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. 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...
  2. 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...
  3. 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...
  4. R

    Using the Divergence Theorem to Find Flux

    Let W be the solid bounded by the paraboloid x = y^2 + z^2 and the plane x = 16. Let = 3xi + yj + zk a. Let S1 be the paraboloid surface oriented in the negative x direction. Find the flux of the vector field through the surface S1. b. Let S be the closed boundary of W. Use the Divergence...
  5. jegues

    Surface Integral (Divergence Theorem?)

    Homework Statement See figure attached for problem statement. Homework Equations The Attempt at a Solution See figure attached for my attempt. What I decided to do was add a surface z=0 so that S became a closed surface. Then I preformed the integration using divergence...
  6. W

    Did i calculate this divergence theorem correclty?

    Homework Statement what is the divergence of <y,z,x>? Homework Equations The Attempt at a Solution is the answer 0? seems too easy, lol, because the actual question is "compute the surface integral for F dot prod dS over domain T where T is the unit sphere and F = <y,z,x>"...
  7. D

    Physical Examples of Divergence Theorem

    Homework Statement This problem I have been set is to find real life applications of divergence theorem. I have to show the equivalence between the integral and differential forms of conservation laws using it. 2. The attempt at a solution I have used div theorem to show the equivalence...
  8. C

    Gravitational flux and divergence theorem

    Hi. I've been reading PF for quite a while and have decided to ask my first question. Please be gentle. (I'm a retired computer programmer, not a student)... I've been learning Gauss' divergence theorem and I understand what "flux density" is when considering things like fluid transport or...
  9. R

    Divergence Theorem & Neumann Problem Explained

    I've tried to make sense of this conjecture but I can't wrap my head around it. We've been learning about the divergence theorem and the Neumann problem. I came across this question. Use the divergence theorem and the partial differential equation to show that...
  10. Saladsamurai

    Divergence Theorem: Show e\rho Integral Equality

    Homework Statement This is from a fluid mechanics text. There are no assumptions being made (i.e., no constants): Show that \frac{\partial{}}{\partial{t}}\int_V e\rho \,dV + \int_S e\rho\mathbf{v}\cdot\mathbf{n}\,dA = \rho\frac{De}{Dt}\,dV\qquad(1) where e and \rho are scalar quantities...
  11. L

    Energy Momentum Tensor and Divergence Theorem

    In the notes attached to this thread: https://www.physicsforums.com/showthread.php?t=457123 On page 110, how has he gone from equation (369) to eqn (370). He claims to have done it by "integration by parts using the divergence theorem to eliminate derivatives of \delta g_{ab} if present". (The...
  12. S

    I'm not completely sure that this is right, but it seems like it should be.

    Homework Statement Let D be the region x^2 + y^2 + z^2 <=4a^2, x^2 + y^2 >= a^2, and S its boundary (with outward orientation) which consists of the cylindrical part S1 and the spherical part S2. Evaluate the ux of F = (x + yz) i + (y - xz) j + (z -((e^x) sin y)) k through (a) the whole...
  13. H

    Divergence theorem and surface integrals

    Homework Statement Consider the following vector field in cylindrical polar components: F(r) = rz^2 r^ + rz^2 theta^ By directly solving a surface integral, evaluate the flux of F across a cylinder of radius R, height h, centred on the z axis, and with basis lying on the z = 0 plane. Using the...
  14. D

    Laplace and Divergence theorem

    Homework Statement Use Divergence theorem to determine an alternate formula for \int\int u \nabla^2 u dx dy dz Then use this to prove laplaces equation \nabla^2 u = 0 is unique. u is given on the boundary.Homework Equations u \nabla^2 u = \nabla * (u \nabla u) -(\nabla u)^2 The Attempt at...
  15. C

    Gauss Divergence Theorem - Silly doubt - Almost solved

    Homework Statement The problem statement has been attached with this post. Homework Equations I considered u = ux i + uy j and unit normal n = nx i + ny j. The Attempt at a Solution I used gauss' divergence theorem. Then it came as integral [(dux/dx) d(omega)] + integral...
  16. P

    Neumann Problem: Use the divergence theorem to show it has a solution

    From Partial Differential Equations: An Introduction, by Walter A. Strauss; Chapter 1.5, no.4 (b). Homework Statement "Consider the Neumann problem (delta) u = f(x,y,z) in D \frac{\partial u}{\partial n}=0 on bdy D." "(b) Use the divergence theorem and the PDE to show that...
  17. E

    Why is the Divergence Theorem failing for this scalar function?

