Divergence Definition and 775 Threads

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. Vectronix

    Vector Calculus: Question about the origin of the term 'divergence'

    Why is the divergence operation called the 'divergence?' What is the significance of this operation on a vector-valued function? And what about "the curl?" The curl seems self-explanatory (at least it does in electrodynamics), but I need someone to expound on 'the curl' as well.
  2. 0

    How Does a Narrow Slit Affect Laser Beam Divergence?

    what is the effect of placing a slit of say one tenth of an inch infront of a laser beam (3 inch dia )with a divergence of say one degree upon the divergence of the emanating beam?(slit placed symmetrically along the beam axis) ? would the emanating beam also be of the same divergence ?
  3. B

    Comparing Series Divergence: \(\sum\frac{n^2-\arctan(n)}{n^3+\sin(n)}\)

    \sum (n^2-arctan(n)) / (n^3 + sin(n)) n=0 to ∞ I know this series diverges, but how would you use the comparison test to compare it to (n^2 / n^3 meaning the harmonic series 1/n) Thank you very much
  4. J

    Understanding Divergence: Exploring the Definition and Its Applications

    Statement: The definition of the Divergence is given by the following, \nabla \cdot \vec{V} \equiv lim_{\Delta v \rightarrow 0}(\frac{\int \int _{surface}\vec{V} \cdot \vec{ds}}{\Delta v}), where v is the unit volume.Relevant questions: The expression \vec{V} \cdot \vec{ds} on the right side...
  5. J

    Why Does Divergence of Electric Flux Equal Volume Charge Density?

    Question: Can someone remind me why the divergence of the electric flux is equal to the volume charge density, \nabla \bullet \vec{D} = \rho_{v} (where \vec{D} is the electric flux density). Thoughts: The divergence measures the flow of a field out of a region of space. The del operator takes...
  6. B

    How would you determine the convergence or divergence of this Series

    Homework Statement sum 1/n*sin (1/n), n=1..infinity I tried the limit comparison test, but I always get 0. The ratio test is impossible Comparison to the harmonic series cannot be used because 1/n*sin (1/n) is smaller than 1/n Can you guys help? Thanks
  7. R

    Understanding Gauss' Law to Divergence and Charge Determination

    I think after weeks of study, I'm finally getting a handle on Gauss' Law. A few ? though. The equation does not specifically state that there is not charge inside the surface. One may think that it doesn't matter if there or isn't...it's alway zero. How come that is not state in the...
  8. W

    Convergence or Divergence? Exploring the Limit of a Sum Series

    1. \sum(\sqrt{k^{2}+1}-\sqrt{k^{2}}) from K=0 to K=\infty 2. Hi all. I need some help here. I have to use a test to determine whether the sum series diverges or converges 3. I thought it was the divergence test because I thought that the limit of the sum didn't approach zero...
  9. H

    Divergence Theorem Explained: Learn the Basics

    http://img60.imageshack.us/img60/9696/21249035.jpg I am stuck at the last step. Can anyone give some hints? Thanks in advanced.
  10. B

    Evaluating \int\int_{\sigma} F.n ds with Divergence Theorem

    Homework Statement Use the divergence theorem to evaluate \int\int_{\sigma}F . n ds Where n is the outer unit normal to \sigma we have F(x,y,z)=2x i + 2y j +2z k and \sigma is the sphere x^2 + y^2 +z^2=9 Homework Equations \int\int_{s}F . dA = \int\int\int_{R}divF dV The...
  11. R

    Divergence theorem - mass flux

    Homework Statement Water in an irrigation ditch of width w = 3.0 m and depth d = 2.0 m flows with a speed of 0.40 m/s. For each case, sketch the situation, then find the mass flux through the surface: (a) a surface of area wd, entirely in the water, perpendicular to the flow; (b) a surface...
  12. P

    Solving a Tensor Field: Divergence of P = 0

    The following isn't actually a homework problem, but this seems to be the natural place to ask questions of this sort. Without further ado, I have a spherically symmetric tensor field, which is written with the help of dyadics as P(r) = P_n(r)\mathbf{e}_r\mathbf{e}_r +...
  13. J

