Divergence Definition and 775 Threads

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. J

    Is there a divergence theorem for higher dimensions and what is it called?

    The fundamental theorem of calculus is basically the divergence theorem but dealing with a ball in R^1 instead of a ball in R^3. The fundamental theorem of Calculus relates the stuff inside the ball to its boundary, just like how the divergence theorem relates the stuff inside a volume with its...
  2. G

    Non-compact Divergence Theorem: Is it Applicable to Scattering Problems?

    Are there versions of the divergence theorem that don't require a compact domain? In my reference literature the divergence theorem is proved under the assumption that the domain is compact.
  3. J

    Understanding Divergence: Solving the Mystery of Vector Functions | Jim L.

    Been working my way thru H.M. Schey-been out of college for 50 yrs. This problem has me stumped. F (of x,y,z)= i (f of x) + j (f of y)+k f (-2z). F is a vector function, and i,j,k are unit vectors for x,y,z axis. The problem is to find Div F., and then show it is 0 for the point c,c...
  4. E

    Divergence and Curl: Can a Non-Constant Function Have Both 0?

    I am trying to think of a non-constant function whose divergence and curl is 0. It seems like this is impossible to me. Any hints?
  5. E

    Understanding Divergence Transformations in 2D Rotations

    divergence question show that the divergence transforms as a vector under 2D rotations. I am so confused abouth what this question wants me to do. Obviously the divergence is not invariant under rotations. Consider the divergence of the function f(x,y) = x^2 * x-hat. The divergence is...
  6. E

    Divergence Explained: Velocity & Density in Fluid Flow

    My book says that divergence can be understood in the context of fluid flow as the rate at which density flows out of a given region. It says that if F(x,y,z) is the velocity of a fluid, then that is the interpretation of the divergence. I fail to understand where the density comes in when we...
  7. S

    What is the physical significance of the divergence?

    Hello; I remember the days of muti variable calculus. The man said that divergence is equal to del dot the vector field. So on the exam he gave us a vector field, and I did del dot the given vector field and won big time. The other day I decided my concentration would be...
  8. R

    Convergence or divergence of log series

    Homework Statement I'm supposed to evaluate the following series or show if it diverges: SUM (sigma) log [(x+1)/x] Homework Equations Drawing a blank...:confused: The Attempt at a Solution I'm unsure how to start this. We've gone over all sorts of tests for convergence (ratio, comparison...
  9. S

    Why Is There No Generalized Function for div (r̂ / r²)?

    We know that div \; (\hat{r} / r ) = 4 \pi \delta (r) Why is there no generalized function (distribution) for div \; (\hat{r} / r^2) = ??
  10. K

    First test of divergence lim n-> n / 8^n

    Homework Statement n / 8^nHomework Equations The Attempt at a Solution It converges to 8 / 49? Not sure how. First test of divergence lim n-> n / 8^n. infinity / infinity = 1. BUT bottom grows fast. Using L`Hospital lim n -> 1 / 3*8^n*ln(2) ---> goes to 0 Tried to use the ratio test [ 1 /...
  11. G

    How can i do the divergence of a matrix 3x3?

    How can i do the divergence of a matrix 3x3?
  12. H

    Divergence Theorem - Confused :s (2 problems)

    Question Evaluate both sides of the divergence theorem for V =(x)i +(y)j over a circle of radius R Correct answer The answer should be 2(pi)(R^2) My Answer the divergence theorem is **integral** (V . n ) d(sigma) = **double intergral** DivV d(tau) in 2D. Where (sigma)...
  13. H

    Zero divergence in an enclosed point charge

    Why does an enclosed point charge have zero divergence/flux? Mathematically I can see that when the divergence operator is applied to E=q/r^2 (pointing in the r direction) I get zero, but what is the physical explanation for what is going on? I am confused because other enclosed electric fields...
  14. L

    Divergence of a partition function

    Let us consider a collection of non-interacting hydrogen atoms at a certain temperature T. The energy levels of the hydrogen atom and their degeneracy are: En = -R/n² gn = n² The partition function in statistical physics is given by: Z = Sum(gn Exp(-En/kT), n=1 to Inf) This...
  15. S

