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. S

    Convergent sequence property and proving divergence

    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...
  2. A

    Surface integral or Divergence Theorem confused?

    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...
  3. A

    Flux through a box? And divergence as a limit?

    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...
  4. K

    Problem interpreting the divergence result

    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...
  5. K

    Calculating the Divergence of a Vector Field

    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...
  6. Z

    Divergence of Spherical Coordinates

    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...
  7. Krizalid1

    MHB An interesting series divergence

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

    Calculating divergence using covariant derivative

    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...
  9. M

    Homo Sapien vs. Chimpanzee - Divergence Timeline

    What is wrong with this logic? Homo Sapien 1. 46 (diploid) chromosomes 2. 32,185 genes 3. 3,079,843,747 bases (DNA bases A,C,T,G) 4. Homo sapines diverged from chimpanzees 6 million years ago. 5. There is a 3% difference in the genetic makeup of a homo sapien and a chimpanzee. 6...
  10. N

    Determining of a sequence is convergent or divergence

    Homework Statement For x_{n} given by the following formula, establish either the convergence or divergence of the sequence X = (x_{n}) x_{n} := (-1)^{n}n/(n+1)Homework Equations The Attempt at a Solution This is for my real analysis class. I tried to use the squeeze theorem, but didn't get...
  11. F

    Does the Complex Series Sum of (n!)^3/(3n)! * z^n Diverge?

    \displaystyle\sum_{n = 1}^{\infty}\frac{(n!)^3}{(3n)!}z^n , \ z\in\mathbb{C} By the ratio test, \displaystyle L = \lim_{n\to\infty}\left|\frac{[(n + 1)!]^3 z^{n + 1} (3n)!}{[3(n + 1)]! (n!)^3 z^n}\right| = \lim_{n\to\infty}\left|\frac{z (n + 1)^2}{3}\right| = \infty. Therefore, the series...
  12. N

    Why we do not consider the divergence due to mass-shell in QED?

    Please teach me this: Why we do not consider the divergences in loops in QED when p^{2}→m^{2} but only consider the soft photon when k^{2}→0(IR divergence) and UV divergence? Thank you very much in advance.
  13. DryRun

    Evaluate using divergence theorem

    Homework Statement http://s1.ipicture.ru/uploads/20120120/eAO1JUYk.jpg The attempt at a solution \int\int \vec{F}.\hat{n}\,ds=\int\int\int div\vec{F}\,dV where dV is the element of volume. div\vec{F}=3 Now, i need to find dV which (i assume) is the hardest part of this problem. I've drawn the...
  14. DryRun

    Green's, Gauss divergence and Stoke's theorems

    Homework Statement What's the difference between Green's theorem, Gauss divergence theorem and Stoke's theorem? The attempt at a solution I'm struggling to understand when i should apply each of those theorems. Here is what i understand. Please correct my statements below, if needed. Green's...
  15. H

    How does the del operator change with incompressibility assumption?

    I'm trying to understand why the del operator is working a certain way. So in my literature there is a term: \nabla \cdot \rho_a \mathbf{v} but then after saying that \rho_a=w_a\rho the term can somehow become \rho (\mathbf{v}\cdot \nabla w_a) I do not understand how nabla and the...
  16. S

    How to find flux through certain sides of a surface with divergence theorem.

    Homework Statement Given a vector field \textbf{F} and a composite (with this I mean cuboids, cylinders, etc. and not spheres for example) surface S, how do I calculate the flux through only some of the sides of S? I am interested in a general way to do this, but right now I am struggling with...
  17. K

    Cancellation of infrared divergence

    Infrared contribution of vertex correction gives an infinity, the resolution is to add infrared bremsstrahlung contributions as well, I can follow the math, but I'm not so convinced by the justification of this resolution given by Peskin(and Weinberg, and whatever I can find on internet)...
  18. D

    Problem applying divergence theorem to wave equation

    I'm an undergrad doing research in PDE and my adviser gave me some material to read over the holiday. But I'm getting stuck at the beginning where the divergence theorem is applied to a calculation. Maybe somebody can help me? Without getting too detailed about the context of the problem...
  19. N

    Confirming divergence theorm example

    Hello, I am having trouble confirming that the flux integral is equal to the divergence over a volume. I am making a silly mistake & its just one of those days that I can't eyeball it. Here is the problem. I want to compute the flux integral for \vec{ F}=x\hat i+y\hat j-z\hat k...
  20. T

    Determining convergence or divergence

    Homework Statement Use a valid convergence test to see if the sum converges. Ʃ(n^(2))/(n^(2)+1) Homework Equations Well, according to p-series, I'd assume this sum diverges, but I don't know which test to use. The Attempt at a Solution I probably can't do this, but I was think...
  21. R

    Proof involving divergence and gradients

    del^2(\Phi^2)=( 2\Phidel^2)(2||grad\Phi||^2) typing out my entire solution will take me ages so I'm going to verbally explain what I've done. I tried to work on the right side of the equation to compress it and make it equal to the left side. it just isn't working. I took the magnitude of the...
  22. T

    Divergence and convergence question

    Is the sum of 2 divergent series Ʃ(an±bn) divergent? From what I have learned is that it is not always divergent. Is this true? I believe that is what the picture i included is saying, but i maybe miss interpreting it. Also, is the product of 2 divergent series divergent or convergent?
  23. T

    Divergence and rotational equal to zero - solutions?

