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. PLAGUE

    I What's the physical meaning of Curl of Curl of a Vector Field?

    So, curl of curl of a vector field is, $$\nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$$ Now, curl means how much a vector field rotates counterclockwise. Then, curl of curl should mean how much the curl rotate counterclockwise. The laplacian...
  2. chwala

    Find the divergence and curl of the given vector field

    Been long since i studied this area...time to go back. ##F = x \cos xi -e^y j+xyz k## For divergence i have, ##∇⋅F = (\cos x -x\sin x)i -e^y j +xy k## and for curl, ##∇× F = \left(\dfrac{∂}{∂y}(xyz)-\dfrac{∂}{∂z}(-e^y)\right) i -\left(\dfrac{∂}{∂x}(xyz)-\dfrac{∂}{∂z}(x \cos...
  3. walkeraj

    B A Magnetic Misconception on Divergence 0/Closed Field Lines?

    Question: Can we ultimately atttribute no work or net zero work done by a magnetic force to the closed magnetic field lines that results in Divergence zero of a magnetic field? That is, is it a misconception to say that closed magnetic field lines imply magnetic force will always result in no...
  4. Z

    I In ##\nabla\cdot\vec{E}## why can ##\nabla## pass through the integral?

    We have $$\vec{E}(\vec{r})=\frac{1}{4\pi\epsilon_0}\int_V\frac{\rho(\vec{r}')}{\eta^2}\hat{\eta}d\tau'\tag{1}$$ A few initial observations 1) I am using notation from the book Introduction to Electrodynamics by Griffiths. When considering point charges, this notation uses position vectors...
  5. PhysicsRock

    Divergence of ##\vec{x}/\vert\vec{x}\vert^3##

    As you can see in the homework statement, I am asked to calculate what's effectively the divergence of the vector field ##\vec{F} = \vec{x}/\vert\vec{x}\vert^3## over ##\mathbb{R}^3##. I have done that, the calculation itself isn't that difficult after all. However, I can't make sense of the...
  6. L

    Divergence of the Electric field of a point charge

    Hi, unfortunately, I am not sure if I have calculated the task correctly The electric field of a point charge looks like this ##\vec{E}(\vec{r})=\frac{Q}{4 \pi \epsilon_0}\frac{\vec{r}}{|\vec{r}|^3}## I have now simply divided the electric field into its components i.e. #E_x , E-y, E_z#...
  7. Vanilla Gorilla

    B Solving for the Nth divergence in any coordinate system

    Preface We know that, in Cartesian Coordinates, $$\nabla f= \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} + \frac{\partial f}{\partial z}$$ and $$\nabla^2 f= \frac{\partial^2 f}{\partial^2 x} + \frac{\partial^2 f}{\partial^2 y} + \frac{\partial^2 f}{\partial^2 z}$$ Generalizing...
  8. chwala

    I Understanding the nth-term test for Divergence

    There are several examples that i have looked at which are quite clear and straightforward, e.g ##\sum_{n=0}^\infty 2^n## it follows that ##\lim_{n \rightarrow \infty} {2^n}=∞## thus going with the theorem, the series diverges. Now let's look at the example below; ##\sum_{n=1}^\infty...
  9. P

    Series investigation: divergence/convergence

    Hi everyone! It's about the following task: show the convergence or divergence of the following series (combine estimates and criteria). I am not sure if I have solved the problem correctly. Can you guys help me? Is there anything I need to correct? I look forward to your feedback.
  10. chwala

    Determine Convergence/Divergence of Sequence: f(x)=ln(x)^2/x

    ##a_n= \left[\dfrac {\ln (n)^2}{n}\right]## We may consider a function of a real variable. This is my approach; ##f(x) =\left[\dfrac {\ln (x)^2}{x}\right]## Applying L'Hopital's rule we shall have; ##\displaystyle\lim_ {x\to\infty} \left[\dfrac {\ln (x)^2}{x}\right]=\lim_ {x\to\infty}\left[...
  11. M

    How to apply divergence free (∇.v=0) in nodal finite element method?

