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. T

    The Divergence of a Polar Vector Function

    Homework Statement Find the divergence of the function ##\vec{v} = (rcos\theta)\hat{r}+(rsin\theta)\hat{\theta}+(rsin\theta cos\phi)\hat{\phi}## Homework Equations ##\nabla\cdot\vec{v}=\frac{1}{r^2}\frac{\partial}{\partial r}(r^2v_r)+\frac{1}{r sin\theta}\frac{\partial}{\partial...
  2. davidge

    Prove divergence of the Series

    Homework Statement Given ##b_n = 1 / n## if ##n## odd and ##b_n = 1 / n^2## if ##n## even, show that the series $$\sum_{n=1}^{\infty} (-1)^n b_n$$ diverges. Homework Equations Did'nt find any for this problem The Attempt at a Solution I assumed that ##\sum_{n=1}^{\infty} (-1)^n b_n =...
  3. P

    I Divergence of v x B = Divergence of E in the v=0 frame?

    Consider a scenario where in one frame R, I have a magnet at rest and a solid slab of charges with an arbitrarily large mass moving at velocity v. The overall acceleration of the slab is trivial, however, the v x B exerted on the slab is divergent, thus compressive/tensile stresses are exerted...
  4. terryds

    Divergence of electrostatic field?

    Homework Statement By Gauss' law, how is it able to obtain ## \nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0} ## ? By Coulomb's law, ##\vec{E} = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} \hat{r}## I calculate the divergence of ##\frac{1}{r^2} \hat{r}## and get the result is zero That means the...
  5. C

    Calculating divergence as a function of radius

    {Moderator note: Member advised to retain and use the formatting template when starting a thread in the homework sections} Hey guys Question: Calculate the divergence as a function of radius for each of the following radially symmetrical fields in which the magnitude of the field vector: (a)...
  6. G

    A Divergent Diagrams in the Standard Model

    It is my understanding that the task of enumerating all of the divergent diagrams in a quantum field theory can be reduced to analyzing a hand full of diagrams (well, at the moment I know that this is at least true for QED and phi^4 theory), and that all other divergent diagrams are divergent...
  7. M

    MHB Divergence Theorem and shape of hyperboloid

    Hello! I have been doing a previous exam task involving the divergence theorem, but there is a minor detail in the answer which i can't fully understand. I have a figur given by ${x}^{2} +{y}^{2} -{z}^{2} = 1$ , $z= 0$ and $z=\sqrt{3}$ As i have understood this is a hyperboloid going from...
  8. Y

    I How does divergence calculate all of flow through a surface?

    I'd like to use 2-d for simplification. Divergence is the rate of change of a component of a field F as you travel along that component's direction. So Fx represents the part of F flowing the X direction and same with Fy along the Y direction, and so divergence is calculated by dFx/dx +...
  9. K

    Divergence theorem with inequality

    Homework Statement F(x,y,z)=4x i - 2y^2 j +z^2 k S is the cylinder x^2+y^2<=4, The plane 0<=z<=6-x-y Find the flux of F Homework Equations The Attempt at a Solution What is the difference after if I change the equation to inequality? For example : x^2+y^2<=4, z=0 x^2+y^2<=4 , z=6-x-y...
  10. Vitani11

    Calculate the volume integral of divergence over a sphere

    Homework Statement For the vector field F(r) = Ar3e-ar2rˆ+Br-3θ^ calculate the volume integral of the divergence over a sphere of radius R, centered at the origin. Homework Equations Volume of sphere V= ∫∫∫dV = ∫∫∫r2sinθdrdθdφ Force F(r) = Ar3e-ar2rˆ+Br-3θ^ where ^ denote basis (unit vectors)...
  11. N

    A Comparing Kullback-Leibler divergence values

    I’m currently evaluating the "realism" of two survival models in R by comparing the respective Kullback-Leibler divergence between their simulated survival time dataset (`dat.s1` and `dat.s2`) and a “true”, observed survival time dataset (`dat.obs`). Initially, directed KLD functions show that...
  12. M

    Vector Analysis Problem Involving Divergence

    Homework Statement [/B] Let f and g be scalar functions of position. Show that: \nabla f \cdot \nabla(\nabla ^2 g)-\nabla g \cdot \nabla(\nabla ^2f) Can be written as the divergence of some vector function given in terms of f and g. Homework Equations [/B] All the identities given at...
  13. K

    How can I verify the Divergence Theorem for F=(2xz,y,−z^2)