    Hi everyone, so let me introduce the scalar function \Phi = -(x2+y2+z2)(-1/2) which some of you may recognize as minus one over the radial distance from the origin. When I compute \nabla2\Phi is get 0. Now if I do the following integral on the surface S of the unit sphere x2+y2+z2= 1 ...
  18. V

    Divergence Theorem: Check Function w/y^2, 2x+z^2, 2y

    Homework Statement Check the divergence theorem using the function: \mathbf{v} = y^2\mathbf{\hat{x}} + (2xy + z^2) \mathbf{\hat{y}} + (2yz)\mathbf{\hat{z}} Homework Equations \int_\script{v} (\mathbf{\nabla . v }) d\tau = \oint_\script{S} \mathbf{v} . d\mathbf{a} The Attempt at a Solution...
  19. K

    What Went Wrong in My Verification of the Divergence Theorem?

    Homework Statement Let the surface, G, be the paraboloid z = x^2 + y^2 be capped by the disk x^2 + y^2 \leq 1 in the plane z = 1. Verify the Divergence Theorem for \textbf{F}(x,y,z) = 2x\textbf{i} - yz\textbf{j} + z^2\textbf{k} Homework Equations I have solved the problem using the...
  20. S

    Vector Calculus - Divergence theorem

    Homework Statement 1. Consider a cube with vertices at A=(0,0,0) B=(2,0,0) C=(2,2,0) D=(0,2,0) E=(0,0,2) F=(2,0,2) G=(2,2,2) H=(0,2,2) A)Calculate the flux of the vector fieldF=xi through each face of the cube by taking the normal vectors pointing outwards. B)Verify Gauss's divergence theorem...
  21. H

    Solving Gauss Divergence Theorem on a Closed Surface

    Homework Statement Verify Gauss Divergence Theorem ∭∇.F dxdydz=∬F. (N)dA Where the closed surface S is the sphere x^2+y^2+z^2=9 and the vector field F = xz^2i+x^2yj+y^2zk The Attempt at a Solution I have tried to solve the left hand side which appear to be (972*pi)/5 However, I...
  22. 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.
  23. 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...
  24. 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...
  25. 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...
  26. 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...
  27. 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...
  28. 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...
  29. 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...
  30. 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...
  31. 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...
  32. 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...
  33. H

    Divergence Theorem Explained: Learn the Basics

    http://img60.imageshack.us/img60/9696/21249035.jpg I am stuck at the last step. Can anyone give some hints? Thanks in advanced.
  34. B

    Evaluating \int\int_{\sigma} F.n ds with Divergence Theorem

    Homework Statement Use the divergence theorem to evaluate \int\int_{\sigma}F . n ds Where n is the outer unit normal to \sigma we have F(x,y,z)=2x i + 2y j +2z k and \sigma is the sphere x^2 + y^2 +z^2=9 Homework Equations \int\int_{s}F . dA = \int\int\int_{R}divF dV The...
  35. R

    Divergence theorem - mass flux

    Homework Statement Water in an irrigation ditch of width w = 3.0 m and depth d = 2.0 m flows with a speed of 0.40 m/s. For each case, sketch the situation, then find the mass flux through the surface: (a) a surface of area wd, entirely in the water, perpendicular to the flow; (b) a surface...
  36. J

    Evaluating Integral F dot dA using Divergence Theorem

    Homework Statement use the divergence theorem to evaluate the integral F dot dA F = (2x-z)i + x2yj + xz2k s is the surface enclosing the unit cube and oriented outward Homework Equations The Attempt at a Solution is the the region from -1 to 1 for x y and z div F = x2 +...
  37. T

    Compute Surface Integral: Divergence Theorem & F=(xy^2,2y^2,xy^3)

    Use the divergence theorem to compute the surface integral F dot dS , where F=(xy^2, 2y^2, xy^3) over closed cylindrical surface bounded by x^2+z^2=4 and y is from -1 to 1. I've tried doing it and got 32pi/3 (i guess its wrong, so how to do it?) Is it ok to compute Div F in terms of xyz and...
  38. X

    Easy divergence theorem problem

    Evaluate the flux integral using the Divergence Theorem if F(x,y,z)=2xi+3yj+4zk and S is the sphere x^2+y^2+z^2=9 answer is 324pi so far i took the partial derivitavs of i j k for x y z and added them to get 9. so i have the triple integral of 9 dzdxdy i think u have to use polar...
  39. J

    Is Multiplying Divergence by Area Correct in Divergence Theorem Problems?

