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.
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.
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...
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...
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...
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...
I am trying to understand “divergence” by considering a one-dimensional example of the vector y defined by:
. the parabola: y = -1 + x^2
The direction of the vector y will either be to the right ( R) when y is positive or to the Left (L).
The gradient = dy/dx = Divergence = Div y = 2 x
x...
Given $$\vec E = -\nabla \phi$$ there $$\vec d \rightarrow 0, \phi(\vec r) = \frac {\vec p \cdot \vec r} {r^3}$$ and ##\vec p## is the dipole moment defined as $$\vec p = q\vec d$$
It's quite trivial to show that ##\nabla \times \vec E = \nabla \times (-\nabla \phi) = 0##. However, I want to...
according to continuity equation (partial ρ)/(partial t) +divergence J = 0 . there is such a situation that there is continuous water spreads out from the center of a sphere with unchanged density ρ, and at the center dm/dt = C(a constant), divergence of J = ρv should be 0 anywhere except the...
I want to visualize the concept of divergence of a vector field.I also have searched the web.Some says it is
1.the amount of flux per unit volume in a region around some point
2.Divergence of vector quantity indicates how much the vector spreads out from the certain point.(is a...
Homework Statement
[1] is the one-speed steady-state neutron diffusion equation, where D is the diffusion coefficient, Φ is the neutron flux, Σa is the neutron absorption cross-section, and S is an external neutron source. Solving this equation using a 'homogeneous' material allows D to be...
This is more of a general question, but I've encountered this kind of exercises a lot in my current preperations for my exam:
There are two cases but the excercise is pretty much the same:
Compute
$$(1) \space \operatorname{div}\vec{A}(\vec{r}) \qquad , where \thinspace...
Homework Statement
Show that $$\frac{(-1)^nn!}{z^n}$$ is divergent.
Homework Equations
We can use the ratio test, which states that if, $$\lim_{n\to\infty}\bigg|\frac{a_{n+1}}{a_n}\bigg|>1$$ a series is divergent.
The Attempt at a Solution
Applying the ratio test, we find that...
Homework Statement
Determine whether the following series converge, converge conditionally, or converge absolutely.
Homework Equations
a) Σ(-1)^k×k^3×(5+k)^-2k (where k goes from 1 to infinity)
b) ∑sin(2π + kπ)/√k × ln(k) (where k goes from 2 to infinity)
c) ∑k×sin(1+k^3)/(k + ln(k))...
Hello PF,
I was reading through “A First Course in General Relativity” by Schutz and I got to the part where he derives the divergence formula for a vector:$$V^α { } _{;α} = \frac {1} {\sqrt{-g}} ( \sqrt{-g} V^α )_{,α}$$I’m having trouble with a couple of the steps he made. So we start with the...
Basically a case where a positive charge q is placed in space which for convenience is taken as the origin. This electric field must have a large positive divergence but yet when evaluated mathematically we get 0. Also when we find divergence, we find it for a point right ? or is it possible to...
I just learned that an incompressible fluid must have zero divergence within a given control volume. Given that the divergence of a fluid at a point(x,y,z) can be found by taking the scalar sum of the of the x, y, z acceleration vectors at the given point, wouldn't this mean that water flowing...
Hi Folks,
Was just curious as to what is the gradient of a divergence is and is it always equal to the zero vector. I am doing some free lance research and find that I need to refresh my knowledge of vector calculus a bit. I am having some difficulty with finding web-based sources for the...
Homework Statement
Find the divergence of the function ##\vec{v} = (rcos\theta)\hat{r}+(rsin\theta)\hat{\theta}+(rsin\theta cos\phi)\hat{\phi}##
Homework Equations
##\nabla\cdot\vec{v}=\frac{1}{r^2}\frac{\partial}{\partial r}(r^2v_r)+\frac{1}{r sin\theta}\frac{\partial}{\partial...
Consider a scenario where in one frame R, I have a magnet at rest and a solid slab of charges with an arbitrarily large mass moving at velocity v. The overall acceleration of the slab is trivial, however, the v x B exerted on the slab is divergent, thus compressive/tensile stresses are exerted...
Homework Statement
By Gauss' law, how is it able to obtain ## \nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0} ## ?
By Coulomb's law, ##\vec{E} = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} \hat{r}##
I calculate the divergence of ##\frac{1}{r^2} \hat{r}## and get the result is zero
That means the...
It is my understanding that the task of enumerating all of the divergent diagrams in a quantum field theory can be reduced to analyzing a hand full of diagrams (well, at the moment I know that this is at least true for QED and phi^4 theory), and that all other divergent diagrams are divergent...
Homework Statement
I have a couple of series where I need to find out if they are convergent (absolute/conditional) or divergent.
Σ(n3/3n
Σk(2/3)k
Σ√n/1+n2
Σ(-1)n+1*n/n^2+9
Homework Equations
Comparison Test
Ratio Test
Alternating Series Test
Divergence Test, etc
The Attempt at a...
Homework Statement
Find te gradient of the following function f(r) = rcos(##\theta##) in spherical coordinates.
Homework Equations
\begin{equation}
\nabla f = \frac{\partial f}{\partial r} \hat{r} + (\frac{1}{r}) \frac{\partial f}{\partial \theta} \hat{\theta} + \frac{1}{rsin\theta}...
Homework Statement
The laser beam is not a point source. It is known that it has a rectangular shape with a divergence of 30 mrad x 1 mrad. I would like to know how large my laser lobe will be at a distance of 250 mm from the laser source.
