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. Y

    Question in divergence and curl

    Let ##\vec {F}(\vec {r}')## be a vector function of position vector ##\vec {r}'=\hat x x'+\hat y y'+\hat z z'##. Question is why: \nabla\cdot\vec {F}(\vec{r}')=\nabla\times\vec {F}(\vec{r}')=0 I understand ##\nabla## work on ##x,y,z##, not ##x',y',z'##. But what if \vec {F}(\vec {r}')=\frac...
  2. Y

    Divergence in spherical coordinates.

    I want to verify: \vec A=\hat R \frac{k}{R^2}\;\hbox{ where k is a constant.} \nabla\cdot\vec A=\frac{1}{R^2}\frac{\partial (R^2A_R)}{\partial R}+\frac{1}{R\sin\theta}\frac{\partial (A_{\theta}\sin\theta)}{\partial \theta}+\frac{1}{R\sin\theta}\frac{\partial A_{\phi}}{\partial \phi}...
  3. L

    Flux/ Divergence Theorem interpretation

    Hello, I am approaching the end of my multivariable/ vector analysis "Calc III" class and have a question about flux. My book states that flux, ∫∫ F \bullet N dS measures the fluid flow "across" a surface S per unit time. Now, the divergence theorem ∫∫∫ divF dV measures the "same...
  4. Z

    Determine the convergence or divergence of the sequence

    Homework Statement Determine the convergence or divergence of the sequence with the given nth term. If the sequence converges find its limit. an = (1*3*5*...*(2n-1))/(2n)n Homework Equations lim n->infinity an = L The Attempt at a Solution The answer in the book shows: 1/2n *...
  5. P

    Divergence of dyadic product of a dyadic and vector

    I would like to take divergence of the following expression ∇.((xi × xj × xk)/r^3), which is a triadic. here × denote a dyadic product and r=mod(r vector) and xi, xj and xk are the components of r vector. So, the above eq. can also be written as ∇.((xixjxk ei×ej×ek)/r^3), where ei...
  6. Z

    Calculus: Absolute/Conditional Convergence or Divergence Question

    Homework Statement This is the problem: http://i.imgur.com/AtISqng.jpg its the first series, not the second one with the cos Homework Equations The Attempt at a Solution so anyways i did this question but i just had one doubt. in every other question like this that I've done, the series...
  7. marellasunny

    Why is the trace of jacobian=the divergence?

    What is the theoretical connection intuitively justifying that the trace of the jacobian=the divergence of a vector field? I know that this also equals the volume flow rate/original volume in the vector field but leaving that aside, what is the mathematical background behind establishing this...
  8. C

    Self energy logarithmic divergence due to chiral symmetry.

    In peskin at page 319 right above equation (10.6) he writes "If the constant term in a taylor expansion of the self energy were proportional to the cutoff ##\Lambda##, the electron mass shift would also have a term proportional to ##\Lambda##. But the electron mass shift must actually be...
  9. D

    Calculating flux via divergence theorem.

    Homework Statement Compute the flux of \vec{F} through z=e^{1-r^2} where \vec{F} = [x,y,2-2z]^T and r=\sqrt{x^2+y^2} . EDIT: the curve must satisfy z\geq 0 .Homework Equations Divergence theorem: \iint\limits_{\partial X} \Phi_{\vec{F}} = \iiint\limits_X \nabla\cdot\vec{F}\,dx\,dy\,dz...
  10. U

    Volume of paraboloid using divergence theorem (gives zero)

    Homework Statement A surface S in three dimensional space may be specified by the equation f(x, y, z) = 0, where f(x, y, z) is a real function. Show that a unit vector nˆ normal to the surface at point (x0, y0, z0) is given by Homework Equations The Attempt at a Solution r...
  11. Lebombo

    Improper Integrals and Series (convergence and divergence)

    Is it safe to say when an integral has an infinite boundary \int_n^∞ a_{n} and the limit yields a finite number, then the integral is said to converge. And when a series has an upper limit of infinity \sum_n^{∞}a_{n} and the limit yields a finite number, then the series is said to diverge.
  12. B

    MHB Convergence and Divergence - Calculus 2

    Alright, I have my final for Calc 2 on Monday. I am only stuck on two sections of problems because I am terrible at convergence and divergence/alternating series. I have questions for each one really. They are as follows: 4) a. Can I simply factor out the alternating series and apply a...
  13. B

    Can You Solve the Divergence Theorem for a Cylindrical Vector Field?

