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.
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:
$$\iint_S \vec F\cdot\hat n\, dS = \iiint_D \nabla\cdot\vec F\,dV$$
$$= \iiint_D \frac{\partial...
Greetings!
here is the following exercice
I understand that when we follow the traditional approach, (prametrization of the surface) we got the answer which is 8/3
But why the divergence theorem can not be used in our case? (I know it's a trap here)
thank you!
I wanted to ask about a step I couldn't understand in Tong's notes$$\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...
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...
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...
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' =...
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...
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...
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...
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 ·...
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...
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...
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)##
##\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|}...
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)
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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]...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...