I tried using spherical co-ordinates to solve this. But I am not sure...

Homework Statement (x_n) is a sequence and x_1 > 2. From then on, x_{n+1} = x_n + 1/x_n Prove that (x_n) is divergent. Homework Equations n/a The Attempt at a Solution I first tried assuming that a limit existed, but I didn't get a contradiction. (I had x = 2 + 1/x, x = (2 \pm...

I just happened to read two papers that pretend that the quadratic divergence of the Higgs mass is not a problem. The first is "Vacuum energy: Quantum Hydrodynamics vs Quantum Gravity" http://arxiv.org/abs/gr-qc/0505104 (Update: this is now the correct paper from arxiv) where Volovik says that...

Homework Statement show that the definition of the invariant divergence divA = 1/√g ∂i (√g Ai) is equivalent to the other invariant definition divA = Ai;i Ai;k = ∂Ai/∂xk + ГiklAl Гkij = gkl/2 (∂gil/∂xj+∂glj/∂xi-∂gij/∂xl) Homework Equations g is the metric tensor...

Why is \nabla\cdot\vec{A}=\frac{1}{r}\frac{\partial}{\partial r}(rA_{r})+\frac{1}{r}\frac{\partial}{\partial \theta}(A_{\theta}) Where \vec{A}=A_{r}\hat{r}+A_{\theta}\hat{\theta} And \nabla=\hat{r}\frac{\partial}{\partial r}+\hat{\theta}\frac{1}{r}\frac{\partial}{\partial \theta} Instead of...

I've also posted this in the Physics forum as it applies to some physical aspects as well. --- I want to know if I'm on the right track here. I'm asked to prove the following. a) \nabla \cdot (\vec{A} \times \vec{B}) = \vec{B} \cdot (\nabla \times \vec{A}) - \vec{A} \cdot (\nabla \times...

Homework Statement Here is a link to the problem: http://www.brainmass.com/homework-help/physics/electromagnetic-theory/68800 The Attempt at a Solution To find the divergence 1/r^2*d(r)*(r^2*r^2*cos(theta)) +[1/r*sin(theta)]*d(theta)*(sin(theta)*r^2*cos(phi))...

What is the Divergence? is it only the Partial derivatives? Lets say I have a vector field: F=x^2i+y^2j+z^2k, the divergence is F=2xi+2yj+2zk? And if it is, than what is the gradient?:confused:

Hello, I was wondering if anyone could explain the troubling divergence here of the differential cross-section for rutherford scattering for \theta = 0. I know it must have something to do with the fact that the em force extends to infinity, which makes sense to me for the total cross section...

Homework Statement This is from my textbook Engineering Electromagnetics by John Buck and William Hayt 7th Edn, pg 238 in the chapter titled "The Steady Magnetic Field": Homework Equations Divergence theorem: \oint_S \textbf{B} \cdot d\textbf{S} = \int_{\mbox{vol}} \nabla \cdot...

http://img403.imageshack.us/img403/9478/roffelsw8.png I really can't understand the last sentence, how do they get that the sum has to be smaller than k/2?

Homework Statement Is the series from n=1 to infinity of 3/n converging or diverging? Homework EquationsThe Attempt at a Solution Since 3/n is not a geometric series, my guess is that we can just use the Test for Divergence and take it's limit to see if it's converging or diverging. As...

Dear friends, How is the divergence in curvilinear coordinates of a second order mixed tensor defined? I mean, shall I contract the covariant or the contravariant index?? And for both cases which is the physical meaning? \nabla_i N^i_j or \nabla_j N^i_j ? Thanks a lot, Enzo

"Helmholtz equation" Neumann and divergence Hello, I'm trying to solve the following elliptic problem : S = B - \mu\nabla^2 B Where S(x,y) and B(x,y) are 3 component vectors. I have \nabla\cdot S = 0 and I want B such that \nabla\cdot B = 0 everywhere. I'm using finite differences on a...

Does anyone know of any analytical expression for the upper bound on the Kullback–Leibler divergence for a discrete random variable? What I am looking for is the bound expressed as 0 <= S_KL <= f(k) Where k is the number of distinguishable outcomes. Ultimately I am also looking for...

Homework Statement The nth term for a sequence is the square root of [n/ (n^4 + 1)] Investigate whether it is convergence or divergence. Homework Equations Ratio test and integral test The Attempt at a Solution Ratio test will fail for this question, since no conclusion can be...

Homework Statement v = (a.r)r where r=xi+yj+zk and a is a constant vector show \nabla.v = 4(a.r) I let a= ai+bJ+ck then (a.r) = ax+by+cz then this (a.r)r = ax^{2}i+by^{2}j+cz^{2}k \nabla.v = da1/dx+da2/dz+da3/dz =2ax+2by+2cz which is equal to 2(a.r) am i wrong or the book?

I am slightly unsure about how the divergence can be increased by the use of either bi-concave or plano-concave lenses. I understand the general theory behind it but am having trouble putting numbers to it. e.g. if you have a laser beam with a diameter of 2mm and a divergence of 2mrad what...

Homework Statement Hi, I'm trying to follow the proof for the statement \nabla . u = 0 I'm basing it off this paper: http://delivery.acm.org/10.1145/1190000/1185730/p1-bridson.pdf?key1=1185730&key2=4151929021&coll=GUIDE&dl=GUIDE&CFID=25582973&CFTOKEN=82107744 (page 7, 8) In...

Homework Statement Suppose a_n > 0, s_n =a_1 + ... + a_n, and \sum a_n diverges, a) Prove that \sum \frac{a_n}{1+a_n} diverges. Homework Equations The Attempt at a Solution Comparison with a_n fails miserably.

As the thread title suggests, I'm having trouble realizing when the divergence theorem is applicable and when it is not. In some examples, I am instructed not to use it because it doesn't hold but on others I can use it. My first instinct was that it doesn't apply when the vector field isn't...

Homework Statement Lets say that I have some sequence (a_n) which converges to 0 at infinity and that for all n a_{n+1} < a_n but the sequence (a_n) diverges. Now I know that the series (cos(n) a_n) converges but can I use the following argument to prove that |cos(n) a_n| doesn't...

Homework Statement Well I am studying for my final which is in a couple of days, and I am stuck on this topic of convergence of improper integrals. I've been doing a couple and I thought I was doing them correctly but my answers do not go with the answers I was given. So I am stressing out...

Homework Statement Let \vec{F}=xyz\vec{i}+(y^{2}+1)\vec{j}+z^{3}\vec{k} And let S be the surface of the unit cube in the first octant. Evaluate the surface integral: \int\int_{S} \nabla\times \vec{F} \cdot \vec{n} dS using: a) The divergence theorem b) Stoke's theorem c)...

I need help identifying if it converges or diverges or conditionally converges. \Sigma(-1)^{k}\frac{(k+4)}{(k^{2}+k)} First I want to test for absolute convergence, and comparing this limit to 1/k I get that it diverges. Since it diverges, I need to test it now using the Alternating...

Homework Statement Let D be an area in R^3 and S be its surface. D fulfills the Divergence theorem. Let N be the unit normal on S and let the volume, V, be known. Let (\overline{x},\overline{y}, \overline{z}) coordinates of the centre of mass of D be known (and the density delta is...