Divergence Definition and 53 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. 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.
  2. 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...
  3. 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...
  4. 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}...
  5. 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...
  6. 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...
  7. B

    I Divergence with Chain Rule

    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...
  8. 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...
  9. R

    Divergence of an Electric Field due to an ideal dipole

    Given $$\vec E = -\nabla \phi$$ there $$\vec d \rightarrow 0, \phi(\vec r) = \frac {\vec p \cdot \vec r} {r^3}$$ and ##\vec p## is the dipole moment defined as $$\vec p = q\vec d$$ It's quite trivial to show that ##\nabla \times \vec E = \nabla \times (-\nabla \phi) = 0##. However, I want to...
  10. W

    I The continuity equation and the divergence

    according to continuity equation (partial ρ)/(partial t) +divergence J = 0 . there is such a situation that there is continuous water spreads out from the center of a sphere with unchanged density ρ, and at the center dm/dt = C(a constant), divergence of J = ρv should be 0 anywhere except the...
  11. Hawkingo

    I What is the physical meaning of divergence?

    I want to visualize the concept of divergence of a vector field.I also have searched the web.Some says it is 1.the amount of flux per unit volume in a region around some point 2.Divergence of vector quantity indicates how much the vector spreads out from the certain point.(is a...
  12. E

    Divergence operator for multi-dimensional neutron diffusion

    Homework Statement [1] is the one-speed steady-state neutron diffusion equation, where D is the diffusion coefficient, Φ is the neutron flux, Σa is the neutron absorption cross-section, and S is an external neutron source. Solving this equation using a 'homogeneous' material allows D to be...
  13. maxknrd

    I More elegant way to solve divergence of arbitrary dotproduct

    This is more of a general question, but I've encountered this kind of exercises a lot in my current preperations for my exam: There are two cases but the excercise is pretty much the same: Compute $$(1) \space \operatorname{div}\vec{A}(\vec{r}) \qquad , where \thinspace...
  14. V

    Show that a series is divergent

    Homework Statement Show that $$\frac{(-1)^nn!}{z^n}$$ is divergent. Homework Equations We can use the ratio test, which states that if, $$\lim_{n\to\infty}\bigg|\frac{a_{n+1}}{a_n}\bigg|>1$$ a series is divergent. The Attempt at a Solution Applying the ratio test, we find that...
  15. E

    Convergence of a series

    Homework Statement Determine whether the following series converge, converge conditionally, or converge absolutely. Homework Equations a) Σ(-1)^k×k^3×(5+k)^-2k (where k goes from 1 to infinity) b) ∑sin(2π + kπ)/√k × ln(k) (where k goes from 2 to infinity) c) ∑k×sin(1+k^3)/(k + ln(k))...
  16. Pencilvester

    I Deriving the divergence formula

    Hello PF, I was reading through “A First Course in General Relativity” by Schutz and I got to the part where he derives the divergence formula for a vector:$$V^α { } _{;α} = \frac {1} {\sqrt{-g}} ( \sqrt{-g} V^α )_{,α}$$I’m having trouble with a couple of the steps he made. So we start with the...
  17. ubergewehr273

    I Divergence of ##\frac {1} {r^2} \hat r##

    Basically a case where a positive charge q is placed in space which for convenience is taken as the origin. This electric field must have a large positive divergence but yet when evaluated mathematically we get 0. Also when we find divergence, we find it for a point right ? or is it possible to...
  18. UMath1

    Divergence of downhill flowing water

    I just learned that an incompressible fluid must have zero divergence within a given control volume. Given that the divergence of a fluid at a point(x,y,z) can be found by taking the scalar sum of the of the x, y, z acceleration vectors at the given point, wouldn't this mean that water flowing...
  19. J

    I What is the gradient of a divergence and is it always zero?

    Hi Folks, Was just curious as to what is the gradient of a divergence is and is it always equal to the zero vector. I am doing some free lance research and find that I need to refresh my knowledge of vector calculus a bit. I am having some difficulty with finding web-based sources for the...
  20. T

    Polar Divergence of a Vector

    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...
  21. 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...
  22. 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...
  23. 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...
  24. 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...
  25. 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}...
  26. 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...
  27. Dave-o

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

    Homework Statement ∇ . r = 3, ∇ x r = 0 Homework Equations The Attempt at a Solution So far I've gotten up to ∇(∇^2 r)
  28. 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...
  29. 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...
  30. 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...
  31. 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\...
  32. 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...
  33. 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...
  34. 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}...
  35. P

