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. M

    Divergence Test for an Infinite Series (General question)

    This might seem like a rudimentary question but when trying to prove divergence (or even convergence) of an infinite series does the series always have to start at n = 1? For example would doing a test for \sum^{∞}_{n=1}\frac{1}{n} be any different from \sum^{∞}_{n=0}\frac{1}{n}
  2. M

    MHB Apply the divergence theorem to calculate the flux of the vector field

    Hey! :o I have the following exercise: Apply the divergence theorem to calculate the flux of the vector field $\overrightarrow{F}=(yx-x)\hat{i}+2xyz\hat{j}+y\hat{k}$ at the cube that is bounded by the planes $x= \pm 1, y= \pm 1, z= \pm 1$. I have done the following...Could you tell me if this...
  3. P

    Divergence Theorem Problem

    Homework Statement Use the divergence theorem (and sensible reasoning) to show that the E field a distance r outside a long, charged conducting cylinder of radius r0 which carries a charge density of σ Cm-2 has a magnitude E=σr0/ε0r. What is the orientation of the field? Homework Equations...
  4. M

    MHB Apply the divergence theorem for the vector field F

    Hey! :o Apply the divergence theorem over the region $1 \leq x^2+y^2+z^2 \leq 4$ for the vector field $\overrightarrow{F}=-\frac{\hat{i}x+\hat{j}y+\hat{k}z}{p^3}$, where $p=(x^2+y^2+z^2)^\frac{1}{2}$. $\bigtriangledown...
  5. S

    MHB Convergence and Divergence

    Determine whether the integral is Divergent or Convergent\int^0_{-\infty} \frac{1}{3 - 4x} dx I did a u substitution and got \lim_{a\to\infty} -\frac{1}{4}\sqrt{3} + \frac{1}{4}\sqrt{3 - 4a} So is because the -\infty is under the square root is it going to be divergent? I have...
  6. P

    Understanding the divergence theorem

    I'm having some trouble understanding what divergence of a vector field is in my "Fields and Waves" course. Divergence is defined as divE=∇E = (∂Ex/∂x) + (∂Ey/∂y) + (∂Ez/∂z). As far as I understand this gives the strength of vector E at the point(x,y,z). Divergence theorem is defined as ∫∇Eds...
  7. T

    Can the Divergence of a Bessel Integral Be Prevented?

    Hi, I would like to confirm my intuition about a bessel integral from you guys. The integral is: Integrate[ (1/r) * J[2,2*pi*phi*r] ] from 0 → ∞ with respect to r. J[2,2*pi*phi*r] is a second order bessel. Integrals with 1/x from 0 to Inf are divergent. Sure enough, this one is going...
  8. S

    Applying the Divergence Theorem to the Volume of a Ball with a Given Radius

    Homework Statement let Bn be a ball in Rn with radius r. ∂Bn is the boundary. Use divergence theorem to show that: V(Bn(r)) = (r/n) * A (∂Bn(r)) where V(Bn) is volume and A(∂Bn) is surface area. Homework Equations consider the function: u = x1 ^2 + x2 ^2 +...+ xn ^2 The...
  9. S

    Divergence Theorem-Electromagnetism

    Homework Statement Homework Equations The divergence theorem is quoted on the problem sheet. The Attempt at a Solution I am struggling with the last question (2)c)). I have tried to put the continuity equation into the divergence theorem and have got: ∫S J.ds=-d/dt∫V ρdV But...
  10. M

    Divergence theorem (determining the correct direction for normal vecto

    The problem is in the paint doc.. My question is why is the base vector aR have a negative sign attached to it?
  11. M

    Exploring Divergence of Function Arguments and Its Impact on Physics

    Hi folks -- could anyone think of a justification of the idea that if a function's arguments diverge (i.e. are taken to infinity), there's a high probability that the function too will diverge? This would be really helpful for thinking about fundamental theories in particle physics, so any...
  12. bcrowell

    Maxwell's equations from divergence of stress-energy tensor?

