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. H

    Problem on divergence and curl

    Hi Guys,, i have just started to study Divergence and curl but this is not at all enetering into my mind...Pls help me out understand this...This also has Divergence and Stokes theorm ..pls help me grasp it...Thx in advance... The Divergence Theorem and Stokes's Theorem provide the...
  2. M

    Divergence Theorem for the Curl

    Hi, I'm having trouble proving the following result: \int_{V} (\nabla\times\vec{A}) dV = -\int_{S} (\vec{A}\times\vec{n}) dS I'm not sure how I should Stokes' and/or the Divergence Theorem in proving this, or if you should use them at all. Thanks in advance.
  3. F

    Summing an Infinite Series: Can We Prove Divergence?

    1+(1/2)+(1/3)+(1/4)+(1/5)...+(1/n) can sum one prove this series is divergent? or just tell me what the expression for sum to infintiy in terms of n is?
  4. M

    Is My Calculation of the Divergence of a Vector Correct?

    Hi, I'm doing a problem of finding the divergence of a radius vector from the origin to any point in Cartesian, cylindrical, and spherical coordinates. The answers look kind of strange to me. I just want to make sure what I did was correct. To find: \nabla\cdot \vec{r} Cartesian: r = (x...
  5. cepheid

    Divergence of a Radial Vector Field

    Something we did in electrostatics that's a source of confusion for me: We learned to use caution when taking the divergence of the (all important) radial vector field: \vec{v} = \frac{1}{r^2} \hat{r} Applying the formula in spherical coords gave zero...a perplexing result. The...
  6. dink

    Curl and Divergence (flux, and what not)

    I'm having a bit of difficulty with this problem: \vec{\nabla} \times \vec{G} = \vec{F} where \vec{\nabla} \cdot \vec{F} = 0 and \vec{F} = <y, z, x> . Find \vec{G} . I'm really at a loss how to solve this. I know the solution must be quick and easy because it was on a quiz. What...
  7. M

    Divergence and solenoidal vector fields

    I want to find which values of n make the vector field \underline{F} = {|\underline{r}|}^n\underline{r} solenoidal. So I have to evaluate the divergence of this vector field I think, then show for which values of n it is zero? Im starting by substituting: \underline{r} = \sqrt{x^2...
  8. A

    Methods for convergence divergence

    I have \sum_{n=1}^\infty{\frac{1}{n^2+n+1}} and I need to show that it converges or diverges. I choose to do the comparison test making A_n=\sum_{n=1}^\infty{\frac{1}{n^2+n+1}} and B_n=\sum_{n=1}^{\infty}\frac{1}{n^2+n} so far so good? Okay well \lim_{n\rightarrow0}B_n=0 so does A_n...
  9. G

    Solving Constants a & b for Electric Field E - Curl & Divergence

    Hi All, Given electric field E=c(2bxy,x^2+ay^2), I need to determine the constants a and b such that CURL E = 0 and DIV E = 0. I'm also given a path from (0,0) , (1,0) and (1,1). Ok so the curl = 0+0+cx(2-b) = 0 and the divergence = 2cy(b+a) = 0 How do I solve for a and b at this...
  10. G

    Can the Surface Integral of a Zero Divergence Electric Field Be Non-Zero?

    I'm not sure why this question comes to mind now, since I haven't had an E&M class for a few months now, but nonetheless. Place some charge at the origin. Surround the charge with a spherical Gaussian surface and calculate the surface integral. You obviously get a non-zero result(Gauss's...
  11. P

    Evaluating a Vector Field Through a Surface with the Divergence Theorem

    ok this probley seems simple but i just need to see how to do it, ok well how do u evaluate this... find the flux of the vector field... \vec{F}=<x,y,z> throught this surface above the xy-plane.. z = 4-x^2-y^2 how do u evaluate this with surface integrals method and the divergence...
  12. P

    Understanding Divergence and Curl on a 3D Surface

    Can anyone please explain what Diverange and Curl actually physically represent on a 3d surface, i know what the operators are, but what do they actually mean? Thanks all
  13. S

    Divergence and Stoke's Theorems in 2D

    Could I get a demonstration of why they are the same? I have the two equations which the two theorems reduce to in two dimensions, and it's pretty tantalizing because they are virtually the same, but differ in a nice symmetrical way. But I can't for the life of me show that they are the same (I...
  14. T

    How can I limit the divergence of my laser using a telescope?

    I have read that you can limit the divergence of a laser by sending it backwards through a telescope. I have not yet been able to do this with my telescope, but seem to have some success using less powerfull lenses. Would anyone be willing to help me conduct this expiriment. The problem I am...
  15. D

    Help with using the Divergence Theorem

    Hi! We are nearing the end of our course --- culminating in Stokes and Divergence Theorems for surface integrals, and I am having some difficulty with the following 1. F(x,y,z) = <x^3y, -x^2y^2, -x^2yz> where S is the solid bounded by the hyperboloid x^2 + y^2 - z^2 =1 and the planes z...
  16. C

    Problem with the Divergence Theorem

    I was wondering if someone could give me a hand here with 2b) on the following link. http://www.am.qub.ac.uk/users/j.mccann/teaching/ama102/2003/assignments/assign_8.pdf For part a) I got it to be equal to 3x^2+3y^2+3z^2+2y-2xy, and I'm hoping that's right! However, for part b) I can't...
  17. C

    Proving the Divergence of a Sequence

    Hey there, one of our h/work questions is to prove that a certain sequence is divergent, where xn=(-1)^n for every natural n. I started off by assuming that it was infact convergent so wrote that mod(xn/l)<e where e is any real number greater than zero, and this holds for any n>no. But from...
  18. J

    How Does the Divergence Theorem Apply to a Vector Field on a Unit Sphere?

    I need help evaluating both sides of the divergence theorem if V=xi+yj+zk and the surface S is the sphere x^2+y^2+z^2=1, and so verify the divergence theorem for this case. Is the divergence theorem the triple integral over V (div V) dxdydz= the double integral over S (V dot normal)dS? If so...
  19. J

    Evaluating Divergence Thm: V, S & Verify for x^2+y^2+z^2=1

    I need help evaluating both sides of the divergence theorem if V=xi+yj+zk and the surface S is the sphere x^2+y^2+z^2=1, and so verify the divergence theorem for this case. Is the divergence theorem the triple integral over V (div V) dxdydz= the double integral over S (V dot normal)dS? If so...
  20. M

    Proving the Divergence of A+B Series

    hi. how can I prove that if A is a converges and B diverges that the Sum of these series (A +B) diverges.. ( A = a1 + a2 + a3 + ... B = b1 + b2 + b3 + ...) if the series start from n=1
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