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. Clvrhammer

    A Stationary solutions to Klein Gordon equation in spherical symmetry

    I am trying to prove that in spherically symmetric spacetimes there are no nontrivial time-independent solutions to the Klein-Gordon equation (with mass ##= 0##) (**is this even true?**). My Ansatz is as follows: A spherically symmetric spacetime has metric $$g = g_{tt} \, dt^2 + g_{tr} \, dt...
  2. jv07cs

    I Conditions for applying Gauss' Law

    To apply the Divergence Theorem (DT), at least as it is stated and proved in undergrad calculus, it is required for the vector field ##\vec{F}## to be defined both on the surface ∂V, so that we can evaluate the flux through this surface, and on the volume V enclosed by ∂V, so that we can...
  3. Kostik

    A Dirac "GTR" Eq. 27.11 -- how to show that a boundary term vanishes?

    In Dirac's "General Theory of Relativity", p. 53, eq. (27.11), Dirac is deriving Einstein's field equations and the geodesic equation from the variation ##\delta(I_g+I_m)=0## of the actions for gravity and matter. Here ##p^\mu=\rho v^\mu \sqrt{-g}## is the momentum of an element of matter. He...
  4. Leo Liu

    How to use the divergence theorem to solve this question

    The correct answer is ##\frac{\pi a^2 h} 2## by using the standard approach. However when I tried using the divergence theorem to solve this problem, I got a different answer. My work is as follows: $$\iint_S \vec F\cdot\hat n\, dS = \iiint_D \nabla\cdot\vec F\,dV$$ $$= \iiint_D \frac{\partial...
  5. A

    Why can't I use the divergence theorem?

    Greetings! here is the following exercice I understand that when we follow the traditional approach, (prametrization of the surface) we got the answer which is 8/3 But why the divergence theorem can not be used in our case? (I know it's a trap here) thank you!
  6. E

    B Confusion about Divergence Theorem Step in Tong's Notes

    I wanted to ask about a step I couldn't understand in Tong's notes$$\int_M d^n x \partial_{\mu}(\sqrt{g} X^{\mu}) = \int_{\partial M} d^{n-1}x \sqrt{\gamma N^2} X^n = \int_{\partial M} d^{n-1}x \sqrt{\gamma} n_{\mu} X^{\mu}$$we're told that in these coordinates ##\partial M## is a surface of...
  7. B

    Understanding the electric field of a sphere with a hole

    Here's an image. O and O' are the respective centers, a is the distance between them, r is the distance from the center of the sphere to P, and r' = r - a, the distance from O' to P. The approach (which I don't understnad) given is to use Gauss' Law and superposition, so that we calculate the...
  8. fluidistic

    Gauss' divergence theorem and thermoelectricity contradiction

    I get a nonsensical result. I am unable to understand where I go wrong. Let's consider a material with a temperature independent Seebeck coefficient, thermal conductivity and electrochemical potential to keep things simple. Let's assume that this material is sandwiched between 2 other materials...
  9. TheGreatDeadOne

    Using the Divergence Theorem on the surface of a sphere

    The integral that I have to solve is as follows: \oint_{s} \frac{1}{|r-r'|}da', \quad\text{ integrating with respect to r '}, integrating with respect to r' Then I apply the divergence theorem, resulting in: \iiint \limits _{v} \nabla \cdot \frac{1}{|r-r'|}dv' =...
  10. M

    Divergence Theorem Verification: Surface Integral

    Hi, I just had a quick question about a step in the method of calculating the surface integral and why it is valid. I have already done the divergence step and it yields the correct result. Method: Let us calculate the normal: ## \nabla (z + x^2 + y^2 - 3) = (2x, 2y, 1) ##. Just to double...
  11. M

    Divergence Theorem in Curvilinear Coordinates: Questions & Explanations

    Hi, I was trying to gain an understanding of a proof of the divergence theorem in curvilinear coordinates. I have found these online notes here and am looking at the proof on pages 4-5. The method intuitively makes sense to me as opposed to other proofs which fiddle around with vector...
  12. A

    Understanding the Divergence Theorem

    Good day all my question is the following Is it correct to (after calculation the new field which is the curl of the old one)to use the divergence theroem on the volume shown on that picture? The divergence theorem should be applied on a closed surface , can I consider this as closed? Thanks...
  13. J

    Divergence Theorem Problem Using Multiple Arbitrary Fields

    My main issue with this question is the manipulation of the two arbitrary fields into a single one which can then be substituted into the divergence theorem and worked through to the given algebraic forms. My attempt: $$ ∇(ab) = a∇b + b∇a $$ Subsituting into the Eq. gives $$ \int dS ·...
  14. dRic2

    I What is the role of the divergence theorem in deriving local laws in physics?