    If I start with the stress-energy tensor T^{\mu\nu} of the electromagnetic field and then apply energy-momentum conservation \partial_\mu T^{\mu\nu}=0, I get a whole bunch of messy stuff, but, e.g., with \nu=x part of it looks like -E_x \nabla\cdot E, which would vanish according to Maxwell's...
  13. F

    A seemingly simple exercise on the divergence theorem

    Here is the problem statement: I thought it's a straightforward exercise on the divergence theorem, yet it looks like \operatorname{div} f = 0 . So the surface integral is zero? Am I missing some sort of a trick here? The exercise isn't supposed to be that easy. Any hints are very appreciated!
  14. J

    Curl and divergence of units vectos

    Hellow! I'd like to know what results the curl and divergence of unit vectos bellow: https://www.physicsforums.com/attachment.php?attachmentid=65279&stc=1&d=1388593339 I just know that ∇·x = 0 ∇·y = 0 ∇·z = 0 ∇×x = 0 ∇×y = 0 ∇×z = 0
  15. N

    Divergence of a rank-2 tensor in Einstein summation

    Homework Statement Hi When I want to take the divergence of a rank-2 tensor (matrix), then I have to apply the divergence operator to each column. In other words, I get \nabla \cdot M = (d_x M_{xx} + d_y M_{yx} + d_zM_{zx}\,\, ,\,\, d_x M_{xy} + d_y M_{yy} + d_zM_{zy}\,\,,\,\, d_x M_{xz} +...
  16. PsychonautQQ

    Evaluate the Flux with Divergence Theorem

    Homework Statement Evaluate the flux where F = <(e^z^2,2y+sin(x^2z),4z+(x^2+9y^2)^(1/2)> in the boundary of the region x^2 + y^2 < z < 8-x^2-y^2 Homework Equations The Attempt at a Solution So using the divergence Theorem, ∇ dot F = 6 ∫∫∫6r dzdrdθ where z is bounded...
  17. T

    Evaluating Volume Integrals and Divergence Theorm

    Homework Statement Evaluate the integral as either a volume integral of a surface integral, whichever is easier. \iiint \nabla .F\,d\tau over the region x^2+y^2+z^2 \leq 25, where F=(x^2+y^2+z^2)(x*i+y*j+z*k) Homework Equations \iiint \nabla .F\,d\tau =\iint F.n\,d\sigma The...
  18. J

    Determine Divergence of Ʃ ((n!)^n) /(4^(4n))

    Homework Statement Ʃ ((n!)^n) /(4^(4n)) Homework Equations Root test? The Attempt at a Solution Can you do ... the root test so then you will get rid of exponents n and you have (n!)/n^4 then take the limit and you get ∞ so the original sum is divergent?
  19. M

    How to Apply the Divergence Theorem to a Non-Closed Surface?

    Homework Statement . Let ##C## be the curve in the plane ##xz## given in polar coordinates by: ##r(\phi)=\frac{4√3}{9}(2-cos(2\phi)), \frac{π}{6}≤\phi≤\frac{5π}{6}## (##\phi## being the angle between the radius vector and the positive z-semiaxis). Let ##S## the surface obtained by the...
  20. [Quadratic]

    How Do You Solve a Complex Divergence Problem Using Spherical Coordinates?

    Homework Statement Apologies for the attachment. Homework Equations Limit definition of the divergence as seen in attachment Volume of a sphere: \frac{4}{3}\pi r^{3}The Attempt at a Solution The first thing I did was parameterize the vector function F(x,y,z) = <xy,x,y+z> My parameterization is...
  21. U

    Component of vector parallel to boundary while calculating divergence

    So when we calculate divergence (especially referring to the gauss divergence theorem), why aren't the components of the vector field parallel to the boundary considered? I mean even of, say fluid, is traveling parallel to the boundary when it comes out, fluid is exiting, or diverging out...
  22. A

    Dirac Delta function and Divergence

    Homework Statement The Potential V(r) is given: A*e^(-lambda*r)/r, A and lambda are constants From this potential, I have to calculate: E(r), Rho(r) -- charge density, and Q -- total charge. Homework Equations The Attempt at a Solution I know that E(r) is simply minus...
  23. O

    Divergence formula derivation ?

    Homework Statement How to get equation 1 from the thumbnail? h1 h2 h3 doesn't have to be constant. The most I can try is equation 2 . Please guide thanks. Homework Equations The Attempt at a Solution
  24. M

    Proving the divergence of a Harmonic Series

    Homework Statement Prove that Hn converges given that: H_{n}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n} The Attempt at a Solution First I supposed that the series converges to H...
  25. A

    Does the Series \(\sum_{n=1}^{\infty} \frac{n+2^n}{n+3^n}\) Converge or Diverge?