    Divergence, Vector Calculus

    Got it. Thank you ;)
  36. N

    Verify Divergence Theorem

    Homework Statement Verify the divergence theorem for the function V = xy i − y^2 j + z k and the surface enclosed by the three parts (i) z = 0, s < 1, s^2 = x^2 + y^2, (ii) s = 1, 0 ≤ z ≤ 1 and (iii) z^2 = a^2 + (1 − a^2)s^2, 1 ≤ z ≤ a, a > 1. Homework Equations [/B]...
  37. M

    Confusion with the divergence of E fields

    Suppose I have electric field of the form ##\mathbf{E} = 3x\mathbf{i} + 3y\mathbf{j}##. Calculating the charge density gives me ##\rho = \epsilon_0 \nabla\cdot\mathbf{E} = 6\epsilon_0##. But now if I turn one of the components of the field in the opposite direction, for example ##\mathbf{E} =...
  38. H

    Intuition for divergences in sunset diagram

    What is the intuition behind divergences for the sunset diagram? I know that there is quadratic divergence by why no quartic divergence or higher?
  39. Titan97

    Finding flux from electric field

    Homework Statement If ##\vec{E}=k\frac{x\hat i +y\hat j}{x^2+y^2}##, find flux through a sphere of radius R centered at origin. Homework Equations ##\int E.da=\int(\nabla\cdot E)\cdot da## The Attempt at a Solution I was able to solve this problem without finding divergence of electric field...
  40. D

    Electric field inside and around a hollow sphere

    Hi everyone, I am wondering if anybody could help me out. For my study I got the following question but I got stuck in part C (see image below). I Found at A that due to symmetry all components which are not in Ar direction will get canceled out I found at B that there is only charge density at...
  41. SquidgyGuff

    Stoke's and Gauss's Theorum in proving div(curlA)=0

    Homework Statement The problem puts forth and identity for me to prove: or . It says that I can use "straight-forward" calculation to solve this using the definition of nabla or I can use Gauss's and Stoke's Theorum on an example in which I have a solid 3D shape nearly cut in two by a curve...
  42. T

    Divergence of radial unit vector field

    Sorry if this was addressed in another thread, but I couldn't find a discussion of it in a preliminary search. If it is discussed elsewhere, I'll appreciate being directed to it. Okay, well here's my question. If I take the divergence of the unit radial vector field, I get the result: \vec...
  43. L

    Convergence of improper integrals

    What is the difference between \int_{-\infty}^{\infty} \frac{x}{1+x^2}dx and \lim_{R\rightarrow \infty}\int_{-R}^{R} \frac{x}{1+x^2}dx ? And why does the first expression diverge, whilst the second converges and is equal to zero?
  44. C

    Convergence or Divergence of Factorial Series

    How can I find out if 1/n! is divergent or convergent? I cannot solve it using integral test because the expression contains a factorial. I also tried solving it using Divergence test. The limit of 1/n! as n approaches infinity is zero. So it follows that no information can be obtained using...
  45. S

    Intuitive interpretation of some vector-dif-calc identities

    Dear All, I am studying electrodynamics and I am trying hard to clearly understand each and every formula. By "understand" I mean that I can "truly see its meaning in front of my eyes". Generally, I am not satisfied only by being able to prove or derive certain formula algebraically; I want to...
  46. C

    Divergence Operator on the Incompressible N-S Equation

    Hello All, If I apply the Divergence Operator on the incompressible Navier-Stokes equation, I get this equation: $$\nabla ^2P = -\rho \nabla \cdot \left [ V \cdot \nabla V \right ]$$ In 2D cartesian coordinates (x and y), I am supposed to get: $$\nabla ^2P = -\rho \left[ \left( \frac...
  47. cvex

    How to get the laplacian of a scalar field?

    Hi, I am trying to calculate the laplacian of a scalar field but I might actually need something else. So basically I am applying reaction diffusion on a 2d image. I am reading the neighbours, multiplying them with these weights and then add them. This works great. I don't know if what I am...
  48. genxium

    How to compute the divergence of retarded scalar potential

    I'm learning time-dependent Maxwell's Equations and having difficulty understanding the following derivative: Given f(\textbf{r}, \textbf{r}', t) = \frac{[\rho(\textbf{r}, t)]}{|\textbf{r} - \textbf{r}'|} where \textbf{r} = x \cdot \textbf{i} + y \cdot \textbf{j} + z \cdot \textbf{k}, in...
  49. 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...
  50. 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...