Divergence Definition and 775 Threads

  1. E

    What is the name for testing divergence in rational functions?

    After doing my homework on testing for convergence/divergence in infinite series, I noticed that if you are testing for divergence of a rational function, if the difference in the degree of the functions (bottom - top) \leq 1, then it is divergent, and if the difference in the degree of the...
  2. A

    Alternating Series Test/Test for Divergence

    So I've been practicing several series that can be solved using the alternating series test, but I've came to a question that's been bothering me for sometime now. If a series fails the alternating series test, will the test for divergence always prove it to be divergent? Typically, in...
  3. D

    Divergence Theorm example for 28 Nov 12:00

    Homework Statement Let S be a smooth surface enclosing the volume V, and let \vec{n} to be the unit outward normal. Using the Divergence Theorm show that: ∫∫ x \vec{r} ° \vec{n} dS = 4 * ∫∫∫ x dV, where \vec{r}=(x,y,z) Homework Equations Divergence theorm...
  4. D

    Divergence theorm example for 28 Nov 12:00

    Homework Statement Hi. I think it will be easiest to understand if I upload a word document as it has some complicated characters. The bold letters stand for vectors. http://www.sendspace.pl/file/1f830d4ff025d966f71b62c Homework Equations Divergence theorm...
  5. L

    Divergence Theorem on a surface without boundary

    Reading through Spivak's Calculus on Manifolds and some basic books in Analysis I notice that the divergence theorem is derived for surfaces or manifolds with boundary. I am trying to understand the case where I can apply the divergence theorem on a surface without boundary.
  6. D

    What does divergence in the reflectivity mean?

    Recently I was reading some paper on design surface plasmon polaritons(so called "SPPs") on corrugated surfaces, mainly these two papers listed below: 1.Pendry, J. B., et al. (2004). "Mimicking surface plasmons with structured surfaces." Science 305(5685): 847-848. 2.Garcia-Vidal, F. J...
  7. P

    Yes, I meant c>0. Thank you for clarifying!

    Hi, How may I show that 2^(n^2)/n! converges to infinity?
  8. B

    What Methods Can Be Used to Prove Sequence Divergence?

    I'm trying to understand divergence of a sequence (not series). What methods can I use to prove divergence? I know that convergence can be proven using various methods, such as squeeze theorem and sum, difference, product and quotient rule etc. Could I use the following to prove divergence...
  9. D

    Electric field of a line charge with the divergence theorem

    Hi, on page 63 of David J. Griffiths' "Introduction to Electrodynamics" he calculates the electric field at a point z above a line charge (with a finite length L) using the electric field in integral form. E_z = \frac{1}{4 \pi \epsilon_0} \int_{0}^{L} \frac{2 \lambda z}{\sqrt{(z^2 + x^2)^3}}...
  10. A

    Evaluate both sides of divergence theorem

    Homework Statement NOTE: don't know see the phi symbol so I used theta. this is cylindrical coordinates not spherical. Given the field D = 6ρsin(θ/2)ap + 1.5ρcos(θ/2)aθ C/m^2 , evaluate both sides of the divergence theorem for the region bounded by ρ=2, θ=0 to ∏, and z = 0 to 5...
  11. Y

    Divergence Theorem: Multiplied by Scalar Field

    Homework Statement Homework Equations Definitely related to the divergence theorem (we're working on it): The Attempt at a Solution I'm a bit confused about multiplying a scalar field f into those integrals on the RHS, and I'm not sure if they can be taken out or not. If they can be, I...
  12. C

    Divergence of a Curl - Then Integrate By Parts

    Homework Statement ∫Bdot[∇×A]dV=∫Adot[∇×B]dV Prove this by integration by parts. A(r) and B(r) vanish at infinity. Homework Equations I'm getting stuck while trying to integrate by parts - I end up with partial derivatives and dV, which is dxdydz? The Attempt at a Solution I...
  13. A

    Convert Divergence equation into matrix form

    Dear All, I am working on electrical modeling which I cannot change divergence to matrix form to solve it with finite difference or finite element, anybody can help me? J Current density E Electrical fi eld I Current...
  14. B

    Divergence of 1/r^2; delta dirac's role

    Homework Statement Given \nabla\frac{1}{r}, show \nabla\bullet\nabla\frac{1}{r} = -4πδ(r), where δ(r) is the delta dirac function.The Attempt at a Solution I've used divergence theorem and also solved the equation itself, so I know that outright solving is zero and the divergence theorem gives...
  15. bcrowell

    Experimental tests of zero divergence for stress-energy?