    Homework Statement Verify the divergence theorem by computing both integrals for the vector field F = <x^3, y^3, z^2> over a cylindrical region define by x^2+y^2 ≤ 9. Homework Equations Divergence Theorem, and Flux Integrals. The Attempt at a Solution I did the divergence...
  14. B

    Finding The Divergence Of A Vector Field

    Homework Statement Find The Divergence Of The Vector Field: < ex2 -2xy, sin(y^2), 3yz-2x> Homework Equations I know that divergence is ∇ dot F. The Attempt at a Solution When I did it by hand I got 2xex2 + 2ycos(y2) + 3y However wolfram alpha says it should be 2xex2 +...
  15. E

    Divergence in polar; not asking for derivation

    I understand how to derive the divergence in polar. Ok, so I have the formula. But what I am confused about is this: say u=(0,0,w(r)), so as you can see the third component of this vector u is in fact a function of r. If I plug this into the polar divergence formula, I get zero, fine. But...
  16. K

    Showing that the Einstein Tensor has zero divergence

    Homework Statement We have R_{iklm;n}+R_{iknl;m}+R_{ikmn;l} \equiv 0 Show that by multiplying above with g^{im}g^{kn} we'll get \left( R^{ik}-\frac{1}{2} g^{ik} R \right)_{;k} 2. The attempt at a solution g^{im}g^{kn} \left( R_{iklm;n}+R_{iknl;m}+R_{ikmn;l} \right) \equiv 0...
  17. N

    Can You Help Me Solve the Infinitely Large Integral ∫|r|-3dV in Gauss Law?

    Hey. I want to use integrals-math to get from Gauss law in divergence form to the one in integral form. I know you can do it by simply accepting ∇*E dV = ρ/ε => ∫ ∇*E dV= ∫ρ/εdV = Q/ε = ∫E*dA, but I want to do it another way. I want to begin with ∫∇*E*dV and end up with Q/ε. So: E =...
  18. S

    When Is the Divergence Test Applicable for Series?

    Homework Statement Not really a problem, more of a general question. When exactly can you use the Divergence test. Does it only work on both series and sequences?Homework Equations The series Diverges if lim ƩAn ≠ 0 The Attempt at a Solution If you take the lim of the series n^3/2n^3 ≠ 0 there...
  19. W

    Does the given series converge absolutely or conditionally?

    Homework Statement Determine either absolute convergence, conditional convergence or divergence for the series.Homework Equations \displaystyle \sum^{∞}_{n=1} \frac{(-1)^n}{5n^{1/4} + 5}The Attempt at a Solution It converges conditionally i know, but i can't figure out how. 1. I applied the...
  20. R

    Divergence of tensor times vector

    (My question is simpler than it looks at first glance.) Here is Reynolds Transport Theorem: $$\frac{D}{Dt}\int \limits_{V(t)} \mathbf{F}(\vec{x}, t)\ dV = \int \limits_{V(t)} \left[ \frac{\partial \mathbf{F}}{\partial t} + \vec{\nabla} \cdot (\mathbf{F} \vec{u}) \right] \ dV$$ where boldface...
  21. K

    Surface Integral With Divergence Thm

    Homework Statement Let ##\mathit{F}(x,y,z) = (e^y\cos z, \sqrt{x^3 + 1}\sin z, x^2 + y^2 + 3)## and let ##S## be the graph of ##z = (1-x^2-y^2)e^{(1-x^2-3y^2)}## for ##z \ge 0##, oriented by the upward unit normal. Evaluate ##\int_{S} \mathit{F} \ dS##. (Hint: Close up this surface and use the...
  22. A

    Proof of Divergence of a Series

    Prove that the series \displaystyle\sum_{k=1}^{\infty}\sqrt[k]{k+1}-1 diverges. I thought that I could show the n^{th} term was greater than \frac{1}{n} but this is turning out to be more difficult than I imagined. Is there a neat proof that n^n>(n+1)^{n-1}?
  23. R

    Divergence Theorem: Does Multiplying div F Multiply Volume?

