In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.
As an example, consider air as it is heated or cooled. The velocity of the air at each point defines a vector field. While air is heated in a region, it expands in all directions, and thus the velocity field points outward from that region. The divergence of the velocity field in that region would thus have a positive value. While the air is cooled and thus contracting, the divergence of the velocity has a negative value.
I understand mathematically several ways to test whether an infinite series converges or diverges. However, I came across one particular equation that is stumping me, ## \sum_{n=1}^{\infty} 1/n ##. I understand how to mathematically apply series tests to show it diverges. But intuitively, I...
Theorem
1. If a series ##{a_n}## converges, then the sequence ##{a_n}## converges to ##0##.
Now, the contra does not apply, and my question is why? i.e if the the sequence ##{a_n}## converges to ##0## then the series may or may not converge correct? and if it does not converge to ##0## then it...
I am currently studying a section from \textit{Electricity and Magnetism} by Purcell, pages 81 and 82, and need some clarification on the following concept. Here’s what I understand so far:
1. The integral of a function $ \mathbf{F} $ over a surface \( S \) is equal to the sum of the integrals...
For this problem,
Let ##a_n = \frac{1}{n(\ln n)^p}##
##b_n = \frac{1}{(n \ln n)^p} = \frac{1}{(n^*)^p}##
We know that ##\sum_{2 \ln 2}^{\infty} \frac{1}{(n^*)^p}## is a p-series with ##n^* = n\ln n##, ##n^* \in \mathbf{R}##
Assume p-series stilll has the same property when ##n^* \in...
So, curl of curl of a vector field is, $$\nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$$
Now, curl means how much a vector field rotates counterclockwise. Then, curl of curl should mean how much the curl rotate counterclockwise.
The laplacian...
Been long since i studied this area...time to go back.
##F = x \cos xi -e^y j+xyz k##
For divergence i have,
##∇⋅F = (\cos x -x\sin x)i -e^y j +xy k##
and for curl,
##∇× F = \left(\dfrac{∂}{∂y}(xyz)-\dfrac{∂}{∂z}(-e^y)\right) i -\left(\dfrac{∂}{∂x}(xyz)-\dfrac{∂}{∂z}(x \cos...
Question: Can we ultimately atttribute no work or net zero work done by a magnetic force to the closed magnetic field lines that results in Divergence zero of a magnetic field? That is, is it a misconception to say that closed magnetic field lines imply magnetic force will always result in no...
We have
$$\vec{E}(\vec{r})=\frac{1}{4\pi\epsilon_0}\int_V\frac{\rho(\vec{r}')}{\eta^2}\hat{\eta}d\tau'\tag{1}$$
A few initial observations
1) I am using notation from the book Introduction to Electrodynamics by Griffiths. When considering point charges, this notation uses position vectors...
As you can see in the homework statement, I am asked to calculate what's effectively the divergence of the vector field ##\vec{F} = \vec{x}/\vert\vec{x}\vert^3## over ##\mathbb{R}^3##. I have done that, the calculation itself isn't that difficult after all. However, I can't make sense of the...
Hi,
unfortunately, I am not sure if I have calculated the task correctly
The electric field of a point charge looks like this ##\vec{E}(\vec{r})=\frac{Q}{4 \pi \epsilon_0}\frac{\vec{r}}{|\vec{r}|^3}## I have now simply divided the electric field into its components i.e. #E_x , E-y, E_z#...
There are several examples that i have looked at which are quite clear and straightforward, e.g
##\sum_{n=0}^\infty 2^n##
it follows that
##\lim_{n \rightarrow \infty} {2^n}=∞##
thus going with the theorem, the series diverges.
Now let's look at the example below;
##\sum_{n=1}^\infty...
Hi everyone!
It's about the following task: show the convergence or divergence of the following series (combine estimates and
criteria).
I am not sure if I have solved the problem correctly. Can you guys help me? Is there anything I need to correct? I look forward to your feedback.
##a_n= \left[\dfrac {\ln (n)^2}{n}\right]##
We may consider a function of a real variable. This is my approach;
##f(x) =\left[\dfrac {\ln (x)^2}{x}\right]##
Applying L'Hopital's rule we shall have;
##\displaystyle\lim_ {x\to\infty} \left[\dfrac {\ln (x)^2}{x}\right]=\lim_ {x\to\infty}\left[...
I know how to apply boundary condition like Dirichlet, Neumann and Robin but i have been struggling to apply divergence free condition for Maxwells or Stokes equations in nodal finite element method. to overcome this difficulties a special element was developed called as edge element but i don't...
Homework Statement:: Tell me if a sequence or series diverges or converges
Relevant Equations:: Geometric series, Telescoping series, Sequences.
If I have a sequence equation can I tell if it converges or diverges by taking its limit or plugging in numbers to see what it goes too?
Also if I...
Im having trouble understanding the divergence of this vector function. I am just getting lost at calculating the divergence. I get that 1/r^2 is a constant so you can pull it out, but where does r^2 x 1/r^2 come from?
What is the index notation of divergence of product of 4th rank tensor and second rank tensor?
What is the index notation of divergence of 3rd rank tensor and vector?
div(a:b) = div(c^transpose. d)
Where a = 4th rank tensor, b is second rank tensor, c is 3rd rank tensor and d is a vector.
I am wondering if it is possible to use principals of diffraction to cause a collimated beam of light (laser) to become divergent. I see that zone plates are most always used for focusing the light from a source, unless they are used in reverse. This is why zone plates are seemingly always...
