Divergence Definition and 746 Threads

Homework Statement Homework Equations Definitely related to the divergence theorem (we're working on it): The Attempt at a Solution I'm a bit confused about multiplying a scalar field f into those integrals on the RHS, and I'm not sure if they can be taken out or not. If they can be, I...

Homework Statement ∫Bdot[∇×A]dV=∫Adot[∇×B]dV Prove this by integration by parts. A(r) and B(r) vanish at infinity. Homework Equations I'm getting stuck while trying to integrate by parts - I end up with partial derivatives and dV, which is dxdydz? The Attempt at a Solution I...

Homework Statement Given \nabla\frac{1}{r}, show \nabla\bullet\nabla\frac{1}{r} = -4πδ(r), where δ(r) is the delta dirac function.The Attempt at a Solution I've used divergence theorem and also solved the equation itself, so I know that outright solving is zero and the divergence theorem gives...

We expect the stress-energy tensor to have zero divergence, because this is required for local conservation of energy-momentum, which has been verified to high precision in laboratory and solar system experiments. The standard review article is Will, "The Confrontation between General Relativity...

We were given an electric field defined by Kr^3 , and asked to calculate what the total flux would be given a sphere of a radius R. I had already calculated the divergence of E to be equal to 5kr^2 . So the first integral is calculating what the divergence over the area of the sphere is...

say you have a function f(x,y) \nablaf= \partialf/\partialx + \partialf/\partialy however when y is a function of x the situation is more complicated first off \partialf/\partialx = \partialf/\partialx +(\partialf/\partialy) (\partialy/\partialx) ( i wrote partial of y to x in case y was...

I work with a grid-based code, this means that all of my quantities are defined on a mesh. I need to compute, for every point of the mesh the divergence of the velocity field. All I have is, for every cell of my mesh, the values of the 3-d velocity in his 26 neighbors. I call neighbors the...

Hey guys, was wondering if anyone could help walk me through this problem. I am fairly sure it converges by the ratio test. Thank you. http://www.wolframalpha.com/share/clip?f=d41d8cd98f00b204e9800998ecf8427e1sgqilv5iv

why do most modern books claim the divergence of the E field is ρ/ε_{0} but in more classical books, and when you actually derive it mathematically you arrive at 4πρ

Please Someone explain why: 1.div(F×G)=GcurlF-FcurlG 2.curl(F×G)=F.divG-G.divF+(G.∇).F-(F.∇).G

Homework Statement Verify that the infinite series diverges. I have the series from n=1 to infinity of (2^(n)+1/2^(n+1) Homework Equations Nth term test(This is the way the book did it but I did it used the geometric series test and I just want to verify if my Algebra was correct)...

I have been studying gradience; divergence and curl. I think i understand them in cartesian coordinate system; But i don't understand how do they get such complex stuffs out of nowhere in calculating divergence in spherical and cylindrical coordinate system. Any helps; links or suggestion...

One can easily prove that \nabla \cdot f is invariant under a rotation of the reference frame, however I would like to prove that the divergence operator itself is invariant (same principle, different approach). In other words I want to prove that \mathbf \nabla = \mathbf e_x...

Homework Statement use divergence theorem to evaluate ∫s∫F dot n dA if F=[sinh yz, 0, y4] , S: r=[u,cosv,sinv], -4≤u≤4 , 0≤v≤pi The Attempt at a Solution Instructor surprised us with this one, I have no idea how to attempt. I know that ∫vdiv v dV=∫sn dot v dA, which is the...

Hi all I basically have two questions that are very closely related to each other about divergence, specifically the divergence of a vector function 1/r2\widehat{r} First, I will be referencing pages 17, 18, and 45 from the 3rd edition of Intro to Electrodynamics. The first question...

So I am wondering about one thing. The charged propagators in weak theory are W+- bosons. The mathematical expression for them, while drawing the Feynman diagrams is: -i\frac{g_{\mu\nu}-\frac{q_\mu q_\nu}{m_W^2}}{q^2-m_w^2}. The problems that are usually given to me are simple and involve...

When a vector field representing a physical quantity (e.g. B) has ∇\cdotB = 0 what is then the physical interpretation of this? Some people have said that the field doesn't diverge away from anything, but as far as I can tell magnetic field can easily get weaker and weaker the further you go...

Homework Statement Compute the flux of F=xi+yj+zk through the curved surface of the cylinder x2+y2=1 bounded below by the plane x+y+z=1, above by the plane x+y+z=7, and oriented away from the z-axis. Homework Equations div(F) = (dF/dx) + (dF/dy) + (dF/dz) The Attempt at a Solution...