    Homework Statement F = xi + yj + zk, s = x^2 + y^2 + z^2 Homework Equations The Attempt at a Solution div F = 1+1+1=3 area of sphere = 4pi i can just multiply them to get 12pi as an answer right?
  40. J

    Evaluating ∫∫(∇xF).n dS: Divergence vs. Stokes' Theorem

    Given F = xyz i + (y^2 + 1) j + z^3 k Let S be the surface of the unit cube 0 ≤ x, y, z ≤ 1. Evaluate the surface integral ∫∫(∇xF).n dS using a) the divergence theorem b) using Stokes' theorem --- Since the divergence theorem involves a dot product rather than a curl,how would it...
  41. E

    Calculating Flux and Applying the Divergence Theorem

    Homework Statement http://img16.imageshack.us/img16/88/fluxm.th.jpg Homework Equations The Attempt at a Solution I've tried to find the divergence of F and I got 3x^2 + 3y^2 + 3z^2 and as this is a variable I need to set up the integral... how do I set the integral
  42. N

    Finding the flux (Divergence Theorem)

    Homework Statement By using divergence theorem find the flux of vector F out of the surface of the paraboloid z = x^2 + y^2, z<=9, when F = (y^3)i + (x^3)j + (3z^2)kHomework Equations Divergence theorem equation stated in the attempt partThe Attempt at a Solution
  43. H

    Verification of the Divergence Theorem

    The question I was given asks to verify the divergence theorem by showing that both sides of the theorem show the same result. With the divergence theorem obviously being \iint_S\mathbf{F}\cdot\mathbf{n}\,dS = \iiint_V \nabla\cdot\mathbf{F}\,dV . The vector field is...
  44. S

    Quick Divergence Theorem question

    Homework Statement Use the divergence theorem in three dimensions \int\int\int\nabla\bullet V d\tau= \int \int V \bullet n d \sigma to evaluate the flux of the vector field V= (3x-2y)i + x4zj + (1-2z)k through the hemisphere bounded by the spherical surface x2+y2+z2=a2 (for z>0)...
  45. E

    Divergence theorem in four(or more) dimension

    I don't know whether it was proved or can be prove. I don't know whether it is useful. maybe it can be used in string theory or some other things. any comment or address will be appreciated.
  46. L

    Proving Divergence Theorem using Gauss' Theorem

    I need to show that, using Gauss' Theorem (Divergence Theorem), i.e. integration by parts, that: \int_V dV e^{-r} \nabla \cdot (\frac{\vec{\hat{r}}}{r^2}) = \int_V dV \frac{e^{-r}}{r^2} any ideas on where to start?
  47. L

    Using Gauss' Theorem to Show Integral Convergence in Divergence Theorem

    This may well be the wrong place to post this so apologies for that if it's the case. Anyway, I'm stuck on this question, any help appreciated Use Gauss' Theorem to show that: (i) If \psi($\mathbf{r}$) ~ \frac{1}{r} as r \rightarrow \infty , then, \int_V {\psi \nabla^{2} \psi}...
  48. Saladsamurai

    Check Divergence Theorem on Unit Cube

    Homework Statement Check the Divergence Theorem \int_V(\nabla\cdot\bold{v})\,d\tau=\oint_S\bold{v}\cdot d\bold{a} using the function \bold{v}=<y^2, 2xy+z^2, 2yz> and the unit cube below. Now when I calculate the divergence I get (\nabla\cdot\bold{v})=2y+2x+2y but Griffith's...
  49. J

    Divergence theorem over a hemisphere

    I was told this problem could be done with divergence theorem, instead of as a surface integral, by adding the unit disc on the bottom, doing the calculation, then subtracting it again. Homework Statement Homework Equations The Attempt at a Solution for del . f I get i + j =...
  50. Saladsamurai

    Surface Integral using Divergence Theorem

    Homework Statement Evaluate the surface Integral I=\int\int_S\vec{F}\cdot\vec{n}\,dS where \vec{F}=<z^2+xy^2,x^100e^x, y+x^2z> and S is the surface bounded by the paraboloid z=x^2+y^2 and the plane z=1; oriented by the outward normal.The Attempt at a Solution...
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