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.

View More On Wikipedia.org
Recent contents Top users
  1. D

    Four divergence of stress energy tensor

    Homework Statement Hi, I'm trying to show the four divergence of the stress energy tensor of the sourceless klein gordon equation is zero. I got to the part in the solution where I am left with the equations of motion which is identically zero and 3 other terms. I managed to find a solution...
  2. 159753x

    Simple divergence/Green's theorem question

    I'm exploring the divergence theorem and Green's theorem, but I seem to be lacking some understanding. I have tried this problem several times, and I am wondering where my mistake is in this method. The problem: For one example, I am trying to find the divergence of some vector field from a...
  3. Abner

    Radius of Divergence: Find R & Interval of Convergence

    Homework Statement I have this problem to consider the power series, \sum_{n=1}^{\infty}\frac{(-4)^{n}}{\sqrt{n}}(x+4)^{n} So, i need to find the R and interval of convergence. Homework Equations The Attempt at a Solution This is what i did: \lim_{n\rightarrow \infty}...
  4. U

    Divergence of vorticity vector is zero--intuition behind it

    So mathematically I understand that divergence of curl of something is zero. However, talking specifically about vorticity, this is what it seems to imply to me: When there is vorticity in a fluid, the tiny particles spin around their own axes, so a net circulation is formed around the surface...
  5. G

    Sequence (n)/(n^n) Convergent or Divergent and Limit?

    Homework Statement Is the sequence {(n!)/(n^n)} convergent or divergent. If it is convergent, find its limit. Homework Equations Usually with sequences, you just take the limit and if the limit isn't infinity, it converges... That doesn't really work here. I know I'm supposed to write out the...
  6. L

    Where does the equation for Gaussian beam divergence come from?

    For a Gaussian beam, which has 86% of its power within its beam diameter (spot size 2w0), I've read that beam (angular) divergence is given by 2θ = 4λ/(π[2w0]) Where does this come from? I hate memorizing equations. It makes me feel stupid.
  7. J

    Integrate divergence of a vector over an area

    Hello, I'm hoping somebody can give me some insight on how to solve this problem. This was a solid mechanics exam question and I wasn't able to finish it because I'm rather weak in math. 1. Homework Statement Homework Equations Recall divergence theorem for part ii. ∫div(V)dA = ∫V⋅ndS where...
  8. P

    Divergence of the Stress-Energy Tensor

    Im studying Quantum Field Theory as part of my undergraduate course, and am currently looking at Noether's Theorem which has led me to the following calculation of the divergence of the Stress-Energy Tensor. I'm having difficulty in seeing how we get from line (31) to line (32). Is the 2nd term...
  9. T

    Proof of equivalence between nabla form and integral form of Divergence

    Does anybody knows how you can reach one form of the divergence formula from the other? Or in general, why is the equivalence true?
  10. B

    Differential form of gauss's law.

    I don't understand what charge density is meant in the equation: div E = constant times charge density. I have the derivation in front of me and the last step follows from accepting that the rate of change of the integral of the field divergence per change in volume is the same as the rate of...
  11. S

    Convergence and divergence of a series

    B]1. Homework Statement [/B] Find whether the series is convergent or divergent Homework Equations The Attempt at a Solution By ratio test I have, I would apply L'Hôpital's rule to find the value of limit but before that how do i simplify the expression? It has fractional part both in the...
  12. B

    Simple divergence theorem questions

    So I understand the divergence theorem for the most part. This is the proof that I'm working with http://www.math.ncku.edu.tw/~rchen/Advanced%20Calculus/divergence%20theorem.pdf For right now I'm just looking at the rectangular model. My understanding is that should we find a proof for this...
  13. EnchantedEggs

    How Does Decreased Amplitude Compensate for Field Line Spread in Gauss's Law?

    Has anyone read the book by Daniel Fleisch, 'A Student's Guide to Maxwell's Equations'? I'm having some trouble with Chapter 1, page 36. He's talking about the divergence of an electric field originating from a point charge. Apparently, the divergence of the vector electric field is zero...
  14. N

    Divergence theorem for a non-closed surface?