    The sum is $$\sum_{n=1}^{\infty} \frac{n+2^n}{n+3^n}$$ Is this convergent or divergent? I tried to use the divergent test but the test fail because $a_n = (n+2^n)/(n+3^n) = 0 $ as $n$ goes to infinity. Could someone point me to the right direction? Thanks
  26. N

    How to deduce Gauss' law from Gauss Divergence Law

    Homework Statement Gauss Divergence Law: Gauss' law Can we obtain the Gauss' Law from Gauss Divergence Law? Homework Equations In Spherical coordinates, electric field strength (Q/4\piεr^2,0,0) Then ∇\cdotE=0+0+0=0 The Attempt at a Solution We can not obtain the...
  27. alyafey22

    MHB Proving Divergence of $\cos(n)$ w/ Definition of Limits

    Can we prove using the definition of limits of sequences that \lim \, \cos(n) diverges ? I mean can we use a contradiction or show that two sub-sequences have a different limit ?
  28. T

    Divergence of Curl: Intuitive/Physical Reason

    Can anyone give me an intuitive/physical reason for why the divergence of the curl of a vector field is always zero? I know various methods to prove mathematically that it is so, but have not managed to find a physical reason. In other words, why is the curl of a vector field always incompressible.
  29. M

    Divergence Theorem: Understanding and Applying in Vector Calculus

    hey pf! i had a general question with the divergence theorem. specifically, my text writes \iint_S \rho \vec{V} \cdot \vec{dS} = \iiint_v \nabla \cdot (\rho \vec{V}) where \rho is a scalar, although not necessarily constant! to properly employ the divergence theorem, would i first let \rho...
  30. F

    Sequences, Series, Convergence and Divergence

    Homework Statement Q1 Are the following sequences divergent or convergent as n tends to infinity. a: \frac{5n+2}{n-1} b: tan^{-1}(n) c:\frac{2^n}{n!} Q2 Evaluate:... a: \sum_{n=1}^{\infty} 3^{\frac{n}{2}} b: \sum_{n=1}^{99} (-1)^n Q3 Find whether the following converge or diverge...
  31. Saitama

    Simple Divergence related problem

    Homework Statement Sketch the vector function $$\vec{v}=\frac{\hat{r}}{r^2}$$ and compute its divergence. The answer may surprise you...can you explain it? Homework Equations The Attempt at a Solution I have recently started with Introduction to Electrodynamics by David J...
  32. L

    Using stokes' or divergence theorem to solve integral

    Homework Statement Use either Stokes' theorem or the divergence theorem to evaluate this integral in the easiest possible way. ∫∫V \cdotndσ over the closed surface of the tin can bounded by x2+y2=9, z = 0, z = 5, if V = 2xyi - y2j + (z + xy)k The bolded letters are vectors...
  33. F

    Check for the convergence or divergence of the following series

    Homework Statement Here are some series I'm completely stuck on. 1.sqrt(n)*(1-cos(1/n)) 2. a series in which if n is odd, then an is 1/(n+\sqrt[]{n}) while if n is even, then an is -1/n Homework Equations The Attempt at a Solution For 1., I tried integral test which seemed...
  34. S

    Divergence of r/r^3: Defining w/ Dirac Delta to Fill Gaps

    I have searched the forums and the internet to see various discussions about the divergence of an electric field, or more directly, the divergence of r/r^3. I still don't understand this "spike at r = 0" idea, and really don't believe it. It simply seems to be an idea that fills in the gaps of...
  35. T

    How do I derive the formula for divergence using a prism-shaped volume?

    Homework Statement In deriving the formula div v = \frac{∂v_{x}}{∂x} + \frac{∂v_{y}}{∂y} + \frac{∂v_{z}}{∂z} we used a rectangular solid infinitesimal volume; however, any shape will do (although the calculation gets harder). To see an example, derive the same formula using the prism-shaped...
  36. C

    MHB Sum series- convergence and divergence

    converge or diverge? \sum_{n=1}^{^{\infty }}a_{n} a_{1}= \frac{1}{3}, a_{n+1}= \sqrt[n]{a_{n}} Im having problems to solve this exercise, i would like to see your solutions
  37. D

    Proving divergence of a sequence

    Hello! Please help me to solve following exercise (2.5.8) from Elementary Real Analysis by Thomson-Bruckner: Suppose that a sequence \{s_n\} of positive numbers satisfies the condition s_{n+1} > \alpha s_n for all ##n## where ##\alpha>1.## Show that ##s_n \to \infty.## I can't prove...
  38. W