    As far as I can tell the divergence theorem might be one of the most used theorems in physics. I have found it in electrodynamics, fluid mechanics, reactor theory, just to name a few fields... it's literally everywhere. Usually the divergence theorem is used to change a law from integral form to...
  15. JD_PM

    Checking divergence theorem inside a cylinder and under a paraboloid

    I am checking the divergence theorem for the vector field: $$v = 9y\hat{i} + 9xy\hat{j} -6z\hat{k}$$ The region is inside the cylinder ##x^2 + y^2 = 4## and between ##z = 0## and ##z = x^2 + y^2## This is my set up for the integral of the derivative (##\nabla \cdot v##) over the region...
  16. A

    I Divergence Theorem: Gauss & Cross-Product Integration

    From gauss divergence theorem it is known that ##\int_v(\nabla • u)dv=\int_s(u•ds)## but what will be then ##\int_v(\nabla ×u)dv## Any hint?? The result is given as ##\int_s (ds×u)##
  17. M

    I Why is this volume/surface integration unaffected by a singularity?

    ##\mathbf{M'}## is a vector field in volume ##V'## and ##P## be any point on the surface of ##V'## with position vector ##\mathbf {r}## Now by Gauss divergence theorem: \begin{align} \iiint_{V'} \left[ \nabla' . \left( \dfrac{\mathbf{M'}}{\left| \mathbf{r}-\mathbf{r'} \right|}...
  18. Clara Chung

    I Question about divergence theorem and delta dirac function

    How do you prove 1.85 is valid for all closed surface containing the origin? (i.e. the line integral = 4pi for any closed surface including the origin)
  19. B

    Divergence of the E field at a theoretical Point Charge

    I've been thinking about this problem and would like some clarification regarding the value of the divergence at a theoretical point charge. My logic so far: Because the integral over all space(in spherical coordinates) around the point charge is finite(4pi), then the divergence at r=0 must be...
  20. N

    How Do You Apply the Divergence Theorem to a Vector Field in a Unit Cube?

    Homework Statement Griffiths Introduction to Electrodynamics 4th Edition Example 1.10 Check the divergence theorem using the function: v = y^2 (i) + (2xy + z^2) (j) + (2yz) (k) and a unit cube at the origin. Homework Equations (closed)∫v⋅da = ∫∇⋅vdV The flux of vector v at the boundary of the...
  21. E

    Determining between direct evaluation or vector theorems

    So the main thing I'm wondering is given a question how do we determine whether to use one of the fundamentals theorems of vector calculus or just directly evaluate the integral, and if usage of one of the theorems is required how do we determine which one to use in the situation? Examples are...
  22. M

    MHB Divergence Theorem and shape of hyperboloid

    Hello! I have been doing a previous exam task involving the divergence theorem, but there is a minor detail in the answer which i can't fully understand. I have a figur given by ${x}^{2} +{y}^{2} -{z}^{2} = 1$ , $z= 0$ and $z=\sqrt{3}$ As i have understood this is a hyperboloid going from...
  23. K

    Divergence theorem with inequality

    Homework Statement F(x,y,z)=4x i - 2y^2 j +z^2 k S is the cylinder x^2+y^2<=4, The plane 0<=z<=6-x-y Find the flux of F Homework Equations The Attempt at a Solution What is the difference after if I change the equation to inequality? For example : x^2+y^2<=4, z=0 x^2+y^2<=4 , z=6-x-y...
  24. K

    How can I verify the Divergence Theorem for F=(2xz,y,−z^2)

    Homework Statement Verify the Divergence Theorem for F=(2xz,y,−z^2) and D is the wedge cut from the first octant by the plane z =y and the elliptical cylinder x^2+4y^2=16 Homework Equations \int \int F\cdot n dS=\int \int \int divF dv The Attempt at a Solution For the RHS...
  25. L

    Divergence theorem for vector functions

    Surface S and 3D space E both satisfy divergence theorem conditions. Function f is scalar with continuous partials. I must prove Double integral of f DS in normal direction = triple integral gradient f times dV Surface S is not defined by a picture nor with an equation. Help me. I don't...
  26. X

    I Divergence Theorem not equaling 0

    Why is it possible that ∫∫∫ V f(r) dV ≠ 0 even if f(r) =0
  27. F

    I Integral form of Navier-Stokes Equation

    The Navier-Stokes equation may be written as: If we have a fixed volume (a so-called control volume) then the integral of throughout V yields, with the help of Gauss' theorem: (from 'Turbulence' by Davidson). The definition of Gauss' theorem: Could someone show me how to go from the...
  28. mr.tea