    Using root test and ratio test for divergence

    Homework Statement Does this series converge or diverge? Series from n=1 to infinity n(-3)^(n+1) / 4^(n-1) Homework Equations The Attempt at a Solution Okay, I've tried it both ways. Ratio test: lim n --> inf. ((n+1)*(-3)^(n+1)/4^n) / (n * (-3)^n / 4^(n-1)) Now...
  16. S

    Testing for Convergence or Divergence of 1+sin(n)/10^n Series

    Homework Statement Does the sum of the series from n=1 to infinity of 1+sin(n)/10^n converge or diverge. Homework Equations The Attempt at a Solution I can use the comparison test or the limit comparison test. I'm not sure where to go from here.
  17. B

    Explain why the divergence of each of the functions must be zero

    Homework Statement the question is: Without doing any calculations, explain why the divergence of each of the functions must be zero(Hint: consider what electric field these functions physically correspond to) P.S: I make the unit vectors in bold. A1=x A2=ρ(1/ρ) with ρ>0...
  18. J

    Vector calculus - Divergence Theorem

    Homework Statement Find \int_{s} \vec{A} \cdot d\vec{a} given \vec{A} = ( x\hat{i} + y\hat{j} + z\hat{k} ) ( x^2 + y^2 + z^2 ) and the surface S is defined by the sphere R^2 = x^2 + y^2 + z^2 directly and by Gauss's theorem. Homework Equations \int_{s} \vec{A} \cdot d\vec{a} =...
  19. S

    Divergence- Useful Concept? Why?

    I am studying divergence and curl in my E&M class. I was wondering, why is divergence a useful concept? I mean, for point charges, the divergence is zero everywhere except where the charge is located. Even for charged surfaces, \nabla\cdot E = \frac{\rho}{\epsilon} Loooking at this it seems...
  20. A

    If the divergence of a vector field is zero

    Homework Statement If the divergence of a vector field is zero, I know that that means that it is the curl of some vector. How do I find that vector? Homework Equations Just the equations for divergence and curl. In TeX: \nabla\cdot u=\frac{\partial u_x}{\partial x}+\frac{\partial...
  21. V

    Proof of Divergence for the Harmonic Series.

    Homework Statement Prove the divergence of the harmonic series by contridiction Homework Equations Attached file The Attempt at a Solution I understand what they are doing in the first two lines, however, the lines after assuming the series converges with sum S, confuses me. They...
  22. H

    How to interpret physically the divergence of vector field?

    Hi all. I have difficulty in visualizing the concept of divergence of a vector field. While I have some clue in undertanding, in fluid mechanics, that the divergence of velocity represent the net flux of a point, but I find no clue why the divergence of an electric field measures the charge...
  23. H

    I have difficulty in visualizing the divergence of vector fields.

    Hi all. I have difficulty in visualizing the concept of divergence of a vector field. While I have some clue in undertanding, in fluid mechanics, that the divergence of velocity represent the net flux of a point, but I find no clue why the divergence of an electric field measures the charge...
  24. A

    Using Gauss (Divergence) theorem to find charge distribution on a conductor

    Hi, I hope this is advanced enough to warrant being in this section: I'm supposed to use the Gauss theorem (and presumably his law) to show: 1)The charge on a conductor is on the surface. 2)A closed hollow conductor shields its interior from fields due to charges outside, but doesn't...
  25. M

    Determine the converge or divergence of the sequence

    a_n = (1 + k/n)^n Determine the converge or divergence of the sequence. If it is convergent, find its limit. My professor said to convert the sequence to f(x) and use ln (ln y) and L'Hospital's Rule. Do I have to use ln? Is there another way to find the convergence?
  26. K

    Problem understanding divergence test

    I am going through Boas.Ch-1.on infinite series. Can anyone help? 1.May we use preliminary divergence test for series with +ve and -ve terms?How?For some situation occurs when we are supposed to make out (-1)^infinity
  27. S

    Rigorous Divergence Theorem Proof

    The Background: I'm trying to construct a rigorous proof for the divergence theorem, but I'm far from my goal. So far, I have constructed a basic proof, but it is filled with errors, assumptions, non-rigorousness, etc. I want to make it rigorous; in so doing, I will learn how to construct...
  28. C