The divergence of electric field at a point is proportional to the charge density at the point. Divergence is the rate of change with distance, the rate of change of electric field due to a distant charge is not zero, so how can it be said that the divergence at a point depends only on the...

A ZERO Divergence Vector Field There is theorem that is widely used in physics--e.g., electricity and magnetism for which I have no proof, yet we use this theorem at the drop of a hat. The theorem is this: Given sufficient continuity and differentiability, every vector function A such that...

I had a bit of trouble in testing series like this for convergence $$\sum_{ n=1 }^{ \infty } \frac { 1 }{ 2n+1 } $$ If by the comparison test, ##\frac{ 1 }{ 2n+1 } < \frac{ 1 }{ 2n }## for all of n>0, and ##\lim_{ n \rightarrow \infty} \frac{ 1 }{ 2n }## =0, then the series should be...

Homework Statement Prove the divergence theorem for the vector field A = p = (x,y) and taking the volume V to be the cylinder of radius a with its base centred at the origin, its axis parallel to the z-direction and having height h. I can find the dV side of the equation fine (I think)...

Homework Statement Evaluate the surface integral F * dr, where F=<0, y, -z> and the S is y=x^2+y^2 where y is between 0 and 1. Homework Equations Divergence theorem The Attempt at a Solution I just got out of my calculus final, and that was a problem on it. I used the divergence theorem...

Homework Statement Find the Divergence or Convergence of the series \sum^{∞}_{n=1}\frac{2n^2+3n}{\sqrt{5+n^5}} Homework Equations Ratio Test, Comparison Test, Limit Comparison Test, Integral test etc. The Attempt at a Solution This question was on my final exam and the only question of...

I'm trying to make a little manipulate/interactive box that shows the vector divergence of the E-field coming from a sphere. I have no idea how to start as I'm really new to Mathematica. Does anyone have any pointers? I can't find anything particularly helpful on the Wolfram reference or...

Homework Statement Homework Equations So I have that v \otimes n = \left( \begin{array}{ccc} v_{1}n_{1} & v_{1}n_{2} & v_{1}n_{3} \\ v_{2}n_{1} & v_{2}n_{2} & v_{2}n_{3} \\ v_{3}n_{1} & v_{3}n_{2} & v_{3}n_{3} \end{array} \right) The Attempt at a Solution I've tried applying the...

Homework Statement Determine whether the series is absolutely convergent, conditionally convergent, or divergent. ##\sum _{n=1}\left( -1\right) ^{n}\dfrac {n} {n^{2}+1}## the sum goes to infinity. Homework Equations Theorem for absolute convergence. Test for divergence The Attempt...

Homework Statement Ʃ[(-1)^n (cosn)^2]/√n The Attempt at a Solution i don't have the slightest clue where to start

Homework Statement ##\sum _{n=1}^{\infty }\left[ \left( -1\right) ^{n}\right] \dfrac {\sqrt {n}} {1+2\sqrt {n}}##Homework Equations Alternating Series test, Absolute convergence theorem, p-series, and test for divergence. The Attempt at a Solution The alternating series test tells us that the...

Homework Statement Folks, Verify the divergence theorem for F(x,y,z)=zi+yj+xk and G the solid sphere x^2+y^2+z^2<=16 Homework Equations ##\int\int\int div(F)dV## The Attempt at a Solution My attempt The radius of the sphere is 4 and div F= 1, therefore the integral...

Homework Statement There are 3 parts to this problem: (a) \; \sum^{\infty}_{n=1} \frac{n^4}{4^n} (b) \; \sum^{\infty}_{n=1} \left( \frac{n+8}{n} \right)^n (c) \; \sum^{\infty}_{n=1} \frac{5^n-8}{4^n+11} The attempt at a solution (a) I've used the Ratio test. So, u_n=\frac{n^4}{4^n} and...

Homework Statement ##\sum _{n=1}ne^{-n}## Homework Equations Ratio Test Integral Test The Attempt at a Solution I know that by the ratio test, it converges absolutely. But, I am unable to determine its convergence through the integral test . Could someone help? I thought that the...

Hi, can you tell me which theorem they have used here: http://everything2.com/title/proof+that+the+sum+of+the+reciprocals+of+the+primes+diverges i'm thinking on part: Well, there's an elementary theorem of calculus that a product (1-a1)...(1-ak)... with ak->0 converges to a nonzero value iff...

When the exercise tells me to calculate the flux, how do I know when I need to use each of these theorems (Green's, Stokes or Divergence)? Can anyone tell me the difference between them? I'm a LOT confused about this. If anyone knows any good material about this on internet, it'll help me a...