    Deriving a conservation law using the divergence theorem

    Problem: Material scientists have discovered a new fluid property called "radost" that is carried along with a fluid as it moves from one place to the next (just like a fluid's mass or momentum). Let ##r(x,y,z,t)## be the amount of radost/unit mass in a fluid. Let ##\rho(x,y,z,t)## be the...
  39. G

    Extended divergence theorem

    "Extended" divergence theorem ...which enables us to calculate the outward flux of a singular vector field through a surface S by enclosing it in some other arbitrary surface and looking at the inward flux instead. Is there any other application of this apart from the special case when...
  40. T

    Flux & Divergence Homework Help

    Homework Statement This is a coursework problem. I am having issues understanding the concepts on this one topic - divergence and how it relates to flux. I have attached screenshots that honestly give the best representation of my issue but I will set up the issue I am having...
  41. M

    'Eyeballing' non-zero divergence and curl from vector field diagrams

    Homework Statement Explain whether the divergence and curl of each of the vector fields shown below are zero throught the entire region shown. Justify your answer.https://sphotos-a-ord.xx.fbcdn.net/hphotos-prn2/1185774_4956047513788_517908639_n.jpg Homework Equations N/AThe Attempt at a...
  42. C

    Checking the divergence of a function

    Homework Statement Check the divergence theorem using the vector function V = r^2 \hat{r} + sin(θ) \hat{θ} which is expressed in spherical coordinates. For the volume use a hemisphere of unit radius above the xy-plane (see figure below) (picture not shown, but I integrated r: 0 to 1, theta: 0...
  43. W

    Divergence in spherical coordinates

    Problem: For the vector function \vec{F}(\vec{r})=\frac{r\hat{r}}{(r^2+{\epsilon}^2)^{3/2}} a. Calculate the divergence of ##\vec{F}(\vec{r})##, and sketch a plot of the divergence as a function ##r##, for ##\epsilon##<<1, ##\epsilon##≈1 , and ##\epsilon##>>1. b. Calculate the flux of...
  44. K

    Use the Divergence Theorem to Prove

    Homework Statement Let f and g be sufficiently smooth real-valued (scalar-valued) functions and let u be a sufficiently smooth vector-valued function on a region V of (x1; x2; x3)-space with a sufficiently smooth boundary ∂V . The Laplacian Δf of f: Δf:=∇*∇f=∂2f/∂x21 + ∂2u/∂x22 +...
  45. H

    Derive the divergence formula for spherical coordinates

    Homework Statement The formula for divergence in the spherical coordinate system can be defined as follows: \nabla\bullet\vec{f} = \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 f_r) + \frac{1}{r sinθ} \frac{\partial}{\partial θ} (f_θ sinθ) + \frac{1}{r sinθ}\frac{\partial f_\phi}{\partial...
  46. D

    Find the divergence of the function

    Homework Statement Let V = (sin(theta)cos(phi))/r Determine: (a) ∇V (b) ∇ x ∇V (c) ∇∇V Homework Equations The Attempt at a Solution Uploaded
  47. D

    Evaluate the divergence of the vector field

    Homework Statement Evaluate the divergence of the following vector fields (a) A= XYUx+Y^2Uy-XZUz (b) B= ρZ^2Up+ρsin^2(phi)Uphi+2ρZsin^2(phi)Uz (c) C= rUr+rcos^2(theta)Uphi Homework Equations The Attempt at a Solution Uploaded
  48. D

    Divergence of vector field help

    Hey guys! So I've been trying to get my head around Divergence of a vector field. I do get the general idea, however I thought of a hypothetical situation I can't get my head around. Look at the second vector field on this page, http://mathinsight.org/divergence_idea it has a negative...
  49. B

    Divergence Theorem: Find Delta Function at Origin

    Homework Statement Find the divergence of \vec v = \frac{\hat{v}}{r} Then use the divergence theorem to look for a delta function at the origin. Homework Equations \int ∇\cdot \vec v d\tau = \oint \vec v \cdot da The Attempt at a Solution I got the divergence easy enough...
  50. Y

    How Do You Compute the Divergence of a Vector Function Over a Scalar Field?

    Let ##\vec {F}(\vec {r}')## be a vector function of position vector ##\vec {r}'=\hat x x'+\hat y y'+\hat z z'##. I want to find ##\nabla\cdot\frac {\vec {F}(\vec {r}')}{|\vec {r}-\vec{r}'|}##. My attempt: Let ##\vec {r}=\hat x x+\hat y y+\hat z z##. Since ##\nabla## work on ##x,y,z##, not...
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