    I know how to apply boundary condition like Dirichlet, Neumann and Robin but i have been struggling to apply divergence free condition for Maxwells or Stokes equations in nodal finite element method. to overcome this difficulties a special element was developed called as edge element but i don't...
  12. Vividly

    B Understanding about Sequences and Series

    Homework Statement:: Tell me if a sequence or series diverges or converges Relevant Equations:: Geometric series, Telescoping series, Sequences. If I have a sequence equation can I tell if it converges or diverges by taking its limit or plugging in numbers to see what it goes too? Also if I...
  13. C

    A Divergence, Gradient of higher order tensor

    1.) I have the following equation $$\nabla \cdot \left( \mathbf{A} : \nabla_{s}\mathbf{b} \right) - \frac{\partial^2\mathbf{c}}{\partial t^2} = - \nabla \cdot \left( \mathbf{D}^{Transpose} \cdot \nabla \phi \right )$$ Is my index notation correct? $$(A_{ijkl} b_{k,l}),_{j} - c_{i,tt} = -...
  14. guyvsdcsniper

    What is the explanation for the divergence of the vector function r^hat/r^2?

    Im having trouble understanding the divergence of this vector function. I am just getting lost at calculating the divergence. I get that 1/r^2 is a constant so you can pull it out, but where does r^2 x 1/r^2 come from?
  15. C

    A Index Notation of div(a:b) and div(c^transpose d)

    What is the index notation of divergence of product of 4th rank tensor and second rank tensor? What is the index notation of divergence of 3rd rank tensor and vector? div(a:b) = div(c^transpose. d) Where a = 4th rank tensor, b is second rank tensor, c is 3rd rank tensor and d is a vector.
  16. jorgeluisharo

    Vector calculus — Computing this Divergence

    I really don't know how to proceed if I'm not using an specific coordinate system, Is there a way of doing this using only indices, in general form?
  17. J

    I Using Diffraction (i.e., Fresnel Zone Plate) to defocus/diverge light

    I am wondering if it is possible to use principals of diffraction to cause a collimated beam of light (laser) to become divergent. I see that zone plates are most always used for focusing the light from a source, unless they are used in reverse. This is why zone plates are seemingly always...
  18. M

    I Calculus of Variations on Kullback-Liebler Divergence

    Hi, This isn't a homework question, but a side task given in a machine learning class I am taking. Question: Using variational calculus, prove that one can minimize the KL-divergence by choosing ##q## to be equal to ##p##, given a fixed ##p##. Attempt: Unfortunately, I have never seen...
  19. bob012345

    I Divergence of the Electric field of a charged circular ring

    In a previous thread* the field in a charged ring was discussed and it was shown to be not zero except at the center. In *post #45 a video is referenced that says the field diverges as one gets close to the ring and it was argued that at very close distances the field looks like an infinite line...
  20. Shreya

    Flux density and Divergence of Electric field

    I think Flux density is flux/Volume. Or is it flux/ Area Please be kind to help
  21. Leo Liu

    How to use the divergence theorem to solve this question

    The correct answer is ##\frac{\pi a^2 h} 2## by using the standard approach. However when I tried using the divergence theorem to solve this problem, I got a different answer. My work is as follows: $$\iint_S \vec F\cdot\hat n\, dS = \iiint_D \nabla\cdot\vec F\,dV$$ $$= \iiint_D \frac{\partial...
  22. LCSphysicist

    True or false questions about Divergence and Curl

    ##F = (P,Q,R)## is a field of vector C1 defined on ##V = R3-{0,0,0}## There are a lot of true or false statement here. I am a little skeptical about my answer because it contains a lot of F, but let's go. 1 Rot of F is null in V iff ##\int \int_{S} P dx + Q dy + R dz = 0## for all sphere S...
  23. A

    Why can't I use the divergence theorem?

    Greetings! here is the following exercice I understand that when we follow the traditional approach, (prametrization of the surface) we got the answer which is 8/3 But why the divergence theorem can not be used in our case? (I know it's a trap here) thank you!
  24. B

    I Divergence of first Piola-Kirchoff stress tensor

    Hi everyone, studying the bending of an incompressible elastic block of Neo-Hookean material, one finds out the first Piola-Kirchoff stress tensor as at page 182 (equation 5.93) where $e_r = cos(\theta)e_1 + \sin(\theta)e_2$ and $e_{\theta} = -sin(\theta)e_1 + \cos(\theta)e_2$ How is the...
  25. A

    Divergence in Spherical Coordinate System by Metric Tensor

    The result equation doesn't fit with the familiar divergence form that are usually used in electrodynamics. I want to know the reason why I was wrong. My professor says about transformation of components. But I cannot close to answer by using this hint, because I don't have any idea about "x"...
  26. E

