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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11. 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...
12. 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...
13. 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)##
14. 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|}...
15. 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)
16. 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...
17. E&M: Prove the Divergence Theorem

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...
18. 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...
19. 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...
20. 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...
21. 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...
22. 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...
23. I Divergence Theorem not equaling 0

Why is it possible that ∫∫∫ V f(r) dV ≠ 0 even if f(r) =0
24. 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...
25. 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...
26. 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...
27. 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...
28. 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]...
29. 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...
30. 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...
31. 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...
32. 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...
33. 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...
34. 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...
35. Using divergence theorem?

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...
36. 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...
37. 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...
38. 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...
39. 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...
40. 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...
41. 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...

48. Understanding the divergence theorem

I'm having some trouble understanding what divergence of a vector field is in my "Fields and Waves" course. Divergence is defined as divE=∇E = (∂Ex/∂x) + (∂Ey/∂y) + (∂Ez/∂z). As far as I understand this gives the strength of vector E at the point(x,y,z). Divergence theorem is defined as ∫∇Eds...
49. Applying the Divergence Theorem to the Volume of a Ball with a Given Radius

Homework Statement let Bn be a ball in Rn with radius r. ∂Bn is the boundary. Use divergence theorem to show that: V(Bn(r)) = (r/n) * A (∂Bn(r)) where V(Bn) is volume and A(∂Bn) is surface area. Homework Equations consider the function: u = x1 ^2 + x2 ^2 +...+ xn ^2 The...
50. Divergence theorem (determining the correct direction for normal vecto

The problem is in the paint doc.. My question is why is the base vector aR have a negative sign attached to it?