Homework Statement ##\sum \dfrac {1+2^{n}} {3^{n}}## According to Wolfram Alpha the sum is 5/2. But, I think that my method is fine and shows another result. The Attempt at a Solution ##\sum \dfrac {1+2^{n}} {3^{n}}=\sum \left[ \left( \dfrac {1} {3}\right) ^{n}+\left( \dfrac {2} {3}\right)...

Ʃ n=1 to infinity of cos(n∏) letting an=cos(n∏), I rewrote this as (-1)^n=an. Using the nth term test i let the limit as n->∞ go to infinity. This value bounces back and forth between positive and negative, but I know clearly the value =/= 0, therefore it diverges by the nth term test. Is...

Now for the proof of convergence/divergence of the geometric series we first deduce the Nth partial sum which is given by: \frac{r(1-r^n)}{1-r} Now for 0<r<1 this become \frac{1}{1-r} which clearly converges by AOL At r>1 it's similarly obvious why it diverges. But at r=1, I'm a bit...

Homework Statement F(x,y,z) = (-x+y)i + (y+z)j + (-z+x)k Find divergence Homework Equations The Attempt at a Solution The gradient is -i + j + -k Dotting that with F, I get x - y + y + z + z - x = 2z My book lists the answer as -1. What the heck are they talking...

Homework Statement Folks, have I set these up correctly? THanks Use divergence theorem to calculate the surface integral \int \int F.dS for each of the following Homework Equations \int \int F.dS=\int \int \int div(F)dV The Attempt at a Solution a) F(x,y,z)=xye^z i +xy^2z^3 j-...

Homework Statement show that the volume enclosed by a closed surface S is given by ## \frac{1}{3} \int_{S} \vec{x} \cdot d\vec{A} ## Homework Equationsdivergence theorem The Attempt at a Solution using divergence theorem I get that ##V =\frac{1}{3} \int_{V} \nabla \cdot \vec{x} dV ## but...

[answered] I want to know why this particular approach is wrong so I can learn from my mistakes. Homework Statement a_n = \frac{ln(n^3)}{2n}The Attempt at a Solution For the sake of being time efficient, I will skip writing things like the limit as n approaches infinity etc. a_n =...

considering divergence of a sequence in the reals, a_{n}, if such a sequence → +∞ as → n, then I would like to know what type of sequence this reuqires. (excluding divergence to -∞ for now) so a_n → +∞ iif: \forall M \exists N, \forall n\geqN \Rightarrow a_n \geq M . So is the above...

I feel like I'm missing something obvious, but anyway, in the text it states: lim as n→∞ of an+bn = ( lim as n→∞ of an ) + ( lim as n→∞ of bn ) But say an is 1/n and bn is n. Then the limit of the sum is n/n = 1, but the lim as n→∞ of bn doesn't exist and this property doesn't work...

Homework Statement Find the Volume ∫∫ xy DA where R is the region bounded by by the line y=x-1 and the parabola y^2=2x+6. Homework Equations ∫∫ xy dx dy The Attempt at a Solution first i found the intersection of the above equations . which is (5,4) to (-1,-2) . then i...

Let F=(7z+8)i+2zj+(2z+7)k, and let the point P=(abc), where a, b and c are constants. In this problem we will calculate div F in two different ways, first by using the geometric definition and second by using partial derivatives. (a) Consider a (three-dimensional) box with four of its corners...

Homework Statement I take the divergence of the function: V=x^2 \boldsymbol{\hat {x}}+3xz^2\boldsymbol{\hat {y}}-2xz\boldsymbol{\hat {z}} And get zero. the answer doesn't make sense, since i expect to get a zero divergence only for a function that looks like the one in the drawing attached...

Homework Statement The question is to draw the function: V=\frac {\boldsymbol{\hat {r}}}{r^2} And to compute it's divergence: \nabla \cdot V Homework Equations \nabla\cdot V=\left ( \frac {\partial}{\partial x} \boldsymbol{\hat {x}}+\frac {\partial}{\partial y} \boldsymbol{\hat {y}}+\frac...

Homework Statement Compute the divergence of v = (1/(r^2)) r where r = sin(u)cos(v)i + sin(u)sin(v)j + cos(u)k, r^2 = x^2 + y^2 + z^2 The Attempt at a Solution I can only think to express r as a function of x,y,z and do it. I know there's a simpler way though, but it's driving me...

Prove that $\displaystyle\sum_{n=1}^\infty\frac1{n H_n}=\infty$ where $H_n$ is the n-term of the harmonic sum.

Homework Statement Using the definition of divergence d(i_{X}dV) = (div X)dV where X:M\rightarrow TM is a vector field, dV is a volume element and i_X is a contraction operator e.g. i_{X}T = X^{k}T^{i_{1}...i_{r}}_{kj_{2}...j_{s}}, prove that if we use Levi-Civita connection then the...