# Divergence Definition and 53 Discussions

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.

View More On Wikipedia.org
1. ### Series investigation: divergence/convergence

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.
2. ### B Understanding about Sequences and Series

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...
3. ### I Using Diffraction (i.e., Fresnel Zone Plate) to defocus/diverge light

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...
4. ### Divergence of a radial field ##F=\hat{r}/r^{2+\varepsilon}##

Following (1), \begin{align*} \text{div} F = \vec{\nabla} \cdot \vec{F} &= \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 F_{r}\right) \\ &= \frac{1}{r^2} \frac{\partial }{\partial r} \left( r^2 \frac{1}{r^{2+\varepsilon}}\right) \\ &= \frac{1}{r^2} \frac{\partial}{\partial r}...
5. ### Nabla operations, vector calculus problem

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...
6. ### Divergence of a position vector in spherical coordinates

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...
7. ### I Divergence with Chain Rule

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...
8. ### I A one dimensional example of divergence: Mystery

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...
9. ### Divergence of an Electric Field due to an ideal dipole

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...
10. ### I The continuity equation and the divergence

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...
11. ### I What is the physical meaning of divergence?

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...
12. ### Divergence operator for multi-dimensional neutron diffusion

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...

47. ### How to get the laplacian of a scalar field?

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...
48. ### How to compute the divergence of retarded scalar potential

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...
49. ### Simple divergence/Green's theorem question

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...
50. ### Sequence (n!)/(n^n) Convergent or Divergent and Limit?

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...