    Homework Statement The divergence theorem states that ∫∫∫V div F dV = ∫∫S F(dot)Ndσ Suppose that div F = 1, then ∫∫∫V div F dV = ∫∫S F(dot)Ndσ If divF = 2, does the following hold true?∫∫∫V div F dV = 2∫∫S F(dot)Ndσ Homework Equations Since the divergence theorem computes the volume, if...
  24. H

    Divergence and Radially Symmetric Fields

    Is it possible for a spherically symmetric field, on all of R^3, to have a divergence of 0? (assuming the field is nonzero) Relevant equation: F=f(ρ)a (a is a unit vector of <x,y,z>) and f(ρ) is scalar fxn, and ρ = lal
  25. H

    Vector Potential and Zero Divergence

    I'm trying to understand when a vector field is equal to the curl of a vector potential. Why is it possible that there is always a vector potential with zero divergence? Relevent Equation: B=∇χA I'm trying to understand the proof that the above vector potential A can be one with zero divergence.
  26. A

    Gradient theorem by the divergence theorem

    Hi to all Homework Statement ∫∫∫∇ψdv = ∫∫ψ ds over R over S R is the region closed by a surface S here dv and ψ are given as scalars and ds is given as a vector quantitiy. and questions asks for establishing the gradient theorem by appliying the divergence theorem to each component...
  27. H

    Divergence of inverse square field and Dirac delta

    \nabla \cdot \frac{\mathbf{r}}{|r|^3}=4 \pi \delta ^3(\mathbf{r}) What's the proof for this, and what's wrong with the following analysis? The vector field \frac{\mathbf{r}}{|r|^3}=\frac{1}{r^2}\hat{r} can also be written \mathbf{F}=\frac{x}{\sqrt{x^2+y^2+z^2}^3}\hat{x}+...
  28. S

    Divergence of a cross product help

    I need to prove the identity: \nabla(\vec{A} \times \vec{B})=\vec{B} \bullet(\nabla \times \vec{A}) - \vec{A} \bullet( \nabla \times \vec{B}) I need to prove for an arbitrary coordinate system, meaning I have scaling factors. The proof should be quite straight forward if you use the levi...
  29. P

    Divergence of product of tensor and vector

    I am new to tensor algebra. I have an expression involving a 2nd rank tensor (actually a dyadic) and a vector. I want to take divergence of the product i.e. ∇. (T.V) However, I am not sure if the simple product rule would work here. If I use that ∇. (T.V)= (∇.T).V + T. (∇.V)...
  30. P

    Series convergence vs. divergence

    Simple question: Are there any series which we don't know whether or not they converge?
  31. T

    Divergence of velocity of incompressible fluid in uniform gravity

    Hi! The velocity field as a function of poisition of an incompressible fluid in a uniform acceleration field, such as a waterfall accelerated by gravity can be found as follows: The position is \vec{x}. The velocity field is \vec{v} = \frac{d\vec{x}}{dt}. The constant acceleration field...
  32. P

    Explanation of quadratic divergence in higgs mechanism

    Hi can anyone explain what a quadratic divergence is? and if so how it effects the mass of the scalar field i.e why m^2 = m^2_{0} + \delta m^2, why are these things squared? Also how would this divergence affect the standard model as a natural concept, because from reading books it would...
  33. WannabeNewton

    Divergence theorem question on hyperplanes

    Hi guys, this is in regards to a problem from Wald from the section on linearized gravity. We have a quantity t_{ab} very, very similar to the L&L pseudo tensor and have the quantity (a sort of total energy) E = \int_{\Sigma }t_{00}d^{3}x where \Sigma is a space - like hypersurface of a...
  34. M

    Solving a Divergence Question: What Equation Do I Use?

