Divergence theorem Definition and 184 Threads

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. T

    What is the solution to this Divergence Theorem homework problem?

    Homework Statement evaluate https://instruct.math.lsa.umich.edu/webwork2_files/tmp/equations/71/7816ab9562fbe29a133b96799ed5521.png if https://instruct.math.lsa.umich.edu/webwork2_files/tmp/equations/65/11ed69ea372626e9c4cee674c8dc6f1.png and S is the surface of the region in the first...
  2. B

    Divergence theorem - sign of da

    Homework Statement This is just a general question. My fundamentals aren't very solid because I'm studying on my own at the moment. \int_V (\triangledown \cdot \bold{v}) dV = \int_S \bold{v} d\bold{a} I am trying to find out the sign of the area integral on a surface defined by spherical...
  3. L

    Does the Divergence Theorem Apply to Complex Vector Fields and Hemispheres?

    Homework Statement 2. Verify the divergence theorem for the vector field: F =(r2cosθ) r +(r2cosφ) θ −(r2cosθsinφ) φ using the upper hemisphere of radius R.Homework Equations Is this any close to be correct? The question marks indicate parts I am not sure about please help. Anyone know what are...
  4. T

    Divergence Theorem Homework: Find Divergence

    Homework Statement Here is a link to the problem: http://www.brainmass.com/homework-help/physics/electromagnetic-theory/68800 The Attempt at a Solution To find the divergence 1/r^2*d(r)*(r^2*r^2*cos(theta)) +[1/r*sin(theta)]*d(theta)*(sin(theta)*r^2*cos(phi))...
  5. B

    Divergence Theorem and Incompressible Fluids

    Homework Statement Hi, I'm trying to follow the proof for the statement \nabla . u = 0 I'm basing it off this paper: http://delivery.acm.org/10.1145/1190000/1185730/p1-bridson.pdf?key1=1185730&key2=4151929021&coll=GUIDE&dl=GUIDE&CFID=25582973&CFTOKEN=82107744 (page 7, 8) In...
  6. E

    When does the divergence theorem apply?

    As the thread title suggests, I'm having trouble realizing when the divergence theorem is applicable and when it is not. In some examples, I am instructed not to use it because it doesn't hold but on others I can use it. My first instinct was that it doesn't apply when the vector field isn't...
  7. E

    Stokes and Divergence theorem questions

    Homework Statement Let \vec{F}=xyz\vec{i}+(y^{2}+1)\vec{j}+z^{3}\vec{k} And let S be the surface of the unit cube in the first octant. Evaluate the surface integral: \int\int_{S} \nabla\times \vec{F} \cdot \vec{n} dS using: a) The divergence theorem b) Stoke's theorem c)...
  8. D

    How Does the Divergence Theorem Apply to Vector Fields and Surface Integrals?

    Homework Statement Let D be an area in R^3 and S be its surface. D fulfills the Divergence theorem. Let N be the unit normal on S and let the volume, V, be known. Let (\overline{x},\overline{y}, \overline{z}) coordinates of the centre of mass of D be known (and the density delta is...
  9. J

    Divergence theorem requires a conservative vector field?

    Can anyone tell me whether or not the divergence theorem requires a conservative vector field? On a practice exam my professor gave a vector field that was nonconservative (I checked the curl) and proceeded to perform the divergence theorem to find the flux. On one of my homework problems I...
  10. T

    Calculate Divergence Theorem for F with S and Q

    S\int\int F*Nds F(x,y,z) = (xy^2 + cosz)i + (x^2*y + sinz)j + e^(z)*k s: z = 1/2\sqrt{x^2 + y^2} , z = 8 divF = y^2 + x^2 +e^z Q\int\int\int (y^2 + x^2 + e^k)dV This is as far as I got, I have no idea how to do the limits for this triple integral thanks in advance guys.
  11. T

    Does the divergence theorem work for a specific vector field?