    I Divergence theorem and closed surfaces

    Hi, I have a question about identifying closed and open surfaces. Usually, when I see some exercises in the subject of the divergence theorem/flux integrals, I am not sure when the surface is open and needed to be closed or if it is already closed. I mean for example a cylinder that is...
  29. P

    Applying the divergence theorem to find total surface charge

    Homework Statement Sorry- I've figured it out, but I am afraid I don't know how to delete the thread. Thank you though :) Homework Equations Below The Attempt at a Solution Photo below- I promise its coming! I've started by using cylindrical coordinates, but I wasn't sure if spherical...
  30. C

    I Divergence Theorem and Gauss Law

    Divergence theorem states that $\int \int\vec{E}\cdot\vec{ds}=\int\int\int div(\vec{E})dV$ And Gauss law states that $\int \int\vec{E}\cdot\vec{ds}=\int\int\int \rho(x,y,z)dV$ If $\vec{E}$ to be electric field vector then i could say that $div(\vec{E})=\rho(x,y,z)$ However i can't see any...
  31. N

    Verify Divergence Theorem for V = xy i − y^2 j + z k and Enclosed Surface

    Homework Statement Verify the divergence theorem for the function V = xy i − y^2 j + z k and the surface enclosed by the three parts (i) z = 0, s < 1, s^2 = x^2 + y^2, (ii) s = 1, 0 ≤ z ≤ 1 and (iii) z^2 = a^2 + (1 − a^2)s^2, 1 ≤ z ≤ a, a > 1. Homework Equations [/B]...
  32. F

    Gauss' Theorem - Divergence Theorem for Sphere

    Homework Statement Using the fact that \nabla \cdot r^3 \vec{r} = 6 r^2 (where \vec{F(\vec{r})} = r^3 \vec{r}) where S is the surface of a sphere of radius R centred at the origin. Homework Equations \int \int \int_V \nabla \cdot \vec{F} dV =\int \int_S \vec{F} \cdot d \vec{S} That is meant...
  33. K

    Using the divergence theorem to prove Gauss's law?

    Hello, I've been struggling with this question: Let q be a constant, and let f(X) = f(x,y,z) = q/(4pi*r) where r = ||X||. Compute the integral of E = - grad f over a sphere centered at the origin to find q. I parametrized the sphere using phi and theta, crossed the partials, and got q, but I...
  34. Cygnus_A

    Gradient version of divergence theorem?

    So we all know the divergence/Gauss's theorem as ∫ (\vec∇ ⋅ \vec v) dV = ∫\vec v \cdot d\vec S Now I've come across something labeled as Gauss's theorem: \int (\vec\nabla p)dV = \oint p d\vec S where p is a scalar function. I was wondering if I could go about proving it in the following way...
  35. kostoglotov

    I can't accept the solution manual's explanation

    Homework Statement It's a long winded problem, I'll post a picture and an imgur link Imgur link: http://i.imgur.com/5wvbqO2.jpg Homework Equations Divergence Theorem \iint\limits_S \vec{F}\cdot d\vec{S} = \iiint\limits_E \nabla \cdot \vec{F} \ dV The Attempt at a Solution I'll follow...
  36. L

    Divergence theorem on non compact sets of R3

    So my question here is: the divergence theorem literally states that Let \Omega be a compact subset of \mathbb{R}^3 with a piecewise smooth boundary surface S. Let \vec{F}: D \mapsto \mathbb{R}^3 a continously differentiable vector field defined on a neighborhood D of \Omega. Then...
  37. V

    Divergence Theorem Question (Gauss' Law?)

    If F(x,y,z) is continuous and for all (x,y,z), show that R3 dot F dV = 0 I have been working on this problem all day, and I'm honestly not sure how to proceed. The hint given on this problem is, "Take Br to be a ball of radius r centered at the origin, apply divergence theorem, and let the...
  38. U

    Is the Divergence Theorem Applicable to Open Cylindrical Surfaces?

