What is Divergence theorem: Definition and 181 Discussions

In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a theorem which relates the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed.
More precisely, the divergence theorem states that the surface integral of a vector field over a closed surface, which is called the flux through the surface, is equal to the volume integral of the divergence over the region inside the surface. Intuitively, it states that the sum of all sources of the field in a region (with sinks regarded as negative sources) gives the net flux out of the region.
The divergence theorem is an important result for the mathematics of physics and engineering, particularly in electrostatics and fluid dynamics. In these fields, it is usually applied in three dimensions. However, it generalizes to any number of dimensions. In one dimension, it is equivalent to integration by parts. In two dimensions, it is equivalent to Green's theorem.

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  1. 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!
  2. J

    Volume of a closed surface (divergence theorem)

    If exist a formula for calculate the area of a closed curve: http://en.wikipedia.org/wiki/Green%27s_theorem#Area_Calculation, so, there is a analogous for calculate the volume of a closed surface? I search but I not found...
  3. 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...
  4. 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...
  5. 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...
  6. 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...
  7. 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...
  8. 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...
  9. 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 +...
  10. 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...
  11. 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...
  12. 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...
  13. 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...
  14. 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...
  15. 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...
  16. 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...
  17. 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...
  18. 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...
  19. 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...
  20. 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.
  21. D

    Electric field of a line charge with the divergence theorem

    Hi, on page 63 of David J. Griffiths' "Introduction to Electrodynamics" he calculates the electric field at a point z above a line charge (with a finite length L) using the electric field in integral form. E_z = \frac{1}{4 \pi \epsilon_0} \int_{0}^{L} \frac{2 \lambda z}{\sqrt{(z^2 + x^2)^3}}...
  22. A

    Evaluate both sides of divergence theorem

    Homework Statement NOTE: don't know see the phi symbol so I used theta. this is cylindrical coordinates not spherical. Given the field D = 6ρsin(θ/2)ap + 1.5ρcos(θ/2)aθ C/m^2 , evaluate both sides of the divergence theorem for the region bounded by ρ=2, θ=0 to ∏, and z = 0 to 5...
  23. Y

    Divergence Theorem: Multiplied by Scalar Field

    Homework Statement Homework Equations Definitely related to the divergence theorem (we're working on it): The Attempt at a Solution I'm a bit confused about multiplying a scalar field f into those integrals on the RHS, and I'm not sure if they can be taken out or not. If they can be, I...
  24. S

    Divergence theorem SUPER complex, maybe

    Homework Statement use divergence theorem to evaluate ∫s∫F dot n dA if F=[sinh yz, 0, y4] , S: r=[u,cosv,sinv], -4≤u≤4 , 0≤v≤pi The Attempt at a Solution Instructor surprised us with this one, I have no idea how to attempt. I know that ∫vdiv v dV=∫sn dot v dA, which is the...
  25. J

    Visualizing the Divergence Theorem for a Cylinder

    Homework Statement Prove the divergence theorem for the vector field A = p = (x,y) and taking the volume V to be the cylinder of radius a with its base centred at the origin, its axis parallel to the z-direction and having height h. I can find the dV side of the equation fine (I think)...
  26. B

    Limitations of the divergence theorem

    Homework Statement Evaluate the surface integral F * dr, where F=<0, y, -z> and the S is y=x^2+y^2 where y is between 0 and 1. Homework Equations Divergence theorem The Attempt at a Solution I just got out of my calculus final, and that was a problem on it. I used the divergence theorem...
  27. S

    Using the Divergence Theorem to Solve Vector Calculus Problems

    Homework Statement Homework Equations So I have that v \otimes n = \left( \begin{array}{ccc} v_{1}n_{1} & v_{1}n_{2} & v_{1}n_{3} \\ v_{2}n_{1} & v_{2}n_{2} & v_{2}n_{3} \\ v_{3}n_{1} & v_{3}n_{2} & v_{3}n_{3} \end{array} \right) The Attempt at a Solution I've tried applying the...
  28. B

    Verifying Divergence Theorem on Sphere with F(x,y,z)=zi+yj+xk

    Homework Statement Folks, Verify the divergence theorem for F(x,y,z)=zi+yj+xk and G the solid sphere x^2+y^2+z^2<=16 Homework Equations ##\int\int\int div(F)dV## The Attempt at a Solution My attempt The radius of the sphere is 4 and div F= 1, therefore the integral...
  29. C

    Green's, Stokes and Divergence Theorem

    When the exercise tells me to calculate the flux, how do I know when I need to use each of these theorems (Green's, Stokes or Divergence)? Can anyone tell me the difference between them? I'm a LOT confused about this. If anyone knows any good material about this on internet, it'll help me a...
  30. B

    Divergence Theorem for Surface Integrals

    Homework Statement Folks, have I set these up correctly? THanks Use divergence theorem to calculate the surface integral \int \int F.dS for each of the following Homework Equations \int \int F.dS=\int \int \int div(F)dV The Attempt at a Solution a) F(x,y,z)=xye^z i +xy^2z^3 j-...
  31. A

    Surface integral or Divergence Theorem confused?

    Homework Statement Find the Volume ∫∫ xy DA where R is the region bounded by by the line y=x-1 and the parabola y^2=2x+6. Homework Equations ∫∫ xy dx dy The Attempt at a Solution first i found the intersection of the above equations . which is (5,4) to (-1,-2) . then i...
  32. DryRun

    Evaluate using divergence theorem

    Homework Statement http://s1.ipicture.ru/uploads/20120120/eAO1JUYk.jpg The attempt at a solution \int\int \vec{F}.\hat{n}\,ds=\int\int\int div\vec{F}\,dV where dV is the element of volume. div\vec{F}=3 Now, i need to find dV which (i assume) is the hardest part of this problem. I've drawn the...
  33. S

    How to find flux through certain sides of a surface with divergence theorem.