    We expect the stress-energy tensor to have zero divergence, because this is required for local conservation of energy-momentum, which has been verified to high precision in laboratory and solar system experiments. The standard review article is Will, "The Confrontation between General Relativity...
  16. O

    Divergence Integral doesn't equal surface integral

    We were given an electric field defined by Kr^3 , and asked to calculate what the total flux would be given a sphere of a radius R. I had already calculated the divergence of E to be equal to 5kr^2 . So the first integral is calculating what the divergence over the area of the sphere is...
  17. B

    Chain rule with partial derivatives and divergence

    say you have a function f(x,y) \nablaf= \partialf/\partialx + \partialf/\partialy however when y is a function of x the situation is more complicated first off \partialf/\partialx = \partialf/\partialx +(\partialf/\partialy) (\partialy/\partialx) ( i wrote partial of y to x in case y was...
  18. M

    Discretization of the divergence operator

    I work with a grid-based code, this means that all of my quantities are defined on a mesh. I need to compute, for every point of the mesh the divergence of the velocity field. All I have is, for every cell of my mesh, the values of the 3-d velocity in his 26 neighbors. I call neighbors the...
  19. F

    Does this Divergence Test problem converge?

    Hey guys, was wondering if anyone could help walk me through this problem. I am fairly sure it converges by the ratio test. Thank you. http://www.wolframalpha.com/share/clip?f=d41d8cd98f00b204e9800998ecf8427e1sgqilv5iv
  20. C

    Constants in the Divergence of E

    why do most modern books claim the divergence of the E field is ρ/ε_{0} but in more classical books, and when you actually derive it mathematically you arrive at 4πρ
  21. A

    Question about divergence and curl:

    Please Someone explain why: 1.div(F×G)=GcurlF-FcurlG 2.curl(F×G)=F.divG-G.divF+(G.∇).F-(F.∇).G
  22. M

    Did I correctly prove the divergence of this series?

    Homework Statement Verify that the infinite series diverges. I have the series from n=1 to infinity of (2^(n)+1/2^(n+1) Homework Equations Nth term test(This is the way the book did it but I did it used the geometric series test and I just want to verify if my Algebra was correct)...
  23. P

    Divergence in spherical coordinate system

    I have been studying gradience; divergence and curl. I think i understand them in cartesian coordinate system; But i don't understand how do they get such complex stuffs out of nowhere in calculating divergence in spherical and cylindrical coordinate system. Any helps; links or suggestion...
  24. N

    The divergence operator in a rotated reference frame

    One can easily prove that \nabla \cdot f is invariant under a rotation of the reference frame, however I would like to prove that the divergence operator itself is invariant (same principle, different approach). In other words I want to prove that \mathbf \nabla = \mathbf e_x...
  25. S

    Divergence theorem SUPER complex, maybe

    Homework Statement use divergence theorem to evaluate ∫s∫F dot n dA if F=[sinh yz, 0, y4] , S: r=[u,cosv,sinv], -4≤u≤4 , 0≤v≤pi The Attempt at a Solution Instructor surprised us with this one, I have no idea how to attempt. I know that ∫vdiv v dV=∫sn dot v dA, which is the...
  26. C

    Divergence questions from Griffith's Electrodynamics

    Hi all I basically have two questions that are very closely related to each other about divergence, specifically the divergence of a vector function 1/r2\widehat{r} First, I will be referencing pages 17, 18, and 45 from the 3rd edition of Intro to Electrodynamics. The first question...
  27. D

    The propagator divergence in weak theory

    So I am wondering about one thing. The charged propagators in weak theory are W+- bosons. The mathematical expression for them, while drawing the Feynman diagrams is: -i\frac{g_{\mu\nu}-\frac{q_\mu q_\nu}{m_W^2}}{q^2-m_w^2}. The problems that are usually given to me are simple and involve...
  28. SonOfGod

    Calculation of Divergence dynamic Pressure

    Hello everyone, I am revising for my final examination. I came across this simple problem which I can not solve. the problem is from a textbook (Introduction to Structural Dynamics and Aeroelasticity By Dewey H. Hodges, G. Alvin Pierce). It is the 7th question in the problem sets of the 3rd...
  29. A

    What is the physical interpretation of zero divergence?