    Is there some way we can apply divergence (Gauss') theorem for an open surface, with boundaries? Like a paraboloid that ends at some point, but isn't closed with a plane on the top. I found this at Wikipedia: It can not directly be used to calculate the flux through surfaces with boundaries...
  15. O

    Divergence/Curl of \vec{E}=α\frac{\vec{r}}{r^2}

    Homework Statement Calculate the divergence and curl of \vec{E}=α\frac{\vec{r}}{r^2}Homework Equations Div(\vec{E})=\vec{∇}°\vec{E} Div(\vec{E})=\vec{∇}x\vec{E} Table of coordinate conversions, div, and curl: http://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates The...
  16. deedsy

    Divergence in cylindrical/spherical coordinates

    Homework Statement I'm just having trouble understanding a step in my notes from class.. We're talking about how to derive the divergence in other coordinate systems. Homework Equations So, we are deriving this divergence formula in spherical coordinates \oint \vec{A}\cdot d\vec{A} = \int...
  17. R

    How do I use F and the divergence theorem to find the flux and plot it?

    Homework Statement Homework Equations ∇.F(r) The Attempt at a Solution I keep trying to plug F into the divergence theorem but end up with very ugly answers that I know are not right. Is there a simple way to do this question? Also, how the heck would I plot this for for the...
  18. M

    Divergence of an inverse square field

    Reference to Griffith electrodynamics question:- 1.16 Compute the divergence of an inverse square vector field. Now gradient is (∂/∂r)(r cap) Hence upon taking divergence of inverse square field (r cap)/r^2...We don't get 0. In fact we get (-2)/r^3. But if we write the vector field and...
  19. AbhiFromXtraZ

    Physical Meaning Of Divergence

    Currently I'm reading Magnetostatics. While reading Divergence of B, I fell into a confusion that what divergence really means w.r.t. a coordinate system. Supposed there is a current distribution J at r' w.r.t. some primed coordinate system. And B is defined at position r w.r.t. unprimed...
  20. R

    Does Negative Divergence of Gradient Temperature Lead to the Laplace Equation?

    does negative divergence of gradient tempearature gives to lalace equation...? -div(∇T) = [∂^2T/∂x^2+∂^2T/∂y^2]
  21. Telemachus

    Tensor algebra, divergence of cross product

    Hi there. I wanted to demonstrate this identity which I found in a book of continuum mechanics: ##curl \left ( \vec u \times \vec v \right )=div \left ( \vec u \otimes \vec v - \vec v \otimes \vec u \right ) ## I've tried by writting both sides on components, but I don't get the same, I'm...
  22. F

    MHB Series Divergence: Prove $\frac{a^n}{n^23^n}$ Diverges

    Prove that the series of $\frac{a^n}{n^23^n}$ diverges where $a$ is a complex number with $|a|{\geq}3$
  23. M

    Why Does the Divergence of r-hat Over r-squared Equal Zero?

    Homework Statement How divergence of r (cap)/ |r|^2 is equal to zero? Homework Equations r(cap)= x(cap)+y(cap)+z(cap) |r|^2 as x^2+y^2+z^2 The Attempt at a Solution I tried the problem and end up with with a different solution I took r(cap)= x(cap)+y(cap)+z(cap) |r|^2...
  24. A

    MHB Sequence Divergence and Convergence Questions

    Hey guys, I have a couple more questions. For the first one, taking the limit to infinity obviously equals 0 so it should be convergent, right? Also, for the second one, the limit as n approaches infinity for gives me indeterminate form, so I took the derivative which just gave me ln(n)...
  25. A

    MHB Series Convergence and Divergence III

    Hey guys, I have a few quick questions for the problem set I'm working on at the moment: I'm highly doubtful of my answer for c. I used the roots test instead of the ratio test, which gives 1/n, which I took the limit of to get an interval of [-∞ , ∞] As for a and b, I got [-5,5] and (-∞, ∞)...
  26. A

    MHB Series Convergence and Divergence II

    Hey guys, I have a few more questions for the problem set I'm working on at the moment: I'm unsure about b in particular. I compared the series to 1/(n^3/2), which makes it absolutely convergent by the p-test and comparison test. Do I still have to perform any other tests to confirm absolute...
  27. A

