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. L

    Problem with taking the divergence

    hi, can someone tell me how \nabla\dot(\frac{\widehat{r}}{r^2})=4\pi\delta^3(r) thanks
  2. S

    Divergence Theorem: Find Outward Flux of F (x3,x2y,xy)

    Another question of a practice test. How do I use the Divergence theorem to find the outward flux of the field F = (x3,x2y,xy) out through the surface of the solid U = (x,y,z): 0 < y < 5-z, 0 < z < 4-x2. The answer is 4608/35.
  3. J

    Divergence of curvature scalars * metric

    How can one work out what terms like: (g^{cd}R^{ab}R_{ab})_{;d} are in terms of the divergence of the Ricci curvature or Ricci scalar? One student noted that since: G^{ab} = R^{ab} - \frac12 g^{ab}R {G^{ab}}_{;b} = 0 that we could maybe use the fact that G^{ab}G_{ab} = R^{ab}R_{ab} - \frac12...
  4. B

    How to Derive the Divergence of Magnetic Field in Cylindrical Coordinates?

    Homework Statement Ok well all I am told in the question is that the magnetic field B at a distance r from a straight wire carrying current I has magnitude uoI/2pi r.. The lines of force are circles on the wire and in planes perpendicular to it.. Show that divB = 0 Homework...
  5. S

    Real Analysis: convergence and divergence

    Homework Statement Suppose \sum n converges and an is greater than 0 for all n. Show that the sum of 1/an diverges. Homework Equations The Attempt at a Solution
  6. U

    Divergence theorem in curved space

    I have been contemplating my confusion about my intuition regarding GR and believe I have tracked down the primary source of confusion. The classical theories I have been taught assumed flat space with independent time and used the divergence theorem to derive inverse squared laws for fields...
  7. D

    Divergence of product of killing vector and energy momentum tensor vanishes. Why?

    Hi, in my book, it says: ----------------------- Beacause of T^{\mu\nu}{}{}_{;\nu} = 0 and the symmetry of T^{\mu\nu}, it holds that \left(T^{\mu\nu}\xi_\mu\right)_{;\nu} = 0 ----------------------- (here, T^{\mu\nu} ist the energy momentum tensor and \xi_\mu a killing vector. The semicolon...
  8. B

    Why Do Divergence Theorem and Regular Flux Method Yield Different Results?

    Note: I've attached images of my work at the bottem of this post. I've calculated the flux through a given surface by using The Divergence Theorem and by using the regular flux method. These methods give different results, however. This leads me to assume one of the following is...
  9. T

    Divergence of Complex Harmonic

    Homework Statement WTS divergence of: \sum_{-\infty}^{\infty} \frac{1}{z-n} Homework Equations Basic algebraic manipulation, standard tests of non-convergence? The Attempt at a Solution I have been playing with algebra, perhaps this equivilant (I hope equivilant, anyway)...
  10. 3

    Problem Understanding Divergence in Improper Integrals

    I am having a problem with the definition of divergence in improper integrals. My understanding of the logic behind convergence and divergence is that, for example, as a improper integral approaches infinity the area under the function will be approaching but never reach zero. This implies...
  11. G

    Prove using divergence theorem

    Use the divergence theorem to show that \oint\oints (nXF)dS = \int\int\intR (\nablaXF)dV. The divergence theorem states: \oint\oints (n.F)dS = \int\int\intR (\nabla.F)dV. The difference is switching from dot product to cross product. I have no idea how to start. Can someone please point...
  12. G

    Verifying divergence theorem with an example

    Verify the divergence theorem when F=xi+yj+zk and sigma is the closed surface bounded by the cylindrical surface x^2+y^2=1 and the planes z=0, z=1. I've done the triple integral side of the equation and got 3pi but don't know how to solve the flux side of the equation \oint\ointF.ds. Any...
  13. M

    Use Divergence Theorem to Compute the Flux Integral Just a work check

    Alright so I found div F=3x2+3y2+3z2 The integral then becomes the triple integral of the divergence of F times the derivative of the volume. Changing into spherical coordinates, the integral becomes 3\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{1}p^{4}sin{\phi}dpd{\phi}d{\theta} which ends up...
  14. M

    Why Does the Contraction Term Vanish in the Divergence Theorem on Manifolds?