    Evaluating Integral F dot dA using Divergence Theorem

    Homework Statement use the divergence theorem to evaluate the integral F dot dA F = (2x-z)i + x2yj + xz2k s is the surface enclosing the unit cube and oriented outward Homework Equations The Attempt at a Solution is the the region from -1 to 1 for x y and z div F = x2 +...
  14. T

    Compute Surface Integral: Divergence Theorem & F=(xy^2,2y^2,xy^3)

    Use the divergence theorem to compute the surface integral F dot dS , where F=(xy^2, 2y^2, xy^3) over closed cylindrical surface bounded by x^2+z^2=4 and y is from -1 to 1. I've tried doing it and got 32pi/3 (i guess its wrong, so how to do it?) Is it ok to compute Div F in terms of xyz and...
  15. D

    Finding the curl and divergence

    Homework Statement \vec{F}(x,y,z) = x^2y\vec{i} + y^2z^3\vec{j} + xyz\vec{k} Homework Equations The Attempt at a Solution I got: Curl: (xz - 3y^2z^2)\vec{i} + (-yz)\vec{j} + (-x^2)\vec{k} Div: 2xy + 2yz^3 + xy Are these right?
  16. X

    Easy divergence theorem problem

    Evaluate the flux integral using the Divergence Theorem if F(x,y,z)=2xi+3yj+4zk and S is the sphere x^2+y^2+z^2=9 answer is 324pi so far i took the partial derivitavs of i j k for x y z and added them to get 9. so i have the triple integral of 9 dzdxdy i think u have to use polar...
  17. J

    Is Multiplying Divergence by Area Correct in Divergence Theorem Problems?

    Homework Statement F = xi + yj + zk, s = x^2 + y^2 + z^2 Homework Equations The Attempt at a Solution div F = 1+1+1=3 area of sphere = 4pi i can just multiply them to get 12pi as an answer right?
  18. J

    How Do Stokes' and Divergence Theorems Apply to a Cube's Surface Integral?

    Homework Statement Given F = xyz i + (y^2 + 1) j + z^3 k Let S be the surface of the unit cube 0 ≤ x, y, z ≤ 1. Evaluate the surface integral ∫∫(∇xF).n dS using a) the divergence theorem b) using Stokes' theorem Homework Equations Divergence theorem: ∫∫∫∇.FdV = ∫∫∇.ndS Stokes...
  19. J

    Evaluating ∫∫(∇xF).n dS: Divergence vs. Stokes' Theorem

    Given F = xyz i + (y^2 + 1) j + z^3 k Let S be the surface of the unit cube 0 ≤ x, y, z ≤ 1. Evaluate the surface integral ∫∫(∇xF).n dS using a) the divergence theorem b) using Stokes' theorem --- Since the divergence theorem involves a dot product rather than a curl,how would it...
  20. E

    Flux of Vector Field within Sphere: Find Flux of Given Vector Field

    Homework Statement Suppose \vec{G} is a vector field with the property that div\vec{G} = 5 for 2 \leq ||\vec{r}|| \leq 14 and that the flux of \vec{G} through the sphere of radius 4 centered at the origin is 20\pi . Find the flux of through the sphere of radius 12 centered at the...
  21. E

    Calculating Flux and Applying the Divergence Theorem

    Homework Statement http://img16.imageshack.us/img16/88/fluxm.th.jpg Homework Equations The Attempt at a Solution I've tried to find the divergence of F and I got 3x^2 + 3y^2 + 3z^2 and as this is a variable I need to set up the integral... how do I set the integral
  22. C

    Divergence and Curl of Unit Vectors?