    B Confusion about Divergence Theorem Step in Tong's Notes

    I wanted to ask about a step I couldn't understand in Tong's notes$$\int_M d^n x \partial_{\mu}(\sqrt{g} X^{\mu}) = \int_{\partial M} d^{n-1}x \sqrt{\gamma N^2} X^n = \int_{\partial M} d^{n-1}x \sqrt{\gamma} n_{\mu} X^{\mu}$$we're told that in these coordinates ##\partial M## is a surface of...
  27. SebastianRM

    Relating volumetric dilatation rate to the divergence for a fluid-volume

    in class we derived the following relationship: $$\frac{1}{V}\frac{dV}{dt}= \nabla \cdot \vec{v}$$ This was derived though the analysis of linear deformation for a fluid-volume, where: $$dV = dV_x +dV_y + dV_z$$ I understood the derived relation as: 1/V * (derivative wrt time) = div (velocity)...
  28. fluidistic

    I About divergence, gradient and thermodynamics

    At some point, in Physics (more precisely in thermodynamics), I must take the divergence of a quantity like ##\mu \vec F##. Where ##\mu## is a scalar function of possibly many different variables such as temperature (which is also a scalar), position, and even magnetic field (a vector field)...
  29. F

    MHB Divergence of the Navier Stokes equation

    If not, can someone walk me through the steps to get to the results that my professor got? Thank you.
  30. fluidistic

    Gauss' divergence theorem and thermoelectricity contradiction

    I get a nonsensical result. I am unable to understand where I go wrong. Let's consider a material with a temperature independent Seebeck coefficient, thermal conductivity and electrochemical potential to keep things simple. Let's assume that this material is sandwiched between 2 other materials...
  31. TheGreatDeadOne

    Using the Divergence Theorem on the surface of a sphere

    The integral that I have to solve is as follows: \oint_{s} \frac{1}{|r-r'|}da', \quad\text{ integrating with respect to r '}, integrating with respect to r' Then I apply the divergence theorem, resulting in: \iiint \limits _{v} \nabla \cdot \frac{1}{|r-r'|}dv' =...
  32. G

    Problem about the derivation of divergence for a magnetic field

    Summary:: I am trying to derive that the divergence of a magnetic field is 0. One of the moves is to take the curl out of an integral. Can someone prove that this is addressable Biot Savart's law is $$B(r)=\frac{\mu _0}{4\pi} \int \frac{I(r') \times (r-r')}{|r-r|^3}dl'=\frac{\mu _0}{4\pi}...
  33. F

    I Divergence & Curl -- Is multiplication by a partial derivative operator allowed?

    Divergence & curl are written as the dot/cross product of a gradient. If we take the dot product or cross product of a gradient, we have to multiply a function by a partial derivative operator. is multiplication by a partial derivative operator allowed? Or is this just an abuse of notation
  34. Z

    Electric Field Divergence of Monochromatic Plane Wave: Why is it Zero?

    Why is the divergence of an amplitude of an electric field of a monochromatic plane wave zero?
  35. M

    Divergence Theorem Verification: Surface Integral

    Hi, I just had a quick question about a step in the method of calculating the surface integral and why it is valid. I have already done the divergence step and it yields the correct result. Method: Let us calculate the normal: ## \nabla (z + x^2 + y^2 - 3) = (2x, 2y, 1) ##. Just to double...
  36. P

    Divergence of a radial field ##F=\hat{r}/r^{2+\varepsilon}##

    Following (1), \begin{align*} \text{div} F = \vec{\nabla} \cdot \vec{F} &= \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 F_{r}\right) \\ &= \frac{1}{r^2} \frac{\partial }{\partial r} \left( r^2 \frac{1}{r^{2+\varepsilon}}\right) \\ &= \frac{1}{r^2} \frac{\partial}{\partial r}...
  37. M

    Divergence Theorem in Curvilinear Coordinates: Questions & Explanations

    Hi, I was trying to gain an understanding of a proof of the divergence theorem in curvilinear coordinates. I have found these online notes here and am looking at the proof on pages 4-5. The method intuitively makes sense to me as opposed to other proofs which fiddle around with vector...
  38. K

    Nabla operations, vector calculus problem

    Here is how my teacher solved this: I understand what the nabla operator does, ##∇\cdot\vec v## means that I am supposed to calculate ##\sum_{n=1}^3\frac {d\vec v} {dx_n}## where ##x_n## are cylindrical coordinates and ##\vec e_3 = \vec e_z##. I understand why ##∇\cdot\vec v = 0##, I would get...
  39. A