    Homework Statement Verify the Divergence Theorem for F=(2xz,y,−z^2) and D is the wedge cut from the first octant by the plane z =y and the elliptical cylinder x^2+4y^2=16 Homework Equations \int \int F\cdot n dS=\int \int \int divF dv The Attempt at a Solution For the RHS...
  14. jlmccart03

    Series: Determine if they are convergent or divergent

    Homework Statement I have a couple of series where I need to find out if they are convergent (absolute/conditional) or divergent. Σ(n3/3n Σk(2/3)k Σ√n/1+n2 Σ(-1)n+1*n/n^2+9 Homework Equations Comparison Test Ratio Test Alternating Series Test Divergence Test, etc The Attempt at a...
  15. S

    A Logarithmic divergence of an integral

    I would like to prove that the following integral is logarithmically divergent. $$\int d^{4}k \frac{k^{4}}{(k^{2}-a)((k-b)^{2}-x)((k-y)^{2}-a)((k-z)^{2}-a)}$$ This is 'obvious' because the power of ##k## in the numerator is ##4##, but the highest power of ##k## in the denominator is ##8##...
  16. Dopplershift

    Need Help With Gradient (Spherical Coordinates)

    Homework Statement Find te gradient of the following function f(r) = rcos(##\theta##) in spherical coordinates. Homework Equations \begin{equation} \nabla f = \frac{\partial f}{\partial r} \hat{r} + (\frac{1}{r}) \frac{\partial f}{\partial \theta} \hat{\theta} + \frac{1}{rsin\theta}...
  17. L

    A Gradient Divergence of Nabla Operator Defined

    Nabla operator is defined by \nabla = \sum^3_{i=1} \frac{1}{h_i}\frac{\partial}{\partial q_i}\vec{e}_{q_i} where ##q_i## are generalized coordinates (spherical polar, cylindrical...) and ##h_i## are Lame coefficients. Why then div(\vec{A})=\sum^3_{i=1} \frac{1}{h_i}\frac{\partial}{\partial...
  18. L

    Divergence theorem for vector functions

    Surface S and 3D space E both satisfy divergence theorem conditions. Function f is scalar with continuous partials. I must prove Double integral of f DS in normal direction = triple integral gradient f times dV Surface S is not defined by a picture nor with an equation. Help me. I don't...
  19. SD das

    How Does Zero Divergence and Curl Determine Uniqueness in a Manifold?

    Today when I ask a professor about maxwell eqation He tells me " it seems that the unknowns exceed the number of equations. What are the missing ingredients? The answer is the boundary condition .With appropriate boundary conditions, zero divergence and zero curl will nail down a unique solution...
  20. N

    Beam Divergence from non-circular laser beam

    Homework Statement The laser beam is not a point source. It is known that it has a rectangular shape with a divergence of 30 mrad x 1 mrad. I would like to know how large my laser lobe will be at a distance of 250 mm from the laser source. Homework Equations I think you can use trigonometri...
  21. karush

    MHB Is the series $\sum_{k=1}^{\infty}\frac{\arctan(2k)}{1+4k^2}$ convergent?

    $\tiny{206.f3a.}$ $\textsf{Use the divergence Test to detemine whether the series is divergent}$ \begin{align} \displaystyle &\sum_{k=1}^{\infty}\frac{\arctan(2k)}{1+4k^2}\\ \textit{take limit}\\ =&\lim_{{k}\to{\infty}}\frac{\arctan(2k)}{1+4k^2}\\ \\ =&\frac{\arctan(\infty)}{\infty}...
  22. karush

    MHB Divergence Test: Evaluating I$_{17}$ at $\tiny{206.8.8.17}$

    $\tiny{206.8.8.17}$ $\textsf{Evaluate the following integral, or stat that it diverges.}\\$ \begin{align*} \displaystyle && I_{17}& =\int_{0}^{\infty}36x^8 e^{-x^9}\, dx& &(1)&\\ && &=36\int_{0}^{\infty}\frac{x^8}{e^{x^9}}\, dx & &(2)&\\ \end{align*} $\textit{first what be the recommended...
  23. T

    MHB Series Convergence Or Divergence

    I have $$\sum_{n = 2}^{\infty} \frac{(lnn)^ {12}}{n^{\frac{9}{8}}}$$ I'm trying the limit comparison test, so I let $$ b = \frac{1}{n^{\frac{9}{8}}}$$ and $a = \sum_{n = 2}^{\infty} \frac{(lnn)^ {12}}{n^{\frac{9}{8}}}$ $\frac{a}{b} = (lnn)^ {12}$ therefore I know the limit of this as n...
  24. X

    I Divergence Theorem not equaling 0

    Why is it possible that ∫∫∫ V f(r) dV ≠ 0 even if f(r) =0
  25. Dave-o

    Evaluate: ∇(∇r(hat)/r) where r is a position vector

    Homework Statement ∇ . r = 3, ∇ x r = 0 Homework EquationsThe Attempt at a Solution So far I've gotten up to ∇(∇^2 r)
  26. P

    Magnetostatics: What if "steady" currents were divergent?