    Hi, I'd like to know the solutions for these equations, and how to arrive at them. Is it possible to derive the general form of F(x,y,z) analytically? I'm still studying linear differential equations so I have no clue on what to do with partial differential equations... div F = 0 curl F = 0...
  24. C

    How Do You Apply the Divergence Theorem to a Non-Vector Field?

    Homework Statement Use the Divergence Theorem to evaluate ∫∫S (8x + 10y + z2)dS where S is the sphere x2 + y2 + z2 = 1. Homework Equations ∫∫S F dS = ∫∫∫B Div(F) dV The Attempt at a Solution I dunno, this isn't a vector field so I don't know how to take the divergence of it so...
  25. S

    Divergence free vector fields in R^n

    Prove that every divergence free vector field on R^n, n>1 is of the form: v(x)=SUM dAij/dxi *ej where Aij(x) is smooth function from R^n to R such that Aij(x)=-Aji(x) i.e. matrix $[Aij(x)]$ is skew symmetric for every vector x.
  26. T

    Divergence Free But Not the Curl of Any Vector

    Homework Statement So this is part of a problem set in which I have to show that a vector field is divergence free but not the curl of any vector field. LetF =\frac{<x,y,z>}{(x^2 + y^2 + z^2)^{3/2}} Then F is smooth at every point of R3 except the origin, where it is not defined. (This...
  27. T

    Correct Application of Divergence Theorem?

    Homework Statement http://img593.imageshack.us/img593/5713/skjermbilde20111204kl11.png The Attempt at a Solution I thought it seemed appropriate to use divergence theorem here: I have, div F = 0 + 1 + x = 1+x I let that 0≤z≤c. If, x/a + y/b = 1then y=b(1-x/a) x/a +z/c = 1 then...
  28. B

    Verifying Divergence Theorem with Triple/Surface Integrals

    I am trying to verify the divergence theorem by using the triple integral and the surface integral of the vector field dotted with dS. No trouble per se, I'm not sure though about one thing: I am given a function and six planes (they form a cube). When I set x=0 the vector field is given as...
  29. H

    Green's theorem or divergence theorem?

    Hi, I want to calculate the total flux but I'm not sure if I have to use Green's theorem (2D) or the divergence theorem (3D). The equation below is a modified Reynolds equation describing the air flow in the clearance of porous air bearing. \frac{\partial}{\partial\theta}(PH^3...
  30. A

    Finding the divergence or convergence of a series

    Ʃ ,n=1,∞, (2/n^2+n) Does this series converge or diverge? Im not sure how to start can i use the comparison test here?
  31. L

    Series Convergence and Divergence

    Homework Statement Determine if the following series converges or diverges. If it converges determine its sum. Ʃ1/(i2-1) where the upper limit is n and the index i=2 Homework Equations The General Formula for the partial sum was given: Sn=Ʃ1/(i2-1)=3/4-1/(2n)-1/(2(n+1) The...
  32. C

    Answer: Calc III: Find Div(F) in Terms of r

    Homework Statement Let r = x i + y j + z k and R = |r|. Let F = r/R^p. find div(F) in terms of r.. i can't figure out how to express it in therms of r Homework Equations div(F) = the gradient added together
  33. B

    Writing on Stoke's, Green's, or Divergence theorem

    I suppose this has to go under homework, so here it goes: I'm in Calc III and we won't have enough time to cover the last chapter in the textbook about Stokes theorem, Green's theorem, and the divergence theorem, so instead the teacher wants a 7-page paper on something from that chapter. She...
  34. O

    Switch the divergence coordinate system

    Homework Statement i have the divergence in the (x,y,z) Cartesian as \frac{dA_x}{dx}+\frac{dA_y}{dy}+\frac{dA_z}{dz} and the assignment is to transfer it to cylindrical system (r,{\phi},z), by any way i choose. Homework Equations tried with the chain rule, but i am doing...
  35. W

    Divergence Theorem: Explaining in Simple Words

    HI experts i want to know the physical significance of divergence theorem i.e how volume integral changes to surface integral - how can i explain in simple words.
  36. J