    Hi, I'm having some trouble understanding this theorem in Lang's book, (pp. 497) "Fundamentals of Differential Geometry." It goes as follows: \int_{M} \mathcal{L}_X(\Omega)= \int_{\partial M} \langle X, N \rangle \omega where N is the unit outward normal vector to \partial M , X...
  15. M

    Divergence of a vector field is a scalar field?

    Hello. How can I show the Divergence of a vector field is a scalar field(in E^{3}) ? Should I show that Div is invariant under rotation? x^{i'}=a^{ij}x^{j},V^{'}_{i}(\stackrel{\rightarrow}{x})=a_{ij}v_{j}(\stackrel{\rightarrow}{x}) then \frac{\partial...
  16. T

    Divergence theorem/vector calculus

    Homework Statement I want to show that: Grad dot product with r = 2/r where: r is the unit vector r/r r = xi + yj + zk r is magnitude of r The Attempt at a Solution I think the answer should be 3/r since unit vector r = r/r, r = (x/r)i + (y/r)j + (z/r)k then when I do...
  17. J

    Solving for Divergence and F=∇×A

    Homework Statement A vector field F for which div F = 0, is called incompressible (also called solenoidal). Consider the vector field F(x, y, z) = ⟨y, x + y, −z⟩. (a) (1 point) Show that F is incompressible. (b) (3 points) Find a vector field A such that F=\nabla×A.Homework Equations div F =...
  18. N

    How to calculate divergence of some special fields

    \[ \nabla \cdot \frac{{\vec e_r }}{{r^2 }} = 4\pi \delta (\vec r) \] This can be seen from\[ \nabla \cdot \frac{{\vec e_r }}{{r^2 }} = \frac{1}{{r^2 }}\frac{\partial }{{\partial r}}(r^2 \cdot \frac{1}{{r^2 }}) = \frac{1}{{r^2 }}\frac{\partial }{{\partial r}}(1) = 0(r \ne 0) \] And...
  19. L

    Divergence & Curl: Exploring Meaning & Significance

    What do Divergence and Curl of a vector function actually mean? They are nice to understand as mathematical operators and then we can work on with them, but what do they mean physically and why are they so important in our study of electromagnetism?
  20. L

    Why is the divergence like that?

    Why is the concept of divergence defined to be the sum of the partial derivatives of the x, y and z components of a vector field E with respect to x, y and z, instead of being defined as the sum of the partial derivatives of E itself with respect to x, y and z? What would this operation I just...
  21. C

    Improper integral convergence or divergence.

    Homework Statement Use Comparison Theorem to determine whether the integral is convergent or divergent: integral from 0 to infinity of: arctan(x) / (2 + e^x) Should look like this: http://bit.ly/cAhytV Homework Equations -- The Attempt at a Solution I tried to compare...
  22. N

    Compute area using divergence and flux?

    Compute area using divergence and flux?? Consider the curve given by g(t) =acos^3(t),asin^3(t), where t is [0; 2pi] and a > 0 is a constant. (a) Find the unit tangent and outward normal vectors. (b) Compute the area enclosed by this curve. I have done part a), and I know that flux of F...
  23. N

    Compute area using divergence and flux?

    Compute area using divergence and flux?? Consider the curve given by g(t) =acos^3(t),asin^3(t), where t is [0; 2pi] and a > 0 is a constant. (a) Find the unit tangent and outward normal vectors. (b) Compute the area enclosed by this curve. I have done part a), and I know that flux of F...
  24. S

    Divergence of mass current density

    Hi, i thought a while about the meaning of the following expression \delta t \, \cdot \, \mathrm{div} \, \vec j_m(\vec r,t) \qquad \mathrm{with} \qquad \vec j_m(\vec r,t) = \rho_m(\vec r,t) \cdot \vec v(\vec r, t)Does it indicates the mass, which is produced / annihilated in the Volume...
  25. K

    Divergence of a vector function

    Homework Statement Let's define the radial vector \vec{v}(r) = \hat{r}/r^{2} where \vec{r} = \vec{OP} (O being the origin of our coordinate system and P being our observation point at point (x, y, z)). Using spherical coordinates, demonstrate that \vec{\nabla } \cdot\vec{v}(r) = 0 everywhere...
  26. J

    Gauss Divergence theorem to find flux through sphere with cavity.

