What is Divergence: Definition and 770 Discussions

In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.
As an example, consider air as it is heated or cooled. The velocity of the air at each point defines a vector field. While air is heated in a region, it expands in all directions, and thus the velocity field points outward from that region. The divergence of the velocity field in that region would thus have a positive value. While the air is cooled and thus contracting, the divergence of the velocity has a negative value.

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  1. N

    Using the div-flux theorem (Gauss) to derive divergence in polar coördinates?

    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?
  2. N

    Physical Motivation for Curl and Divergence

    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...
  3. L

    Energy Momentum Tensor and Divergence Theorem

    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...
  4. M

    Divergence and surfaces integral, very hard

    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...
  5. marcus

    Lorentzian spinfoam model free of IR divergence (Muxin Han)

    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...
  6. A

    The divergence of 1/r^2 fields

    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...
  7. W

    C/C++ How to calculate rotation, divergence in C/C++

    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...
  8. A

    Simple Convergence / Divergence Calc 2

    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...
  9. Saladsamurai

    Calculate Divergence using limit definition

    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}} -...
  10. P

    Divergence of second-order Tensor

    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}...
  11. R

    Need help on this series test for convergence or divergence

    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...
  12. M

    Series with divergence: Quick easy question

    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...
  13. P

    Divergence in cylindrical coordinates

    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...
  14. S

    I'm not completely sure that this is right, but it seems like it should be.

    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...
  15. M

    Convergence and Divergence of a series

    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?
  16. H

    Divergence theorem and surface integrals

    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...
  17. D

    Laplace and Divergence theorem

    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...
  18. C

    Gauss Divergence Theorem - Silly doubt - Almost solved

    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...
  19. X

    Divergence of forward Coulomb scattering?

    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...
  20. X

    Divergence of forward Bhabha/Moller scattering

    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...
  21. M

    Calculating Divergence With Spherical Coords

    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.
  22. P

    Neumann Problem: Use the divergence theorem to show it has a solution

    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...
  23. E

    Why is the Divergence Theorem failing for this scalar function?

    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 ...
  24. J

    Can Factorials Cause Divergence? Investigating the Divergence Test for Series

    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...
  25. V

    Divergence Theorem: Check Function w/y^2, 2x+z^2, 2y

    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...
  26. B

    Point charge(Q) has a divergence zero

    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...
  27. T

    Trying to understand the concept of divergence

    \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...
  28. L

    Calculate Div & Curl from V=Kyi-Kxj

    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}...
  29. S

    What is the integral of a vector field with the divergence theorem?

    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...
  30. S

    Divergence operator in cylindrical & sherical

    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
  31. K

    What Went Wrong in My Verification of the Divergence Theorem?

    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...
  32. M

    Proving divergence for a tricky series

    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...
  33. S

    Developing a divergence expression

    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 '...
  34. I

    Inverse/Anti - Divergence? Maxwells Eqns.

    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)...
  35. B

    A convergence and divergence test and a couple integrals

    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...
  36. T

    Boundary conditions, Sturm-Liouville, & Gauss Divergence

    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...
  37. U

    Understanding Divergence & Curl of Vector Fields

    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...
  38. R

    Photon Divergence: Questions & Answers

    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...
  39. Rasalhague

    The Wikipedia article Divergence, in the section Application to

    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...
  40. S

    Vector Calculus - Divergence theorem

    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...
  41. B

    Divergence in spherical coordinates

    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...
  42. H

    Solving Gauss Divergence Theorem on a Closed Surface

    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...
  43. T

    Vector Calculus II: Divergence

    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...
  44. K

    Proving the Divergence Formula for Plane Polars

    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...
  45. B

    Convergence and Divergence of a Series

    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...
  46. L

    Problem with taking the divergence

    hi, can someone tell me how \nabla\dot(\frac{\widehat{r}}{r^2})=4\pi\delta^3(r) thanks
  47. S

    Divergence Theorem: Find Outward Flux of F (x3,x2y,xy)

    Another question of a practice test. How do I use the Divergence theorem to find the outward flux of the field F = (x3,x2y,xy) out through the surface of the solid U = (x,y,z): 0 < y < 5-z, 0 < z < 4-x2. The answer is 4608/35.
  48. J

    Divergence of curvature scalars * metric

    How can one work out what terms like: (g^{cd}R^{ab}R_{ab})_{;d} are in terms of the divergence of the Ricci curvature or Ricci scalar? One student noted that since: G^{ab} = R^{ab} - \frac12 g^{ab}R {G^{ab}}_{;b} = 0 that we could maybe use the fact that G^{ab}G_{ab} = R^{ab}R_{ab} - \frac12...
  49. B

    How to Derive the Divergence of Magnetic Field in Cylindrical Coordinates?

    Homework Statement Ok well all I am told in the question is that the magnetic field B at a distance r from a straight wire carrying current I has magnitude uoI/2pi r.. The lines of force are circles on the wire and in planes perpendicular to it.. Show that divB = 0 Homework...
  50. S

    Real Analysis: convergence and divergence

    Homework Statement Suppose \sum n converges and an is greater than 0 for all n. Show that the sum of 1/an diverges. Homework Equations The Attempt at a Solution
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