    Proof of convergence and divergence

    Prove that if a_{n} > 0 and \sum a_{n} converges, then \sum a_{n}^{2} also converges. So if \sum a_{n} converges, this means that \lim_{n\rightarrow \infty} a_{n} = 0 . Ok, so from this part how do I get to this step: there exists an N such that | a_{n} - 0 | < 1 for all n > N...
  29. Pythagorean

    Is Divergence Commutative in Vector Calculus?

    is (DEL dot A) the same as (A dot DEL)? I know the dot product is commutative, but this involves an operator. if the answer is YES, than why does one of the product rules read like this: DEL X (A X B) = (B dot DELL)A - (A dot DEL)B + A(DEL dot B) - B(DEL dot A) they commute the...
  30. H

    What meaning is divergence and curl?

    What does a divergence calculate? What meaning is it for a vector? And also what is curl? Why we have to definte divergence and curl?
  31. J

    Divergence Problem: Solving for h'(r)

    This problem has me stumped: If r = (x^2 + y^2)^{1/2}, show that div \left( \frac{h(r)}{r^2}(x \vec{i} + y \vec{j}) \right) = \frac{h'(r)}{r} My trouble is with mixing the polar coordinates with the position vector. If I write the above as div \left( \frac{h((x^2 +...
  32. J

    Understanding Divergence Graphically

    I'm trying to get a handle on seeing whether a vector field (in the x-y plane for simplicity) has zero divergence at a point or is divergence free altogether. I'm having some trouble with this. The way I am thinking about it is to mentally draw a little box around my point and then see...
  33. D

    Calculating Divergence of a Function: A Berliner's Struggle

    Hi everybody, I'm preparing myself for the introduction to electrodynamics course and thus I go through vector analysis, but I hardly understand a problem given in my book (Griffiths, Introduction to Electrodynamics 3rd edition, page 18, Problem 1.16): I have to calculate the divergence...
  34. N

    Divergence Theorem Help - Flux of Vector A Through V

    Consider the volume V bounded below by the x-y plane and above by the upper half-sphere x^2 + y^2 + z^2 = 4 and inside the cylinder x^2 + y^2 = 1 Given vector field: A = xi + yj + zk Use the divergence theorem to calculate the flux of A out of V through the spherical cap on the cylinder...
  35. R

    Divergence of (A x B): Proving the Equation for Vector Calculus

    Hi, I'm having trouble proving that: \nabla \cdot (\textbf{A} \times \textbf{B}) = \textbf{B}\cdot (\nabla \times \textbf{A}) - \textbf{A}\cdot (\nabla \times \textbf{B}) This is how I proceeded: \textbf{A} \times \textbf{B} = \overrightarrow{i}(A_y B_z - A_z B_y) + \overrightarrow{j}...
  36. F

    Divergence what am I doing wrong

    I don't understand what I am doing wrong here. I'm supposed to show that this function is divergence free. \vec v = \left( \frac{x}{x^2+y^2}, \frac{y}{x^2+y^2} \right) I ran the divergence through with my TI-89 at it equals 0. But, I want to calculate it by hand, so it would be easier to...
  37. F

    Divergence Theorem - Curve Integrals

    This question is throwing me for a loop. Q: If u = x^2 in the square S = \{ -1<x,y<1\} , verify the divergence theorem when \vec w = \Nabla u : \int\int_S div\,grad\,u\,dx\,dy = \int_C \hat n \cdot grad\,u\,ds If a different u satisfies Laplace's equation in S , what is the net flow...
  38. L

    Test of Divergence Theorem in Cyl. Coord's.

    Ok I am stuck yet again. Below is a synopsis of everything I have done. D. J. Griffiths, 3rd ed., Intro. to Electrodynamics, pg. 45, Problem #1.42(a) and (b): (a) Find the divergence of the vector function: \vec{v} = s(2 + sin^2\phi)\hat{s} + s \cdot sin\phi \cdot cos\phi \hat{\phi} +...
  39. L

    Mysterious Factor of 2 in Divergence Theorem

    Let \vec{v} = r^{2} \hat{r}. Show that the divergence theorm is correct using 0 <= r <= R , 0 <= \theta <= \pi , and 0 <= \phi <= 2\pi . $ \int \nabla \cdot \vec{v} d \tau = \int \vec{v} \cdot d \vec{a} $ First the divergence of \vec{v}. \nabla \cdot \vec{v} = 2r. Then the volume...
  40. M

    What is the result of multiplying a matrix by the divergence of a vector?