    Homework Statement Use the divergence theorem to find the outward flux of a vector field F=\sqrt{x^2+y^2+z^2}(x\hat{i}+y\hat{j}+z\hat{k}) across the boundary of the region 1\leq x^2+y^2+z^2 \leq4 Homework Equations The Gauss Divergence Theorem states \int_D dV \nabla \bullet...
  27. G

    Answer "Find Divergence & Curl of Vector Field A

    Homework Statement find the divergence and curl of the vector field A = (x/(\sqrt{x^2 + y^2 + z^2}))i + (y/(\sqrt{x^2 + y^2 + z^2}))j + (z/(\sqrt{x^2 + y^2 + z^2}))k Homework Statement The Attempt at a Solution Im not going to go through the whole lot but i have done the whole...
  28. E

    Vector calculus: divergence of a cross product

    Homework Statement I need to prove the identity div (a x b) = b dot (curl a) - a dot (curl b) The Attempt at a Solution I've done the proof about 10 times now, and everytime I get the left hand of the identity equal to this: (all the d's are partial derivatives) d(a3b1)/dx -...
  29. A

    Solve Divergence Question: Calculate (B [dot] \nabla)A

    Given two vectors, A and B: A = (x\widehat{x} + 2y\widehat{y} + 3z\widehat{z}) B = (3y\widehat{x} - 2x\widehat{y}) I need to calculate (B [dot] \nabla)A, as part of a problem. The answer should be: \widehat{x}(3y) + \widehat{y}( -4x) I get: (B [dot] \nabla)A = ((3y) \delta /...
  30. B

    Divergence of Newton's law of gravitation

    I am studying vector calculus, and I saw the following result in a physics text: g = -\frac{m}{r^3}\vec{r} r^2 = x^2 + y^2 + z^2 \vec{r} = ix + jy +kz \nabla \cdot g = 0 I'm not sure how this was done. Is the product rule used somehow? What happened to the extra power of r...
  31. S

    Testing Divergence: Alternating Series

    Can the Test for Divergence (limit of an->infinity not equal to zero) be used on an alternating series? For example, if a series has a (-1)^n term. Can we assume that since the limit of that term does not exist, then the series is automatically diverging?
  32. L

    Vector Calc Homework Help Divergence Free Vector Fields

    Homework Statement Let S be the ellipsoid where a,b, and c are all positive constants. x2/(a+1)+y2/(b2)+z2/(c2) = 1 → → → → → Let F = (r - ai) / ||r - ai|| [* r and i are vectors = I tried inserting the arrows] a)Where...
  33. D

    Verifying Divergence Theorem for Given Solid Region and Vector Field?

    Homework Statement Let E be the solid region defined by 0 \leq z \leq 9+x^2+y^2 and x^2+y^2 \leq 16. Let S be the boundary surface of E, with positive (outward) orientation. Also, consider the vector field F(x,y,z)=<x,y,x^4+y^4+z> There are five parts to the problem A) Compute the...
  34. F

    Calculating Divergence Using the Divergence Theorem

    Homework Statement the problem is to calculate \int (\nabla \cdot \vec{F}) d\tau over the region x^2 + y^2 + x^2 \leq 25 where \vec{F} = (x^2 + y^2 + x^2)(x\hat{i} +y\hat{j} + z\hat{k}) in the simplest manner possible.Homework Equations divergence theorem!The Attempt at a Solution...
  35. maverick280857

    Covariant divergence question from Landau and Lifshitz

    Hi everyone, I'm trying to work through section 86 of Landau and Lifgarbagez volume 2 (The Classical Theory of Fields). Basically, I am unable to get equation (86.6) from equations (86.4) and (86.5). I've detailed my working/question in the attached jpg file. I would appreciate any inputs...
  36. R

    Calculating the Divergence of an Electrostatic Field at the Origin

    Homework Statement The electrostatic field of a point charge q is E=\frac{q}{4 \pi \epsilon r^3} r. Calculate the divergence of E. What happens at the origin? Homework Equations The Attempt at a Solution Well the solution is: \nabla.E= \partialEx/\partialx +...
  37. K

    Determine convergence and divergence

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  38. M

    Calculate Beam's Divergence Angle from Earth to Moon

    Homework Statement A tiny laser beam is directed from Earth to moon. If beam's diameter is 2.50 m at the moon, how small much the divergence angle be for the beam? The distance of moon from the Earth is 3.8x10^8 m Homework Equations The Attempt at a Solution
  39. D