Homework Equations
I think you can use trigonometri...
Why must steady currents be non-divergent in magnetostatics?
Based on an article by Kirk T. McDonald (http://www.physics.princeton.edu/~mcdonald/examples/current.pdf), it appears that the answer is that by extrapolating the linear time dependence of the charge density from a constant divergence...
The Navier-Stokes equation is:
(DUj/Dt) = v [(∂2Ui/∂xj∂xi) + (∂2Uj/∂xi∂xi)] – 1/ρ (∇p)
where D/Dt is the material (substantial) derivative, v is the kinematic viscosity and ∇p is the modified pressure gradient (taking into account gravity and pressure). Note that the velocity field is...
I just took a calc 2 test and got 3/8 points on several problems that asked you to show convergence or divergence. The reason being that I didn't use the correct test of convergence? The answer was right, if you get to the point where you know the series converges, then why does it matter which...
Homework Statement
I just want to focus on the divergence outside the cylinder (r >R)
Homework Equations
The Attempt at a Solution
For r > R, I said ∇ * E = p/ε
But that's wrong. The answer is ∇ * E = 0
I'm confused because there is definitely an electric field outside the cylinder (r...
Homework Statement
Determine which of the sequences converge or diverge. Find the limit of the convergent sequences.
1) {asubn}= [((n^2) + (-1)^n)] / [(4n^2)]
Homework Equations
[/B]
a1=first term, a2=second term...an= nth term
The Attempt at a Solution
a) So I found the first couple of...
My question is mostly about notation. I know the general definitions for divergence and curl, which can be derived from the divergence and Stokes' theorems respectively, are:
\mathrm{div } \vec{E} \bigg| _P = \lim_{\Delta V \to 0} \frac{1}{\Delta V} \iint_{S} \vec{E} \cdot \mathrm{d} \vec{S}...
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]...
Suppose I have electric field of the form ##\mathbf{E} = 3x\mathbf{i} + 3y\mathbf{j}##. Calculating the charge density gives me ##\rho = \epsilon_0 \nabla\cdot\mathbf{E} = 6\epsilon_0##.
But now if I turn one of the components of the field in the opposite direction, for example ##\mathbf{E} =...
Homework Statement
If ##\vec{E}=k\frac{x\hat i +y\hat j}{x^2+y^2}##, find flux through a sphere of radius R centered at origin.
Homework Equations
##\int E.da=\int(\nabla\cdot E)\cdot da##
The Attempt at a Solution
I was able to solve this problem without finding divergence of electric field...
Hi everyone,
I am wondering if anybody could help me out. For my study I got the following question but I got stuck in part C (see image below).
I Found at A that due to symmetry all components which are not in Ar direction will get canceled out
I found at B that there is only charge density at...
Homework Statement
The problem puts forth and identity for me to prove: or . It says that I can use "straight-forward" calculation to solve this using the definition of nabla or I can use Gauss's and Stoke's Theorum on an example in which I have a solid 3D shape nearly cut in two by a curve...
Sorry if this was addressed in another thread, but I couldn't find a discussion of it in a preliminary search. If it is discussed elsewhere, I'll appreciate being directed to it.
Okay, well here's my question. If I take the divergence of the unit radial vector field, I get the result:
\vec...
What is the difference between
\int_{-\infty}^{\infty} \frac{x}{1+x^2}dx
and
\lim_{R\rightarrow \infty}\int_{-R}^{R} \frac{x}{1+x^2}dx ?
And why does the first expression diverge, whilst the second converges and is equal to zero?
How can I find out if 1/n! is divergent or convergent?
I cannot solve it using integral test because the expression contains a factorial.
I also tried solving it using Divergence test. The limit of 1/n! as n approaches infinity is zero. So it follows that no information can be obtained using...
Dear All,
I am studying electrodynamics and I am trying hard to clearly understand each and every formula. By "understand" I mean that I can "truly see its meaning in front of my eyes". Generally, I am not satisfied only by being able to prove or derive certain formula algebraically; I want to...
Hello All,
If I apply the Divergence Operator on the incompressible Navier-Stokes equation, I get this equation:
$$\nabla ^2P = -\rho \nabla \cdot \left [ V \cdot \nabla V \right ]$$
In 2D cartesian coordinates (x and y), I am supposed to get:
$$\nabla ^2P = -\rho \left[ \left( \frac...
Hi,
I am trying to calculate the laplacian of a scalar field but I might actually need something else. So basically I am applying reaction diffusion on a 2d image. I am reading the neighbours, multiplying them with these weights and then add them.
This works great. I don't know if what I am...
I'm learning time-dependent Maxwell's Equations and having difficulty understanding the following derivative:
Given f(\textbf{r}, \textbf{r}', t) = \frac{[\rho(\textbf{r}, t)]}{|\textbf{r} - \textbf{r}'|}
where
\textbf{r} = x \cdot \textbf{i} + y \cdot \textbf{j} + z \cdot \textbf{k}, in...
I'm exploring the divergence theorem and Green's theorem, but I seem to be lacking some understanding. I have tried this problem several times, and I am wondering where my mistake is in this method.
The problem:
For one example, I am trying to find the divergence of some vector field from a...
Homework Statement
Is the sequence {(n!)/(n^n)} convergent or divergent. If it is convergent, find its limit.
Homework Equations
Usually with sequences, you just take the limit and if the limit isn't infinity, it converges... That doesn't really work here. I know I'm supposed to write out the...