    So here's my problem. It may be very simple, but I don't know how to do it. Please help. Suppose \tau is a 3x3 matrix with elements listed as (a b c; d e f; g h i). What would be the answer to \nabla\bullet\tau be? Thx -Mark
  41. F

    A: How do you use the divergence theorem to find the flux through a unit sphere?

    This problem is either really easy, or I'm really dumb, and since there are no answers to check my work I figured someone here might want to help :) Q: w=(x,y,z) what is the flux \int \int w \cdot n\,\, dS out of a unit cube and a unit sphere? Compute both sides in the divergence theorem...
  42. L

    Help needed with (the point form of) Gauss's law & divergence of E and

    I need to show that \vec{\nabla} \cdot \vec{E}= \frac{\rho}{\epsilon_0} where \rho is the volume charge density. I know that if I can show that the net flux of the electric field (in three directions xyz) out of the a small gaussian surface in the shape of a cube with faces parallel to...
  43. C

    Is Your Series Convergent or Divergent?

    PLEASE HELP!...Convergence and Divergence! Can n e one help me with this! its determining whether or not a series is convergent or divergent...Thanks http://img.photobucket.com/albums/v211/chu_bear10/Calcy.jpg"
  44. joema

    How to calculate achievable flashlight beam divergence

    Assuming a typical white-light focusing flashlight (e.g, Mag-Lite) with a parabolic reflector focused to produce a convergent/divergent beam, how do I calculate the smallest achievable spot size (i.e, smallest beam cross sectional area) at a given distance? How does this vary with reflector...
  45. S

    *screams in anger* ok, divergence theorem problem

    question says answer in whichever would be easier, the surface integral or the triple integral, then gives me(I'm in a mad hurry, excuse the lack of formatting...stuff) the triple integral of del F over the region x^2+y^2+z^2>=25 F=((x^2+y^2+z^2)(xi+yj+zk)), so del F would be 3(x^2+y^2+z^2)...
  46. P

    Purpose of each of the operators , divergence, gradient and curl?

    purpose of each of the "operators", divergence, gradient and curl? Hi. Can anybody give me a reasonably simple explanation of what the purpose of each of the "operators", divergence, gradient and curl? (I've been looking but I never found something simple to understand) I know how to evaluate...
  47. T

    How Does the Divergence Theorem Apply in Vector Calculus and PDE?

    Suppose D \subset \Re^3 is a bounded, smooth domain with boundary \partial D having outer unit normal n = (n_1, n_2, n_3) . Suppose f: \Re^3 \rightarrow \Re is a given smooth function. Use the divergence theorem to prove that \int_{D} f_{y}(x, y, z)dxdydz = \int_{\partial D} f(x, y...
  48. T

    Verify Divergence Theorem for Q with G(x,y,z) in $\Re^3$

    Let Q denote the unit cube in \Re^3 (that is the unite cube with 0<x,y,z<1). Let G(x,y,z) = (y, xe^z+3y, y^3*sinx). Verify the validity of the divergence theorem. \int_{Q} \bigtriangledown} \cdot G dxdydz = \int_{\partial Q} G \cdot n dS I am not sure how to evaluate the right side. Any...
  49. D

    Vector Field: Showing Divergence & Curl A = 0

    A vector field is difined by A = f(r)r. a) show that f(r) = constant/r^3 if divergence A equal to zero. b) show that curl A is alway equal to zero
  50. S

    Question on Divergence in Cylindrical Coords

    Hey, I have a question on a derivation The following is in my textbook (V = vector): \nabla \cdot V = \frac {1}{r} \frac {\partial{(rV_{r}})}{\partial{r}} + \frac {1}{r} \frac {\partial{V_{\theta}}}{\partial{\theta}} + \frac {\partial{V_{z}}}{\partial{z}} where: \nabla = \hat {r}...
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