    Homework Statement Given a vector field \textbf{F} and a composite (with this I mean cuboids, cylinders, etc. and not spheres for example) surface S, how do I calculate the flux through only some of the sides of S? I am interested in a general way to do this, but right now I am struggling with...
  34. D

    Problem applying divergence theorem to wave equation

    I'm an undergrad doing research in PDE and my adviser gave me some material to read over the holiday. But I'm getting stuck at the beginning where the divergence theorem is applied to a calculation. Maybe somebody can help me? Without getting too detailed about the context of the problem...
  35. T

    Correct Application of Divergence Theorem?

    Homework Statement http://img593.imageshack.us/img593/5713/skjermbilde20111204kl11.png The Attempt at a Solution I thought it seemed appropriate to use divergence theorem here: I have, div F = 0 + 1 + x = 1+x I let that 0≤z≤c. If, x/a + y/b = 1then y=b(1-x/a) x/a +z/c = 1 then...
  36. B

    Verifying Divergence Theorem with Triple/Surface Integrals

    I am trying to verify the divergence theorem by using the triple integral and the surface integral of the vector field dotted with dS. No trouble per se, I'm not sure though about one thing: I am given a function and six planes (they form a cube). When I set x=0 the vector field is given as...
  37. H

    Green's theorem or divergence theorem?

    Hi, I want to calculate the total flux but I'm not sure if I have to use Green's theorem (2D) or the divergence theorem (3D). The equation below is a modified Reynolds equation describing the air flow in the clearance of porous air bearing. \frac{\partial}{\partial\theta}(PH^3...
  38. B

    Writing on Stoke's, Green's, or Divergence theorem

    I suppose this has to go under homework, so here it goes: I'm in Calc III and we won't have enough time to cover the last chapter in the textbook about Stokes theorem, Green's theorem, and the divergence theorem, so instead the teacher wants a 7-page paper on something from that chapter. She...
  39. W

    Divergence Theorem: Explaining in Simple Words

    HI experts i want to know the physical significance of divergence theorem i.e how volume integral changes to surface integral - how can i explain in simple words.
  40. T

    Is it possible to derive Divergence Theorem from Stokes Theorem?

    Is it? If so, can you show how? Thanks :smile:
  41. L

    What is divergence theorem in electrostatics?

  42. A

    Divergence Theorem Homework: Volume & Surface Integral

    Homework Statement Homework Equations The Attempt at a Solution I can get the answer after applying divergence theorem to have a volume integral. But how about about the surface integral? It seems the 4 points given can't form a surface.
  43. J

    Evaluating Surface Integral with Divergence Theorem

    [b]1.The problem asks " use the divergence theorem to evaluate the surface integral \int\int F.ds for F(x,y,z) = <x3y,x2y2,−x2yz> where S is the solid bounded by the hyperboloid x^2 + y^2 - z^2 =1 and the planes z = -2 and z=2. i know that the \int\int F.ds = \int\int\int divFdv...
  44. D

    Using divergence theorem to prove Gauss's law

    i need to prove that div(R/r^3) = 4πδ where R is a vector and r is the magnitude of the vector R. also δ is the dirac delta function. so div(R/r^3) is 0 everywhere except for the origin. i need to show that the volume integral of div(R/r^3) = 4π as well. using the divergence theorem we...
  45. S

    Applying Divergence Theorem to Stokes' Theorem

    Not really a homework problem, just me wondering about this: why is there a problem here? Say you want to use the divergence theorem in conjunction with Stokes' theorem. So, from Stokes' you know: Line integral (F*T ds)= Surface integral (curl(F)*n)dS. And you know that Surface...
  46. V

    Divergence theorem/ Surface integral

    I am not able to find any good reference to answer my question, so I will post here how does divergence theorem translates to 4 dimensional curved spacetime. I understood how volume integral changes but I am not able to understand how surface integral changes. I will be glad if some one...
  47. C

    Calculating Flux through a Sphere using Divergence Theorem

    Evaluate http://webwork.latech.edu/webwork2_files/tmp/equations/93/91cfe28c766cad38444f0213c651281.png where http://webwork.latech.edu/webwork2_files/tmp/equations/59/a56001472f977192637ea927c607a61.png and is the surface of the sphere of radius 6 centered at the origin. Ok so I started by...
  48. M

    Verifying the divergence theorem- half done, help needed

    Homework Statement Verify the divergence theorem for F(x,y,z) = (x,y,2z^2) and T is the region bounded by the paraboloid z=x^2+y^2 and the plane z=1. Homework Equations F(ds) = div(F)dV The Attempt at a Solution I have successfully evaluated the integral and come up with an...
  49. S

    Verification of Divergence Theorem

    Homework Statement F(x,y,z) = (2x-z) i + x2y j + xz2 k and the volume is defined by [0,0,0] and [1,1,1]. Homework Equations flux integral = \int\int\int div F dV The Attempt at a Solution \int\int\int div F dV = \int\int\int (2+x2-2xz)dxdydz = 2 + 1/3 - 1/2 = 11/12 But I...
  50. L

    Calculating Surface Integrals Using the Divergence Theorem

    Homework Statement Evaluate the double integral over M (F \circ dS) where M is the surface of the sphere of radius 3 centered around the origin. (Sorry! I couldn't figure out how to use math symbols!) Homework Equations double integral(F\bulletdS)=triple integral (\nabla\bullet F)dV due...
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