    Why must steady currents be non-divergent in magnetostatics? Based on an article by Kirk T. McDonald (http://www.physics.princeton.edu/~mcdonald/examples/current.pdf), it appears that the answer is that by extrapolating the linear time dependence of the charge density from a constant divergence...
  27. H

    I Proof of convergence & divergence of increasing sequence

    I'm using the book of Jerome Keisler: Elementary calculus an infinitesimal approach. I have trouble understanding the proof of the following theorem. I'm not sure what it means. Theorem: "An increasing sequence <Sn> either converges or diverges to infinity." Proof: Let T be the set of all real...
  28. C

    I Intuition on divergence and curl

    Hi, I'm looking at the following graph, but there are a few things I don't get. For instance: curl should always be zero in circles where the field lines are totally straight (right-most figure) curl should always be non-zero in circles where the field lines are rotating (center figure in 2nd...
  29. D

    A simple problem pertaining to divergence

    Homework Statement The problem states that it is required to prove that ∇f(r)=f'(r)R/r where r is the vector field Homework Equations ∇=(∂/∂x)i + (∂/∂y)j + (∂/∂z)k R= xi + yj +zk r = √(x^2+y^2+z^2) The Attempt at a Solution The trouble that I am having with this problem is the inability to...
  30. F

    Maxwell's equation has well defined divergence?

    Homework Statement How to I explain that maxwell's equation has well defined divergence Homework Equations All four EM Maxwell's equation The Attempt at a Solution I discussed it by showing one of the property of Maxwell's equation that is the Divergence of a Gradient is always zero (With...
  31. T

    MHB Determining the convergence or divergence of a sequence using comparison test

    I have this series: $$\sum_{k = 1}^{\infty} {4}^{\frac{1}{k}}$$ To solve this, I am trying to compare it to this series $$\sum_{k = 1}^{\infty} {4}^{k}$$ So, I can let $a_k = {4}^{\frac{1}{k}} $ and $b_k = {4}^{k}$ These seem to be both positive series and $ 0 \le a_k \le b_k$ Therefore...
  32. T

    MHB Determining the convergence or divergence of a sequence using direct comparison

    I have $$\sum_{n = 2}^{\infty} \frac{{(\ln\left({n}\right)})^{12}}{n^{\frac{9}{8}}}$$ We can compare it to $ \frac{1}{{n}^{\frac{1}{8}}}$. $ \sum_{n = 1}^{\infty} \frac{1}{{n}^{\frac{1}{8}}}$ diverges because $p < 1$ in this case. So, if I can prove that $...
  33. binbagsss

    Curl and Divergence etc algebra manipulating quick question

    ##\nabla p = \rho \nabla \phi ## My textbook says that by taking the curl we get: ## 0=\nabla \rho X \nabla \phi ## ** I don't follow. I understand the LHS is zero, by taking the curl of a divergence. But I'm unsure as to how we get it into this form, from which it is clear that the gradients...
  34. mr.tea

    I Divergence theorem and closed surfaces

    Hi, I have a question about identifying closed and open surfaces. Usually, when I see some exercises in the subject of the divergence theorem/flux integrals, I am not sure when the surface is open and needed to be closed or if it is already closed. I mean for example a cylinder that is...
  35. M

    A Tensor Calculus and Divergence

    Hi PF! I have a question on the dyadic product and the divergence of a tensor. I've never formally leaned this, although I'm sure it's published somewhere, but this is how I understand the operators. Can someone tell me if this is right or wrong? Let's say I have some vector ##\vec{V} = v_x i +...
  36. S

    MHB Proving Divergence of \sum\frac{1}{2n+1}

    Prove that \sum\frac{1}{2n+1} diverges. I understand that \sum\frac{1}{n} i.e. the harmonic series diverges (I say this because of the comparison test, that is, \frac{1}{2n+1}\leq\frac{1}{2n}\leq\frac{1}{n}). However, this doesn't correctly imply that 1/(2n + 1) diverges. Then I decided to use...
  37. C

    I Superficial degree of divergence for scalar theories

    I have a few questions regarding the derivation of the degree of divergence for feynman diagrams. The result is $$D = [g_E] - \sum_{n=3}^{\infty} V_n [g_n]$$ (following notation in Srednicki, ##P118##) I am trying to understand what ##[g_E]## is here? Since in this set up we are summing over...
  38. F

    I Divergence of the Navier-Stokes Equation

    The Navier-Stokes equation is: (DUj/Dt) = v [(∂2Ui/∂xj∂xi) + (∂2Uj/∂xi∂xi)] – 1/ρ (∇p) where D/Dt is the material (substantial) derivative, v is the kinematic viscosity and ∇p is the modified pressure gradient (taking into account gravity and pressure). Note that the velocity field is...
  39. enh89

    Why does it matter what convergence test I use?