Hi,
This isn't a homework question, but a side task given in a machine learning class I am taking.
Question: Using variational calculus, prove that one can minimize the KL-divergence by choosing ##q## to be equal to ##p##, given a fixed ##p##.
Attempt:
Unfortunately, I have never seen...
In a previous thread* the field in a charged ring was discussed and it was shown to be not zero except at the center. In *post #45 a video is referenced that says the field diverges as one gets close to the ring and it was argued that at very close distances the field looks like an infinite line...
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:
$$\iint_S \vec F\cdot\hat n\, dS = \iiint_D \nabla\cdot\vec F\,dV$$
$$= \iiint_D \frac{\partial...
##F = (P,Q,R)## is a field of vector C1 defined on ##V = R3-{0,0,0}##
There are a lot of true or false statement here. I am a little skeptical about my answer because it contains a lot of F, but let's go.
1 Rot of F is null in V iff ##\int \int_{S} P dx + Q dy + R dz = 0## for all sphere S...
Greetings!
here is the following exercice
I understand that when we follow the traditional approach, (prametrization of the surface) we got the answer which is 8/3
But why the divergence theorem can not be used in our case? (I know it's a trap here)
thank you!
Hi everyone,
studying the bending of an incompressible elastic block of Neo-Hookean material, one finds out the first Piola-Kirchoff stress tensor as at page 182 (equation 5.93)
where $e_r = cos(\theta)e_1 + \sin(\theta)e_2$ and $e_{\theta} = -sin(\theta)e_1 + \cos(\theta)e_2$
How is the...
The result equation doesn't fit with the familiar divergence form that are usually used in electrodynamics.
I want to know the reason why I was wrong.
My professor says about transformation of components.
But I cannot close to answer by using this hint, because I don't have any idea about "x"...
I wanted to ask about a step I couldn't understand in Tong's notes$$\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}$$we're told that in these coordinates ##\partial M## is a surface of...
in class we derived the following relationship:
$$\frac{1}{V}\frac{dV}{dt}= \nabla \cdot \vec{v}$$
This was derived though the analysis of linear deformation for a fluid-volume, where:
$$dV = dV_x +dV_y + dV_z$$
I understood the derived relation as: 1/V * (derivative wrt time) = div (velocity)...
At some point, in Physics (more precisely in thermodynamics), I must take the divergence of a quantity like ##\mu \vec F##. Where ##\mu## is a scalar function of possibly many different variables such as temperature (which is also a scalar), position, and even magnetic field (a vector field)...
I get a nonsensical result. I am unable to understand where I go wrong.
Let's consider a material with a temperature independent Seebeck coefficient, thermal conductivity and electrochemical potential to keep things simple. Let's assume that this material is sandwiched between 2 other materials...
The integral that I have to solve is as follows:
\oint_{s} \frac{1}{|r-r'|}da', \quad\text{ integrating with respect to r '}, integrating with respect to r'
Then I apply the divergence theorem, resulting in:
\iiint \limits _{v} \nabla \cdot \frac{1}{|r-r'|}dv' =...
Summary:: I am trying to derive that the divergence of a magnetic field is 0. One of the moves is to take the curl out of an integral. Can someone prove that this is addressable
Biot Savart's law is
$$B(r)=\frac{\mu _0}{4\pi} \int \frac{I(r') \times (r-r')}{|r-r|^3}dl'=\frac{\mu _0}{4\pi}...
Divergence & curl are written as the dot/cross product of a gradient.
If we take the dot product or cross product of a gradient, we have to multiply a function by a partial derivative operator.
is multiplication by a partial derivative operator allowed? Or is this just an abuse of notation
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...
Here is how my teacher solved this:
I understand what the nabla operator does, ##∇\cdot\vec v## means that I am supposed to calculate ##\sum_{n=1}^3\frac {d\vec v} {dx_n}## where ##x_n## are cylindrical coordinates and ##\vec e_3 = \vec e_z##. I understand why ##∇\cdot\vec v = 0##, I would get...
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 ·...
For divergence: We learned to draw a circle at different locations and to see if gas is expanding/contracting. Whenever the y-coordinate is positive, the gas seems to be expanding, and it's contracting when negative. I find it hard to tell if the gas is expanding or contracting as I go to the...
Set ##\epsilon=\frac{1}{2}##. Let ##N\in \mathbb{N}## and choose ##n=N,m=2N##. Then:
##\begin{align*}
\left|s_N-s_{2N}\right|&=&\left|\sum_{l=1}^N \frac{1}{l} - \sum_{l=1}^{2N} \frac{1}{l}\right|\\...
I know the divergence of any position vectors in spherical coordinates is just simply 3, which represents their dimension. But there's a little thing that confuses me.
The vector field of A is written as follows,
,
and the divergence of a vector field A in spherical coordinates are written as...
I am looking at the derivation for the Entropy equation for a Newtonian Fluid with Fourier Conduction law. At some point in the derivation I see
\frac{1}{T} \nabla \cdot (-\kappa \nabla T) = - \nabla \cdot (\frac{\kappa \nabla T}{T}) - \frac{\kappa}{T^2}(\nabla T)^2
K is a constant and T...
At the exam i had this power series
but couldn't solve it
##\sum_{k=0}^\infty (-1)^\left(k+1\right) \frac {k} {log(k+1)} (2x-1)^k##
i did apply the ratio test (lets put aside for the moment (2x-1)^k ) to the series ##\sum_{k=0}^\infty \frac {k} {log(k+1)}## in order to see to what this...