    MHB Series Convergence and Divergence I

    Hey guys, I have a few quick questions for the problem set I'm working on at the moment: I'm mostly unsure of my response for b. For a, I just split the series into two parts and added 6+3 to get 9, and thus the series is convergent. For c, I got 3/5 after taking the limit, which is...
  28. A

    MHB Integral Convergence and Divergence II

    This thread is only for question 5. As for number 5 part a, after tediously expanding the partial fraction expression, I ended up getting c=1, d=0, b=1, and c=1, ultimately resulting in: ln(x) - (1/x^2) + c. I really don't think this looks right. As for 5b, I obtained b=-1, c=-1, a=2, and...
  29. A

    MHB Calculus II Integral Convergence and Divergence Questions

    For a,b, and c respectively, I got divergent (to -infinity), convergent (to π/6), and divergent (to infinity, since the first part's sum is 1/3, but lim negative infinity gives infinity, thus the summation of the two integrals gives a divergent integral). I'm sure these are right, but I'd...
  30. A

    MHB Integral Convergence and Divergence I

    Hello, I'm doubting a couple of my answers for these questions. Some of them seem relatively simple, but there are slight nuances that I'm not sure of. This thread is only for question 4. For 4a, I just used the (a^2) - (x^2) => x=asin(Ø) rule and substituted 3sin(Ø) for x. I ended up...
  31. N

    Laser Beam - Divergence and Solid Angle

    I am somewhat confused about the connection between divergence and solid angle for a beam. I know individually what each term means... but I'm confused as to how (or even if) one can calculate the solid angle of a beam, given the divergence. I have some notes from a previous lecture series I...
  32. T

    Understanding Greene's, Stoke's, and the Divergence theorems

    Hi everyone, first post. Anyway, I am reviewing my math physics, and I am having trouble understanding the Divergence Theorem, Green's Theorem, and Stokes' Theorem. I was able to satisfactorily pass math physics by only being able to regurgitate them, but soon I will be taking e&m, and it...
  33. M

    Divergence angle of light exiting a nano-etched single mode fiber

    I am working with a single mode fiber at 830 nm with a NA of 0.12 (http://www.thorlabs.com/thorcat/19600/P3-830A-FC-2-AutoCADPDF.pdf). One end has a fiber tip coated in silver with a small hole of 200 nm diameter to essentially create a point source. My advisor told me that the fiber has a few...
  34. jdawg

    Alternating Series Test No Divergence?

    Homework Statement Hey! So I just have a quick question. In my notes I wrote down that the alternating series test only proves absolute or conditional convergence, but can not prove divergence. Is this true or did I misunderstand my professor? Homework Equations The Attempt at a...
  35. P

    Divergence Theorem/Surface Gradient

    There is a paper in chemical physics by Overbeek in which he describes the electrostatic energy of a double layer as the "energy of the surface charges and bulk charges in a potential field"; the transformation that he provides appears to be a variant of the divergence theorem in which he...
  36. D

    Number theory - show divergence of ∑1/p for prime p

    1. show that the sum of. The reciprocals of the primes is divergent. I am reposying this here under homework and deleting the inital improperly placed post 2. Theorem i use but don't prove because its assumed thw student has already lim a^1/n = 1. The gist of the approach I took is that∑1/p =...
  37. G

    Series convergence / divergence

    Homework Statement Does the following series converge or diverge? If it converges, does it converge absolutely or conditionally? \sum^{\infty}_{1}(-1)^{n+1}*(1-n^{1/n}) Homework Equations Alternating series test The Attempt at a Solution I started out by taking the limit of ##a_n...
  38. evinda

    MHB Divergence Theorem: Applying to Sphere $\hat{i}x+\hat{j}y+\hat{k}z$

    Hello again! (Wave) I am looking at an exercise of the divergence theorem.. We want to apply the divergence theorem for the sphere $x^2+y^2+z^2=a^2$ in the case when the vector field is $\overrightarrow{F}=\hat{i}x+\hat{j}y+\hat{k}z$.$\displaystyle{\nabla \cdot...
  39. C

    Ricci Tensor Proportional to Divergence of Christoffel Symbol?