    I see identity in one mathematical book div \vec{A}(r)=\frac{\partial \vec{A}}{\partial r} \cdot grad r How? From which equation?
  35. C

    Components of Curl, Divergence

    Homework Statement I'm trying to understand where the Cartesian components of the rotor and the divergence of a vector field derived. I read that the divergence of a vector field is defined by: \vec { \nabla } \cdot \vec { F } =\lim _{ V\rightarrow \left\{ P \right\} }{ \frac { 1 }{ \left| V...
  36. S

    What is the divergence of a unit vector not in the r direction?

    Hi guys, I've run across a problem. In finding the potential energy between two electrical quadrupoles, I've come across the expression for the energy as follows: U_{Q}=\frac{3Q_{0}}{4r^{4}}\left[(\hat{k}\cdot \nabla)(5(\hat{k}\cdot \hat{r})^3-2(\hat{k}\cdot \hat{r})^2-(\hat{k}\cdot...
  37. M

    Divergence of a Magnetic Field not equaling zero

    I have a larger problem involving divergences and curls, but the correct answer requires ∇°B (divergence of B) = 0. I understand the proof of this in Griffiths, but the definition of divergence in cylindrical coordinates is: After using the product rule to split the first term, we get the...
  38. S

    Divergence of a particular serie

    Homework Statement \sum_{n=1}^{\infty}{(-1)^n*n} The Attempt at a Solution Well, this is a series I came upon when analyzing the endpoints of a particular power series, the thing is, my book says it's divergent by the Test for divergence, however, I can't find this result, what I tried to do...
  39. L

    Calculating the average divergence of the flow of fluid in a box.

    Homework Statement A cube of side 5cm is placed in a fluid. For the following velocity values, calculate the average divergence of the flow. Is there a source or sink within this box? The velocities are given in cm/s. The cube is oriented so that the +z axis exits the front face, the +x axis...
  40. S

    What Does the |x=0 = dy dz Mean in the Physical Interpretation of Divergence?

    Hello, I am new to calculus, and am having problems with divergence, I was reading something to explain the physical interpretation of divergence and i got stuck in the very first part. it says that if we have a small volume dxdydz at the origin, and that a fluid flowing into this volume...
  41. G

    Divergence of the curl problem question

    Homework Statement if a vector can be written as the curl of another vector, its divergence vanishes. Can you justify the statement: "any vector field whose divergence vanishes identically can be written as the curl of some other vector"? Homework Equations Prove this by construction. Let...
  42. L

    Divergence theorem example question (Thomas' Calculus)

    Hey guys, I have a general question about example 4 in section 16.8 of the book "Thomas' Calculus, Early Transcedentials". So far I understand the material given in the book without any problems but this particular example is a little bit unclear to me. Homework Statement Given a vector field...
  43. tomwilliam2

    Understanding the Divergence Operator for Time-Varying Vectors

    Homework Statement I'm trying to find the divergence of a vector field (a fluid flow vector), but the vector takes the form u = u(x,y,z,t) The Attempt at a Solution I only really know how to take the divergence of a time-independent vector, so I'm guessing I just take the partial...
  44. mikeanndy

    Understanding Divergence and Curl in Fluid Dynamics

    What is the Physical significance of Divergence and Curl?
  45. M

    Surface integral and divergence theorem over a hemisphere

    Homework Statement Please evaluate the integral \oint d\vec{A}\cdot\vec{v}, where \vec{v} = 3\vec{r} and S is a hemisphere defined by |\vec{r}| \leqa and z ≥ 0, a) directly by surface integration. b) using the divergence theorem. Homework Equations -Divergence theorem in...
  46. 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...
  47. 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...
  48. 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...
  49. 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...
  50. 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.
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