    Understanding the Divergence Theorem

    Good day all my question is the following Is it correct to (after calculation the new field which is the curl of the old one)to use the divergence theroem on the volume shown on that picture? The divergence theorem should be applied on a closed surface , can I consider this as closed? Thanks...
  40. J

    Divergence Theorem Problem Using Multiple Arbitrary Fields

    My main issue with this question is the manipulation of the two arbitrary fields into a single one which can then be substituted into the divergence theorem and worked through to the given algebraic forms. My attempt: $$ ∇(ab) = a∇b + b∇a $$ Subsituting into the Eq. gives $$ \int dS ·...
  41. R

    Finding Scalar Curl and Divergence from a Picture of Vector Field

    For divergence: We learned to draw a circle at different locations and to see if gas is expanding/contracting. Whenever the y-coordinate is positive, the gas seems to be expanding, and it's contracting when negative. I find it hard to tell if the gas is expanding or contracting as I go to the...
  42. Eclair_de_XII

    How to prove divergence of harmonic series by eps-delta proof?

    Set ##\epsilon=\frac{1}{2}##. Let ##N\in \mathbb{N}## and choose ##n=N,m=2N##. Then: ##\begin{align*} \left|s_N-s_{2N}\right|&=&\left|\sum_{l=1}^N \frac{1}{l} - \sum_{l=1}^{2N} \frac{1}{l}\right|\\...
  43. Terrycho

    Divergence of a position vector in spherical coordinates

    I know the divergence of any position vectors in spherical coordinates is just simply 3, which represents their dimension. But there's a little thing that confuses me. The vector field of A is written as follows, , and the divergence of a vector field A in spherical coordinates are written as...
  44. Arman777

    Vector Divergence: Are the Expressions True?

    Do I have to write something like, $$\nabla' \cdot \vec{J} = \frac{\partial J^m(r')}{\partial x'^m} + \frac{\partial J^m(t_r)}{\partial x'^m}$$ $$\nabla \cdot \vec{J} = \frac{\partial J^m(r')}{\partial x^m} + \frac{\partial J^m(t_r)}{\partial x^m} = \frac{\partial J^m(t_r)}{\partial x^m}$$...
  45. B

    I Can the Chain Rule be Applied to Simplify Divergence in Entropy Equation?

    I am looking at the derivation for the Entropy equation for a Newtonian Fluid with Fourier Conduction law. At some point in the derivation I see \frac{1}{T} \nabla \cdot (-\kappa \nabla T) = - \nabla \cdot (\frac{\kappa \nabla T}{T}) - \frac{\kappa}{T^2}(\nabla T)^2 K is a constant and T...
  46. D

    Verify the convergence or divergence of a power series

    At the exam i had this power series but couldn't solve it ##\sum_{k=0}^\infty (-1)^\left(k+1\right) \frac {k} {log(k+1)} (2x-1)^k## i did apply the ratio test (lets put aside for the moment (2x-1)^k ) to the series ##\sum_{k=0}^\infty \frac {k} {log(k+1)}## in order to see to what this...
  47. G

    I A one dimensional example of divergence: Mystery

    I am trying to understand “divergence” by considering a one-dimensional example of the vector y defined by: . the parabola: y = -1 + x^2 The direction of the vector y will either be to the right ( R) when y is positive or to the Left (L). The gradient = dy/dx = Divergence = Div y = 2 x x...
  48. karush

    MHB 11.6.8 determine convergent or divergence by Ratio Test

    Use the Ratio Test to determine whether the series is convergent or divergent $$\sum_{n=1}^{\infty}\dfrac{(-2)^n}{n^2}$$ If $\displaystyle\lim_{n \to \infty} \left|\dfrac{a_{n+1}}{a_n}\right|=L>1 \textit{ or } \left|\dfrac{a_{n+1}}{a_n}\right|=\infty...
  49. N

    I Testing for Divergence using the Integral Test

    Hello all, I was working on some homework regarding testing for convergence and divergence of series and I was having trouble with a particular series (doesn't really matter which one) and tried almost all the methods; then tried the Integral Test, my series met the conditions of the...
  50. S

    A Effect of the driving current on the divergence values of laser diodes

    Dear readers, What happens to the divergence values of a laser diode (along and perpendicular to the emitting surface) when the driving current is increased or the output power is increased? Does the divergence: Increase along one axis? Increase along both axes? Decrease along both axes...
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