    Homework Statement Show divergence theorem works For the vector field E = \hat{r}10e^{-r}-\hat{z}3z Homework Equations \int_{v}\nabla \cdot E dv = \oint_{s} E \cdot ds The Attempt at a Solution \nabla \cdot E = 1/r \frac{d}{dr}(rAr)+1/r\frac{dA\phi}{d\phi}+\frac{dAz}{dz}...
  12. B

    Trouble verifying Divergence Theorem

    i having a some trouble verifying the Divergence theorem for A=y^2zex-2x^3yey+xyz^2ez with respect to V being a unit cube
  13. 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...
  14. 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.
  15. 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)...
  16. 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} =...
  17. 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...
  18. 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...
  19. 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...
  20. 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} +...
  21. 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...
  22. 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...
  23. 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)...
  24. 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...
  25. 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...
  26. Reshma

    Divergence theorem on Octant of a sphere

    Hi everyone! I am having some trouble with this particular problem on Vector Calculus from Griffith's book. The question is: Check the divergence theorem for the vector function(in spherical coordinates) \vec v = r^2\cos\theta\hat r + r^2\cos\phi \hat \theta - r^2\cos\theta\sin\phi\hat...
  27. R

    Why Is There a (dBx/dx)*(dx/2) Term in Divergence Theorem Proof?

    Im having a bit of a problem understanding the crucial part of the divergence theorem from Electromagnetic Fields and Waves by Lorrain and Corson. Ill try descibe the set up of the problem 1st and see if anyone can help me in any way before i continue with the electromagnetism course I am doing...
  28. S

    How did ostrogradsky prove the divergence theorem?

    so we know that the divergence theorm, was proved by Gauss, and also proved by ostrogradsky. but infact, the divergence theorm was discovred by Lagrange...correct? now, did these 3 guys prove it differently? I'm sure it couldn't be exactly the same way right? basically, I've been...
  29. M

    Divergence Theorem for the Curl

    Hi, I'm having trouble proving the following result: \int_{V} (\nabla\times\vec{A}) dV = -\int_{S} (\vec{A}\times\vec{n}) dS I'm not sure how I should Stokes' and/or the Divergence Theorem in proving this, or if you should use them at all. Thanks in advance.
  30. P

    Evaluating a Vector Field Through a Surface with the Divergence Theorem

    ok this probley seems simple but i just need to see how to do it, ok well how do u evaluate this... find the flux of the vector field... \vec{F}=<x,y,z> throught this surface above the xy-plane.. z = 4-x^2-y^2 how do u evaluate this with surface integrals method and the divergence...
  31. D

    Help with using the Divergence Theorem

    Hi! We are nearing the end of our course --- culminating in Stokes and Divergence Theorems for surface integrals, and I am having some difficulty with the following 1. F(x,y,z) = <x^3y, -x^2y^2, -x^2yz> where S is the solid bounded by the hyperboloid x^2 + y^2 - z^2 =1 and the planes z...
  32. C

    Problem with the Divergence Theorem

    I was wondering if someone could give me a hand here with 2b) on the following link. http://www.am.qub.ac.uk/users/j.mccann/teaching/ama102/2003/assignments/assign_8.pdf For part a) I got it to be equal to 3x^2+3y^2+3z^2+2y-2xy, and I'm hoping that's right! However, for part b) I can't...
  33. J

    How Does the Divergence Theorem Apply to a Vector Field on a Unit Sphere?

    I need help evaluating both sides of the divergence theorem if V=xi+yj+zk and the surface S is the sphere x^2+y^2+z^2=1, and so verify the divergence theorem for this case. Is the divergence theorem the triple integral over V (div V) dxdydz= the double integral over S (V dot normal)dS? If so...
  34. J

    Evaluating Divergence Thm: V, S & Verify for x^2+y^2+z^2=1

    I need help evaluating both sides of the divergence theorem if V=xi+yj+zk and the surface S is the sphere x^2+y^2+z^2=1, and so verify the divergence theorem for this case. Is the divergence theorem the triple integral over V (div V) dxdydz= the double integral over S (V dot normal)dS? If so...
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