Divergence Definition and 775 Threads

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...

The series from n=1 to infinity log(n/(n+1)). This was on my quiz, which I got wrong. Here's what I did: lim n-->infinity of log(n/(n+1)) so then that becomes: log(lim n-->infinity n/(n+1)) which becomes the log1, which is 0, so it converges. Whats wrong with my steps?

Homework Statement Consider the following vector field in cylindrical polar components: F(r) = rz^2 r^ + rz^2 theta^ By directly solving a surface integral, evaluate the flux of F across a cylinder of radius R, height h, centred on the z axis, and with basis lying on the z = 0 plane. Using the...

Homework Statement Use Divergence theorem to determine an alternate formula for \int\int u \nabla^2 u dx dy dz Then use this to prove laplaces equation \nabla^2 u = 0 is unique. u is given on the boundary.Homework Equations u \nabla^2 u = \nabla * (u \nabla u) -(\nabla u)^2 The Attempt at...

Homework Statement The problem statement has been attached with this post. Homework Equations I considered u = ux i + uy j and unit normal n = nx i + ny j. The Attempt at a Solution I used gauss' divergence theorem. Then it came as integral [(dux/dx) d(omega)] + integral...

Hi, I have a question about the divergence of forward Coulomb (Bhabha/Moller) scattering. I guess the classical analog of it is the Rutherford cross-section divergence, but that can be explained by the infinite impact parameter. In the QED version - the Bhabha/Moller scattering, it is...

Hi, I have a question about the divergence of forward Bhabha/Moller scattering. I guess the classical analog of it is the Rutherford cross-section divergence, but that can be explained by the infinite impact parameter. In the QED version - the Bhabha/Moller scattering, it is the matrix...

Homework Statement Calculate div v. v= r sin(θ) r + r sin(2θ) cos(φ) θ + r cos(2θ) φ. Homework Equations The Attempt at a Solution I've never had to do a problem like this using spherical coords, so I am not sure where to start. I have the general formula though.

From Partial Differential Equations: An Introduction, by Walter A. Strauss; Chapter 1.5, no.4 (b). Homework Statement "Consider the Neumann problem (delta) u = f(x,y,z) in D \frac{\partial u}{\partial n}=0 on bdy D." "(b) Use the divergence theorem and the PDE to show that...

Hi everyone, so let me introduce the scalar function \Phi = -(x2+y2+z2)(-1/2) which some of you may recognize as minus one over the radial distance from the origin. When I compute \nabla2\Phi is get 0. Now if I do the following integral on the surface S of the unit sphere x2+y2+z2= 1 ...

Homework Statement Show that the following series diverges \sum_{n=1}^{\infty}\frac{n!}{2^{n}} Homework Equations The Divergence Test: In order for a series to be divergent, the following must be true \lim_{n\rightarrow \infty} a_n \neq 0 , or \lim_{n\rightarrow \infty} a_n \nexists...

Homework Statement Check the divergence theorem using the function: \mathbf{v} = y^2\mathbf{\hat{x}} + (2xy + z^2) \mathbf{\hat{y}} + (2yz)\mathbf{\hat{z}} Homework Equations \int_\script{v} (\mathbf{\nabla . v }) d\tau = \oint_\script{S} \mathbf{v} . d\mathbf{a} The Attempt at a Solution...

The D field due to a point charge(Q) has a divergence zero D = (Q/(4*pi*r2)) ar (ar --- unit vector) if we calculate the divergence we get zero. but guass law says that the divergence is a finite value not zero because a charge is present in there...

\operatorname{div}\,\mathbf{F}(p) = \lim_{V \rightarrow \{p\}} \iint_{S(V)} {\mathbf{F}\cdot\mathbf{n} \over |V| } \; dS This is the definition of divergence from wikipedia... The divergence is property of a point in space. Is that right? If the divergence is zero at a point, that...

Determine Div & Curl from a given vector field V=Kyi-Kxj How do I format this? It's been a while since I've done this and every divergence and curl example I look up has the format V(x,y,z)={V1(x,y,z);V2(x,y,z);V3(x,y,z)} Should I reformat my V to be V{x,y}={V1(x,y);V2(x,y)}={Ky,-Kx}...