    I just took a calc 2 test and got 3/8 points on several problems that asked you to show convergence or divergence. The reason being that I didn't use the correct test of convergence? The answer was right, if you get to the point where you know the series converges, then why does it matter which...
  40. The-Mad-Lisper

    Proof for Convergent of Series With Seq. Similar to 1/n

    Homework Statement \sum\limits_{n=1}^{\infty}\frac{n-1}{(n+2)(n+3)} Homework Equations S=\sum\limits_{n=1}^{\infty}a_n (1) \lim\limits_{n\rightarrow\infty}\frac{a_{n+1}}{a_n}\gt 1\rightarrow S\ is\ divergent (2) \lim\limits_{n\rightarrow\infty}\frac{a_{n+1}}{a_n}\lt 1\rightarrow S\ is\...
  41. G

    Divergence and transversal extension integral definitions

    Hi. I am reading a paper about gaussian beams and the author says that gaussian beams have simultaneously minimal divergence and minimal transversal extension. In order to prove it, the author states that \mathrm{divergenece} \propto \int_{-\infty}^{+\infty} \frac{d\,k_{x}}{2\pi}...
  42. 1

    What does divergence of electric field = 0 mean?

    Homework Statement I just want to focus on the divergence outside the cylinder (r >R) Homework Equations The Attempt at a Solution For r > R, I said ∇ * E = p/ε But that's wrong. The answer is ∇ * E = 0 I'm confused because there is definitely an electric field outside the cylinder (r...
  43. Jess Karakov

    Sequence Convergence/Divergence Question

    Homework Statement Determine which of the sequences converge or diverge. Find the limit of the convergent sequences. 1) {asubn}= [((n^2) + (-1)^n)] / [(4n^2)] Homework Equations [/B] a1=first term, a2=second term...an= nth term The Attempt at a Solution a) So I found the first couple of...
  44. P

    Applying the divergence theorem to find total surface charge

    Homework Statement Sorry- I've figured it out, but I am afraid I don't know how to delete the thread. Thank you though :) Homework Equations Below The Attempt at a Solution Photo below- I promise its coming! I've started by using cylindrical coordinates, but I wasn't sure if spherical...
  45. C

    I Divergence Theorem and Gauss Law

    Divergence theorem states that $\int \int\vec{E}\cdot\vec{ds}=\int\int\int div(\vec{E})dV$ And Gauss law states that $\int \int\vec{E}\cdot\vec{ds}=\int\int\int \rho(x,y,z)dV$ If $\vec{E}$ to be electric field vector then i could say that $div(\vec{E})=\rho(x,y,z)$ However i can't see any...
  46. K

    Electric Field Divergence Solution for Moving Source Charge

    I would like to know what is the solution for the divergence of the electric field if the source charge is moving.
  47. debajyoti datta

    A What is the divergence of 1/r^n for positive integer n in physics?

    I have read in Griffiths electrodynamics that divergence of 1/r^2 is delta function and I thought it was the only special case...I have understood the logic there... but a question came in mind...what would happen in general if the function is 1/r^n ...where n is positive integer>0...because the...
  48. Jezza

    Div and curl in other coordinate systems

    My question is mostly about notation. I know the general definitions for divergence and curl, which can be derived from the divergence and Stokes' theorems respectively, are: \mathrm{div } \vec{E} \bigg| _P = \lim_{\Delta V \to 0} \frac{1}{\Delta V} \iint_{S} \vec{E} \cdot \mathrm{d} \vec{S}...
  49. P

    Exploring Divergence in Vector Calculus

    Got it. Thank you ;)
  50. Dopplershift

    Solving Divergence Problem of $\vec{B}(x,y,z)$

    So I have this problem which wants me to find the divergence of: \begin{equation} \vec{B}(x,y,z) = (x^3+y^2z)\hat{x}+(y^3+x^2z)\hat{y} \end{equation} Given that the divergence is given by: \begin{equation} \nabla \cdot \vec{B} = (\hat{x}\frac{\partial}{\partial x}+...
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