    Divergence of an infinite series (using the def of limit)

    Homework Statement Given that a_{n} > 0 and lim(na_{n}) = l with l\neq0, prove that \sum a_{n} diverges.Homework Equations The Attempt at a Solution lim(na_n)=l (with =/= 0), so I can safely say that: \left|na_{n}-l\right| < \epsilon by the definition of limit. Then isn't it also true that...
  40. Z

    Proof of Divergence of Series with Non-Negative Real Numbers

    Homework Statement Suppose (a_n) is a sequence of non-negative real numbers such that the series {\sum_{n=1}}^\infty a_n diverges. Prove that the series {\sum_{n=1}}^\infty \frac{a_n}{1+a_n} must also diverge. Homework Equations The Attempt at a Solution I was thinking about looking at...
  41. H

    What is the Divergence of a Point Charge in Cartesian Coordinates?

    So I am playing around with the differential form of Gauss's Law: \nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0} Starting off simple with a point charge, the electric field is: \vec{E} = \frac{1}{4 \pi \epsilon_0} \frac{q}{r^2} \hat{r} And the divergence, in spherical coordinates...
  42. A

    Deriving the divergence in polar coordinates

    Homework Statement I want to find the divergence operator in polar coordinates (theta and r). I know how to write this operator in cartesian coordinates. The Attempt at a Solution I let F(F1,F2) be a vector field. I calculated the partial derivatives of F1 and F2 with respect to x...
  43. A

    Evaluate the divergence and curl of the following vector

    Homework Statement Evaluate the divergence and curl of the following vectors. A(r) is everywhere parallel to the y-axis with a magnitude A = cx + A0 , where c and A0 are constants. Homework Equations The Attempt at a Solution I can evaluate the div and curl, but i don't know...
  44. P

    Charged cylinder (electric field, rotor, divergence)

    Homework Statement Charged Cylinder has length L and radius a. Charge density increases within the equation rho = rho0(r/a)3 where rho0 is constant and r is distance measured from cylinder´s axis. Outside of the cylinder charge density is zero. a, Calculate the electric field inside and...
  45. S

    Divergence Theorem: Show 0 Integral on Closed Surface

    In h.m. schey, div grad curl and all that, II-25: Use the divergence theorem to show that \int\int_S \hat{\mathbf{n}}\,dS=0, where S is a closed surface and \hat{\mathbf{n}} the unit vector normal to the surface S. How should I understand the l.h.s. ? Coordinatewise? The r.h.s. is not...
  46. J

    What is the infrared divergence problem in quantum field theory?

    To my mind radio waves are different than light because they are described by fields whose energy is not a function of frequency but just the amplitudes. That they always have to surround their source, and are not uniquely associated with individual particles. Light on the other hand is...
  47. Q

    How to Calculate the Divergence of a Tensor in MHD?

    Hi guys, trying to solve a problem in MHD, i realized i need to be able to take the divergence of this following integral, but I don't know how to do it. M is a symmetric rank 2 tensor, r is a vector. The integral is as follows \int_{\partial V} (\textbf{r} d \textbf{S} \cdot...
  48. S

    Where Did I Go Wrong Calculating the Divergence of \(\widehat{r}/r^{2}\)?

    I know that this question was posted before but I just couldn't get it using another way around. So your comments are highly appreciated. In the textbook, \nabla\bullet\left(\widehat{r}/r^{2}\right)=4\pi\delta^{3}\left(r\right). But when I want to calculate the divergence using Catesian...
  49. W

    What is the Divergence Form of Gauss's Law?

    Im really having troubles understanding the divergence form of gauss's law. I have done research on it and am still not able to understand it. it sates that E=\rho/\epsilon or E=rho/epslom, so does that mean that the upside down triangle has no significance, ie does that mean i can simply solve...
  50. M

    Calculating the divergence of r(arrow)/r^a and finding charge density

    Homework Statement This is a three part problem. My first task is to calculate the divergence of \vec{r}/r^{a}. Next, I am to calculate its curl. Then I'm supposed to find the charge density that would produce the field \vec{E}=\frac{q\vec{r}}{4\pi\epsilon_{0}r^{a}} The Attempt at a...
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