    Homework Statement http://img4.imageshack.us/img4/4218/divergenceandcurl.jpg The Attempt at a Solution Totally confused on what the question's asking. Wouldn't the divergence of say x_hat be the partial of x_hat over x which is just 0? So every answer would just be 0 or something? Same...
  23. J

    Series; convergence, divergence

    series; convergence, divergence... Homework Statement 1. sum(infinity,n=1) n!/1.3.5...(2n-1) 2. sum(infinity, n=1) (-1)^n arcsin(-1/n) 3. sum(infinity, n=0) arcsin(1/n^2) / arctan(1/n^2) The Attempt at a Solution 1. i used the ratio test and then i ended up with lim((n+1)(2n-1)/2n+1))...
  24. R

    Convergence - Divergence of a Series

    Homework Statement Test the series for convergence or divergence -1/3+ 2/4 - 3/5 +4/6 - 5/7 + .... Homework Equations I think it's an alternating series The Attempt at a Solution I found that an = (-1) ^n * n/ (n+2) And it approaches 1 as n goes to inifty so the series...
  25. J

    Transforming divergence from cartesian to cylindrical coordinates

    Homework Statement Compute the divergence in cylindrical coordinates by transforming the expression for divergence in cartestian coordinates. Homework Equations F = F_x i + F_y j + F_z k div F = ∂F_x/∂x + ∂F_y/∂y + ∂F_z/∂z ... (divergence in Cartesian coordinates) I need to...
  26. J

    Coordinate systems for divergence

    Homework Statement Compute the divergence in cylindrical coordinates by transforming the expression for divergence in cartestian coordinates. Homework Equations F = F_x i + F_y j + F_z k div F = ∂F_x/∂x + ∂F_y/∂y + ∂F_z/∂z ... (divergence in cartesian coordinates) I need to...
  27. H

    Verification of the Divergence Theorem

    The question I was given asks to verify the divergence theorem by showing that both sides of the theorem show the same result. With the divergence theorem obviously being \iint_S\mathbf{F}\cdot\mathbf{n}\,dS = \iiint_V \nabla\cdot\mathbf{F}\,dV . The vector field is...
  28. B

    Is the Continuity Equation a Restatement of Gauss' Divergence Theorem?

    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...
  29. S

    Quick Divergence Theorem question

    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)...
  30. E

    Divergence theorem in four(or more) dimension

    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.
  31. L

    Determining Field from Div & Curl: Examples & Techniques

    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...
  32. S

    Understanding Divergence: Unit Vectors & Magnitude

    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...
  33. L

    Proving Divergence Theorem using Gauss' Theorem

    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?
  34. A

    The Convergence of the Series (sqrt(k+1) - sqrt(k))/k

    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...
  35. L

    Using Gauss' Theorem to Show Integral Convergence in Divergence Theorem

    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}...
  36. Saladsamurai

    Check Divergence Theorem on Unit Cube

    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...
  37. Fredrik

    Divergence defined from volume element

    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...
  38. K

    Convergence and divergence help

    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...
  39. J

    Divergence theorem over a hemisphere

    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 =...
  40. Saladsamurai

    Surface Integral using Divergence Theorem

    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...
  41. T

    What is the solution to this Divergence Theorem homework problem?

    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...
  42. B

    Divergence theorem - sign of da

    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...
  43. Д

    Proving Divergence of Harmonic Series

    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...
  44. L

    Calculate Flux of Vector Field in Closed Surface | Divergence Problem Solution

    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...
  45. B

    Determining Divergence or Convergence in Series

    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...
  46. I

    Maxwell Equations when current density is time independent and divergence free

    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}...
  47. M

    Does Divergence of a Series Imply Divergence of Its Absolute Values?

    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...
  48. L

    Does the Divergence Theorem Apply to Complex Vector Fields and Hemispheres?

    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...
  49. B

    DGLAP: Kollinear divergence in splitting functions?

    Hello all! I am just preparing for an oral exam in QCD and try to figure out the interplay of UV and IR divergences, regularisation, renormalization and virtual diagrams. As to now, my idea of the whole thing is this: 1) In lowest order no divergences occur 2) In second order we get UV...
  50. F

    Convergence or Divergence of a Series with Multiplication Terms?

    \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!
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