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 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...