    Iteration, linear function. convergence and divergence

    Homework Statement I need to understand and prove the following: That if a>1 the function diverges, except for a special case x_0= b/(1-a). Then if a=-1 diverges for some cases and converges if x_0 is b/2. Again, not to clear on this. Homework Equations lim n →∞...
  37. C

    Divergence of Energy-momentum Tensor

    How do you prove that Maxwell's energy-momentum equation is divergence-free? I don't know whether or not I have to use Lagrangians or Eistein's tensor, or if there's a simlpler way of expanding out the tensor.. ∂_{\mu}T^{\mu\nu}=0...
  38. C

    Divergence of Energy-momentum Tensor

    How do you prove that the energy-momentum tensor is divergence-free? ∂μTμν=0
  39. T

    Convergance and Divergence Could someone go over a routine to help determine them?

    Hey there.. Basically I'm struggling with convergence and divergence of series. I can see if converges and diverges by common sense and thinking through in my head but I struggle to write it down. The definitions in books seem confusing. Are there any steps I can systematically do every...
  40. P

    Does This Alternating Series Diverge?

    Homework Statement \sum_{n=2}^{\infty} \frac{(-1)^n}{\sqrt{n} + (-1)^n}Homework Equations This is in the section covering alternating sequences. Leibniz's rule, conditional/absolute convergence, Dirichlet's test, and Abel's tests were all covered.The Attempt at a Solution I don't know what to...
  41. B

    I am trying to summarise the concept of divergence. Say I have a

    I am trying to summarise the concept of divergence. Say I have a vector field, that is radially spreading outwards from the (0,0), but all vectors are equal in each point. So there are no deviations in magnitude in vectors(is that even possible?), but the field lines are spreading like in...
  42. B

    How Do Divergence Theorem Variants Apply to Vector and Tensor Fields?

    Homework Statement So I got three things to figure out: 1- ∫Curl u dV=∫u χ n dS 2- ∫div Tu dV=∫TT n. udS 3- ∫div θu dV=∫n.θu udS where n defines the outward normal to the boundary S θ is a smooth scalar-valued function u is a smooth vector-valued function T is a smooth tensor-valued function...
  43. T

    Is it possible to derive Divergence Theorem from Stokes Theorem?

    Is it? If so, can you show how? Thanks :smile:
  44. F

    Divergence Simplification/Identities

    Quick question… what does the following simplify to? Can it be written in any other way? \nabla\bullet (a \bullet b)b where a and b are vectors. Thanks,
  45. T

    Divergence, curl of normal vector

    How do you interpret the divergence or curl of the unit normal defined on a surface? This sometimes comes up when applying Stokes' theorem. A simple example would be Surface area = \int_{S} \hat{n} \cdot \hat{n} dA = \int_{V} \nabla \cdot \hat{n} dV where S is the closed surface that...
  46. D

    Show that the vector has zero divergence

    Homework Statement Show that the vector v = \frac{\hat{r}}{r2} (not sure why formatting isn't working?) v = (r-hat) over (r squared) has zero divergence (it is solenoidal) and zero curl (it is irrotational) for r not equal to 0 Homework Equations div(V) = (d/dx)V_x + (d/dy)V_y +...
  47. N

    What is the origin of infrared divergence?

    Please teach me this: It seem that the ultraviolet divergence has origin of we unknow the physics at very small distance(very large momentum,then very small distance).So we must cut off the very large momentum by renormalization procedure.But I do not understand the origin of infrared...
  48. T

    Einstein Tensor Divergence Proof: How to Show it is Divergence-Free?

    Hi everyone! I'm having a lillle problem proving that the einstein tensor is divergence free! I don't know how to begin, i start with \nabla_\mu G^{\mu\nu}=\nabla_{\mu}(R^ {\mu\nu} -\frac{1}{2}g^{\mu\nu}R) i tried to do \nabla_\mu...
  49. I

    Divergence in spherical polar coordinates

    I took the divergence of the function 1/r2\widehat{r} in spherical coordinate system and immediately got the answer as zero, but when I do it in cartesian coordiantes I get the answer as 5/r3. for \widehat{r} I used (xi+yj+zk)/(x2+y2+z2)1/2 what am i missing?
  50. T

    Curl and divergence of the conjugate of an holomorphic function

    I noted that if [itex]f : C \to C[\itex] is holomorphic in a subset [itex]D \in C[\itex], then [itex]\nabla \by \hat{f} = 0, \nabla \dot \hat{f} = 0[\itex]. Moreover, those two expressions are equivalent to the Cauchy-Riemann equations. I'm rewriting this in plaintext, in case latex doesn't...
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