    I'm reading an old article published by Kaluza "On the Unity Problem of Physics" where i encounter an expression for the Ricci tensor given by $$R_{\mu \nu} = \Gamma^\rho_{\ \mu \nu, \rho}$$ where he has used the weak field approximation ##g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}## where...
  40. G

    Convergence / Divergence of a series

    Homework Statement Does the following series converge or diverge? ##∑\frac{n^5}{n^n}## (as n begins from 1 and approaches infinity) Homework Equations Ratio test? The Attempt at a Solution For your reference, thus far I have learned about the geometric series, the limit test...
  41. Feodalherren

    Verify the divergence theorem for a cylinder

    Homework Statement Verify the divergence theorem if \textbf{F} = <1-x^{2}, -y^{2}, z > for a solid cylinder of radius 1 that lies between the planes z=0 and z=2. Homework Equations Divergence theorem The Attempt at a Solution I can do the triple integral part no problem. Where I...
  42. M

    Divergence of curl in spherical coordinates

    Hey pf! I was thinking about how div(curl(f)) = 0 for any vector field f. However, is this true for div and curl in spherical coordinates? It doesn't seem to be. If not, what needs to happen for this to be true in spherical coordinates?? Thanks all!
  43. D

    Proving Divergence of (tn + sn)

    I have a super round about way to prove this, but I'm having trouble finding a succinct proof Let (tn) be diverge and (sn) converge. Show (tn+sn) diverges The way I was doing involved considering that tn was unbounded, then showing it (sn+tn) is divergent. Then I had to consider that...
  44. F

    KL divergence on different domains

    Hallo, I'm trying to compare the distance between two distributions that I got from a Kernel smoothing density estimate (ksdensity in matlab). I was thinking of using the kullback leibler divergence, but I realized that the domains of my distributions are different (see attached). Can I...
  45. QuantumCurt

    Determine the convergence or divergence of the infinite series

    Homework Statement This is for Calculus II. We've just started the chapter on Infinite Series. n runs from 1 to ∞. \Sigma\frac{1}{n(n+3)} The Attempt at a Solution I used partial fraction decomposition to rewrite the sum. \frac{1}{n(n+3)}=\frac{A}{n}+\frac{B}{n+3}...
  46. M

    Help with intuition of divergence, gradient, and curl

    hey pf! i have a few question about the physical intuition for divergence, gradient, and curl. before asking, i'll define these as i have seen them (an intuitive definition). \text{Divergence} \:\: \nabla \cdot \vec{v} \equiv \lim_{V \to 0} \frac{1}{V} \oint_A \hat{n} \cdot \vec{v} da...
  47. S

    MHB More Convergence & Divergence with sequences

    Determine whether the sequence converges or diverges, if it converges fidn the limit. a_n = n \sin(1/n) so Can I just do this: n * \sin(1/n) is indeterminate form so i can use lopitals so: 1 * \cos(1/x) = 1 * 1 = 1 converges to 1?
  48. S

    MHB Tricky question Considering Divergence and Convergence

    Determine whether the sequence Converges or Diverges. Tricky question, so check it out. \frac{n^3}{n + 1} So here is what I did divided out n to get \frac{n^2}{1} = \infty \therefore diverges Now, here is what someone else did. They applied L'Hopitals, and then claimed that 3n^2 = \infty...
  49. S

    MHB Convergence and Divergence with Series

    Determine whether the series is convergent or divergent. \sum^{\infty}_{n = 1} \frac{n - 1}{3n - 1} I ended up with \frac{1}{3} * 1 = \frac{1}{3} , which is 0.333 ... so wouldn't that mean that r < 1? Also wouldn't that mean that it is convergent since r < 1 ? I don't understand why this is...
  50. E

    Showing Divergence Theorem Equivalence

    Homework Statement The problem states that a cube encloses charge. This cube is given in three space by <0,0,0> and <a,a,a>. The electric field is given by: \hat{E}=\frac{4e}{a^{2}e_{0}}[\frac{xy}{a^{2}}\hat{i}+\frac{(y-x)}{a}\hat{j}+\frac{xyz}{a^{2}}\hat{k}]. I am to find the total charge...
Back
Top