    When a vector field representing a physical quantity (e.g. B) has ∇\cdotB = 0 what is then the physical interpretation of this? Some people have said that the field doesn't diverge away from anything, but as far as I can tell magnetic field can easily get weaker and weaker the further you go...
  30. E

    Divergence Theorem/Flux Integral Help

    Homework Statement Compute the flux of F=xi+yj+zk through the curved surface of the cylinder x2+y2=1 bounded below by the plane x+y+z=1, above by the plane x+y+z=7, and oriented away from the z-axis. Homework Equations div(F) = (dF/dx) + (dF/dy) + (dF/dz) The Attempt at a Solution...
  31. V

    Divergence of electric field and charge density

    The divergence of electric field at a point is proportional to the charge density at the point. Divergence is the rate of change with distance, the rate of change of electric field due to a distant charge is not zero, so how can it be said that the divergence at a point depends only on the...
  32. R

    A ZERO Curl and a ZERO divergence

    A ZERO Divergence Vector Field There is theorem that is widely used in physics--e.g., electricity and magnetism for which I have no proof, yet we use this theorem at the drop of a hat. The theorem is this: Given sufficient continuity and differentiability, every vector function A such that...
  33. D

    Series test for convergence or divergence

    I had a bit of trouble in testing series like this for convergence $$\sum_{ n=1 }^{ \infty } \frac { 1 }{ 2n+1 } $$ If by the comparison test, ##\frac{ 1 }{ 2n+1 } < \frac{ 1 }{ 2n }## for all of n>0, and ##\lim_{ n \rightarrow \infty} \frac{ 1 }{ 2n }## =0, then the series should be...
  34. J

    Visualizing the Divergence Theorem for a Cylinder

    Homework Statement Prove the divergence theorem for the vector field A = p = (x,y) and taking the volume V to be the cylinder of radius a with its base centred at the origin, its axis parallel to the z-direction and having height h. I can find the dV side of the equation fine (I think)...
  35. B

    Limitations of the divergence theorem

    Homework Statement Evaluate the surface integral F * dr, where F=<0, y, -z> and the S is y=x^2+y^2 where y is between 0 and 1. Homework Equations Divergence theorem The Attempt at a Solution I just got out of my calculus final, and that was a problem on it. I used the divergence theorem...
  36. C

    Series, find Divergence or Convergence

    Homework Statement Find the Divergence or Convergence of the series \sum^{∞}_{n=1}\frac{2n^2+3n}{\sqrt{5+n^5}} Homework Equations Ratio Test, Comparison Test, Limit Comparison Test, Integral test etc. The Attempt at a Solution This question was on my final exam and the only question of...
  37. M

    Mathematica Vector Divergence in Mathematica

    I'm trying to make a little manipulate/interactive box that shows the vector divergence of the E-field coming from a sphere. I have no idea how to start as I'm really new to Mathematica. Does anyone have any pointers? I can't find anything particularly helpful on the Wolfram reference or...
  38. S

    Using the Divergence Theorem to Solve Vector Calculus Problems

    Homework Statement Homework Equations So I have that v \otimes n = \left( \begin{array}{ccc} v_{1}n_{1} & v_{1}n_{2} & v_{1}n_{3} \\ v_{2}n_{1} & v_{2}n_{2} & v_{2}n_{3} \\ v_{3}n_{1} & v_{3}n_{2} & v_{3}n_{3} \end{array} \right) The Attempt at a Solution I've tried applying the...
  39. K

    Absolute Convergence Theorem and Test for Divergence Connection

    Homework Statement Determine whether the series is absolutely convergent, conditionally convergent, or divergent. ##\sum _{n=1}\left( -1\right) ^{n}\dfrac {n} {n^{2}+1}## the sum goes to infinity. Homework Equations Theorem for absolute convergence. Test for divergence The Attempt...
  40. B