    Homework Statement Homework EquationsThe Attempt at a Solution I thought of using the divergence theorem where I find that ∇.F = 3z thus integral is ∫ ∫ ∫ 3z r dz dr dθ where r dz dr dθ is the cylindrical coordinates with limits 0<=z<=4 0<=r<=3 0<=θ<=2π and solving gives me 216π Can I...
  39. U

    Derivatives in 3D and Dirac Delta

    For a research project, I have to take multiple derivatives of a Yukawa potential, e.g. ## \partial_i \partial_j ( \frac{e^{-m r}}{r} ) ## or another example is ## \partial_i \partial_j \partial_k \partial_\ell ( e^{-mr} ) ## I know that, at least in the first example above, there will be a...
  40. B3NR4Y

    Using the Divergence Theorem to Prove Green's Theorem

    Homework Statement Prove Green's theorem \int_{\tau} (\varphi \nabla^{2} \psi -\psi\nabla^{2}\varphi)d\tau = \int_{\sigma}(\varphi\nabla\psi -\psi\nabla\varphi)\cdot d\vec{\sigma} Homework Equations div (\vec{V})=\lim_{\Delta\tau\rightarrow 0} \frac{1}{\Delta\tau} \int_{\sigma} \vec{V} \cdot...
  41. C

    Triple Integral for Divergence Theorem

    Homework Statement Find the flux of the field F(x) = <x,y,z> across the hemisphere x^2 + y^2 + z^2 = 4 above the plane z = 1, using both the Divergence Theorem and with flux integrals. (The plane is closing the surface) Homework Equations The Attempt at a Solution Obviously, the divergence...
  42. S

    Flux Calculation for Radial Vector Field through Domain Boundary

    Homework Statement Find the outward flux of the radial vector field F(x,y,z) = x i^ + y j^ + z k^ through the boundary of domain in R^3 given by two inequalities x^2 + y^2 + z^2 ≤ 2 and z ≥ x^2 + y^2. Homework Equations Divergence theorem: ∫∫_S F ⋅ n^ = ∫∫∫_D div F dV The Attempt at a...
  43. B

    Simple divergence theorem questions

    So I understand the divergence theorem for the most part. This is the proof that I'm working with http://www.math.ncku.edu.tw/~rchen/Advanced%20Calculus/divergence%20theorem.pdf For right now I'm just looking at the rectangular model. My understanding is that should we find a proof for this...
  44. N

    Divergence theorem for a non-closed surface?

    Is there some way we can apply divergence (Gauss') theorem for an open surface, with boundaries? Like a paraboloid that ends at some point, but isn't closed with a plane on the top. I found this at Wikipedia: It can not directly be used to calculate the flux through surfaces with boundaries...
  45. evinda

    MHB Divergence Theorem: Applying to Sphere $\hat{i}x+\hat{j}y+\hat{k}z$

    Hello again! (Wave) I am looking at an exercise of the divergence theorem.. We want to apply the divergence theorem for the sphere $x^2+y^2+z^2=a^2$ in the case when the vector field is $\overrightarrow{F}=\hat{i}x+\hat{j}y+\hat{k}z$.$\displaystyle{\nabla \cdot...
  46. Feodalherren

    Verify the divergence theorem for a cylinder

    Homework Statement Verify the divergence theorem if \textbf{F} = <1-x^{2}, -y^{2}, z > for a solid cylinder of radius 1 that lies between the planes z=0 and z=2. Homework Equations Divergence theorem The Attempt at a Solution I can do the triple integral part no problem. Where I...
  47. E

    Showing Divergence Theorem Equivalence

    Homework Statement The problem states that a cube encloses charge. This cube is given in three space by <0,0,0> and <a,a,a>. The electric field is given by: \hat{E}=\frac{4e}{a^{2}e_{0}}[\frac{xy}{a^{2}}\hat{i}+\frac{(y-x)}{a}\hat{j}+\frac{xyz}{a^{2}}\hat{k}]. I am to find the total charge...
  48. M

    MHB Apply the divergence theorem to calculate the flux of the vector field

    Hey! :o I have the following exercise: Apply the divergence theorem to calculate the flux of the vector field $\overrightarrow{F}=(yx-x)\hat{i}+2xyz\hat{j}+y\hat{k}$ at the cube that is bounded by the planes $x= \pm 1, y= \pm 1, z= \pm 1$. I have done the following...Could you tell me if this...
  49. P

    How to Show the E Field Outside a Long, Charged Conducting Cylinder?

    Homework Statement Use the divergence theorem (and sensible reasoning) to show that the E field a distance r outside a long, charged conducting cylinder of radius r0 which carries a charge density of σ Cm-2 has a magnitude E=σr0/ε0r. What is the orientation of the field? Homework Equations...
  50. M

    MHB Apply the divergence theorem for the vector field F

    Hey! :o Apply the divergence theorem over the region $1 \leq x^2+y^2+z^2 \leq 4$ for the vector field $\overrightarrow{F}=-\frac{\hat{i}x+\hat{j}y+\hat{k}z}{p^3}$, where $p=(x^2+y^2+z^2)^\frac{1}{2}$. $\bigtriangledown...
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