Homework Statement Evaluate the integral \int\limits_{V=\infty} e^{-r} \left[ \nabla \cdot \frac {\widehat{r}} {r^2} \right] , d^3 xHomework Equations Divergence theorem: \int\limits_{V} \left ( \nabla \cdot A \right ) \, d^3 x = \oint\limits_{S} A \cdot \, da} The Attempt at a Solution I...

look for some proof for the formula of the divergence operator in cylindrical & spherical coordinate is there any on the net ? TNX ! the formula here: http://hyperphysics.phy-astr.gsu.edu/hbase/diverg.html

Homework Statement Let the surface, G, be the paraboloid z = x^2 + y^2 be capped by the disk x^2 + y^2 \leq 1 in the plane z = 1. Verify the Divergence Theorem for \textbf{F}(x,y,z) = 2x\textbf{i} - yz\textbf{j} + z^2\textbf{k} Homework Equations I have solved the problem using the...

Homework Statement This problem takes a bit of background, so please bear with me. The assignment reads: Suppose you have a large supply of books, all the same size, and you stack them at the edge of a table, with each book extending farther beyond the edge of the table than the one...

Hi all, I'm trying to fill in some mathematical steps from a derivation I'm reading in a research paper. I want to see that the following expression \text{Div}\left(\nabla u\frac{ \phi ' \left(|\nabla u|\right)}{|\nabla u|}\right) is equivalent to (\frac{\phi '...

So I have a simple/easy to answer question for any physics buffs out there. I think I'm doing something fundamentally flawed. Can you take the inverse of a divergence? analagous to antiderivative-integral? e.g., I want to find J from the continuity equation with a known \rho(\vec{r},t)...

Here are five separate problems. Show that the series 1/3^(ln(n)) converges and that the series 1/2^(ln(n)) diverges. integral (sqrt(x)*e^-sqrt(x)) dx integral (x/(sqrt(x-1)+2)) dx integral (1/(2+sin(x)+cos(x) integral (1-cos(x))^(5/2)) dx There are no other relevant...

Homework Statement I'm getting through a paper and have a few things I can't wrap my head around. 1. In defining the boundary conditions for a membrane (a function of vector 'r'), the author claims that for a small displacement (u) and a boundary movement (f), the boundary condition can be...

I know how to calculate the divergence and curl of a vector field. What I am lacking is any intuition on what these values mean. example: V= {x, y, z} ∇.V = 3 ∇xV = {0,0,0} F={-y, x, 0} ∇.F = 0 ∇xF = {0,0,2} G={0, 3y, 0} ∇.G = 3 I understand that that the divergence is a measure of how much...

I've been told time and again that any beam of light (think laser) exhibits divergence no matter how perfect. This prompts three questions: 1) Theoretically, if a laser beam is emitted such that each photon is exactly parallel, (obviously more perfect than can be achieved in reality) does the...

The Wikipedia article Divergence, in the section Application to The Wikipedia article Divergence, in the section [i]Application to Cartesian coordinates, says of the del-dot formula for divergence, "Although expressed in terms of coordinates, the result is invariant under orthogonal...

Homework Statement 1. Consider a cube with vertices at A=(0,0,0) B=(2,0,0) C=(2,2,0) D=(0,2,0) E=(0,0,2) F=(2,0,2) G=(2,2,2) H=(0,2,2) A)Calculate the flux of the vector fieldF=xi through each face of the cube by taking the normal vectors pointing outwards. B)Verify Gauss's divergence theorem...

I am stuck on this problem. Use these equations: \textbf{v}(\textbf{r}) = f(r)\textbf{r} \frac{\partial r}{\partial x} = \frac{x}{r} And the chain rule for differentiation, show that: (\nabla\cdot\textbf{v}) = 2f(r) + r\frac{df}{dr} (cylindrical coordinates) Any help greatly...

Homework Statement Verify Gauss Divergence Theorem ∭∇.F dxdydz=∬F. (N)dA Where the closed surface S is the sphere x^2+y^2+z^2=9 and the vector field F = xz^2i+x^2yj+y^2zk The Attempt at a Solution I have tried to solve the left hand side which appear to be (972*pi)/5 However, I...

Homework Statement A smooth vector field F has divF(1,2,3) = 5. Estimate the flux of F out of a small sphere of radius 0.01 centered at the point (1,2,3). Homework Equations Cartesian Coordinate Definition of Divergence: If F= F1i + F2j +F3k, then divF=dF1/dx + dF2/dy + dF3/dz The...

Homework Statement I have to prove the divergence formula for plane polars. The question goes something like: Find the divergence of the vector field F(r,t) = Frer + Ftet where r and t are polar coordinates and er = (cos t, sin t, 0) and et = (- sin t, cos t, 0) (t is theta in the...

Currently, we are covering the topic of convergence and divergence of a series in my calculus 2 class. I was wondering if you could give me in there own words what it means for a series to converge, and what it means for a series to diverge. I know that when a series converges, its limit...