    Convergence or divergence (series)

    Homework Statement Ʃ[(-1)^n (cosn)^2]/√n The Attempt at a Solution i don't have the slightest clue where to start
  41. K

    Test the Series for Convergence or Divergence

    Homework Statement ##\sum _{n=1}^{\infty }\left[ \left( -1\right) ^{n}\right] \dfrac {\sqrt {n}} {1+2\sqrt {n}}##Homework Equations Alternating Series test, Absolute convergence theorem, p-series, and test for divergence. The Attempt at a Solution The alternating series test tells us that the...
  42. B

    Verifying Divergence Theorem on Sphere with F(x,y,z)=zi+yj+xk

    Homework Statement Folks, Verify the divergence theorem for F(x,y,z)=zi+yj+xk and G the solid sphere x^2+y^2+z^2<=16 Homework Equations ##\int\int\int div(F)dV## The Attempt at a Solution My attempt The radius of the sphere is 4 and div F= 1, therefore the integral...
  43. DryRun

    Test series for convergence or divergence

    Homework Statement There are 3 parts to this problem: (a) \; \sum^{\infty}_{n=1} \frac{n^4}{4^n} (b) \; \sum^{\infty}_{n=1} \left( \frac{n+8}{n} \right)^n (c) \; \sum^{\infty}_{n=1} \frac{5^n-8}{4^n+11} The attempt at a solution (a) I've used the Ratio test. So, u_n=\frac{n^4}{4^n} and...
  44. K

    Determine Series' Convergence or Divergence

    Homework Statement ##\sum _{n=1}ne^{-n}## Homework Equations Ratio Test Integral Test The Attempt at a Solution I know that by the ratio test, it converges absolutely. But, I am unable to determine its convergence through the integral test . Could someone help? I thought that the...
  45. K

    Divergence of the sum of the reciprocals of the primes

    Hi, can you tell me which theorem they have used here: http://everything2.com/title/proof+that+the+sum+of+the+reciprocals+of+the+primes+diverges i'm thinking on part: Well, there's an elementary theorem of calculus that a product (1-a1)...(1-ak)... with ak->0 converges to a nonzero value iff...
  46. C

    Green's, Stokes and Divergence Theorem

    When the exercise tells me to calculate the flux, how do I know when I need to use each of these theorems (Green's, Stokes or Divergence)? Can anyone tell me the difference between them? I'm a LOT confused about this. If anyone knows any good material about this on internet, it'll help me a...
  47. K

    Determine Convergence or Divergence. If conv. find the sum:

    Homework Statement ##\sum \dfrac {1+2^{n}} {3^{n}}## According to Wolfram Alpha the sum is 5/2. But, I think that my method is fine and shows another result. The Attempt at a Solution ##\sum \dfrac {1+2^{n}} {3^{n}}=\sum \left[ \left( \dfrac {1} {3}\right) ^{n}+\left( \dfrac {2} {3}\right)...
  48. M

    Is my proof of this sequence's divergence good enough?

    Ʃ n=1 to infinity of cos(n∏) letting an=cos(n∏), I rewrote this as (-1)^n=an. Using the nth term test i let the limit as n->∞ go to infinity. This value bounces back and forth between positive and negative, but I know clearly the value =/= 0, therefore it diverges by the nth term test. Is...
  49. S

    Divergence of the geometric Series at r=1

    Now for the proof of convergence/divergence of the geometric series we first deduce the Nth partial sum which is given by: \frac{r(1-r^n)}{1-r} Now for 0<r<1 this become \frac{1}{1-r} which clearly converges by AOL At r>1 it's similarly obvious why it diverges. But at r=1, I'm a bit...
  50. 1

    What is the divergence of vector field F(x,y,z) = (-x+y)i + (y+z)j + (-z+x)k?

    Homework Statement F(x,y,z) = (-x+y)i + (y+z)j + (-z+x)k Find divergence Homework Equations The Attempt at a Solution The gradient is -i + j + -k Dotting that with F, I get x - y + y + z + z - x = 2z My book lists the answer as -1. What the heck are they talking...
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