Basic details of Divergence and Curl (II)

Hi everybody

This time I will give some explanations about these two operators, very important in engineering and science world. This comes when I faced several exercices focused on understanding the divergence concept. Personally I considered that these concepts would be nice for those working on them.

I commented in the previous post that divergence has a direct relation with a flux balance through a surface. Maybe a rigurous way to proceed is to chech the mathematical definition:

$$\nabla \cdot F = \frac{1}{\bigtriangleup V}\int_S \vec{F} \cdot \vec{n} \cdot dS $$

As it can be seen, divergence is defined as a net flux of a field that pass through a surface of a certain volume.  Let's give an example. We will use spherical coordinates:

 $\vec{F}=r^2 \vec{r}$

Applying the divergence definition for a reference system in spherical coordinates:

$\nabla \cdot \vec{F}=\frac{1}{r^2}\frac{\partial}{\partial r} \left( r^2 \vec{F} \right) = 4r $

This result shows that divergence is always positive in all the field, except in (0,0), that must be 0. As we approach more to the origin, vector field lines decrease.

We can analyze zones where divergence is positive, negative or zero. Recall that divergence is related with a balance in a certain volume. As it can be seen, divergence is always positive in all the field. As we move away from the coordinate system, flux passing through different surfaces surrounding spherical volumes (as we have chosen spherical coordinates, in 2D would be circumferences) is larger.

When we plot a detail of this field and we paint field lines at positions r and r+dr, we can see that field lines are larger when we go away from (0,0). Divergence applied to a diferential of volume between these two surfaces (situated at r and r+dr) must be greater than zero.

But this balance may lead to a have some errors when calculating divergence. We will ilustrate an example of this fact:

$\vec{F}=\frac{1}{r^2} \vec{r}$

Vector field is:

As it can be appreciated, field lines decrease when radius increases. If we make a visual analysis following the other example, it may be deduced that divergence in this field is always smaller than zero for every r, giving a zero value in the infinite. If we make an zoom in a certain zone of the vector field:

It seems that field lines are smaller in S+dS than those placed in S, so balance would be in a first step smaller than zero. However, this is not like that. If we calculate divergence analytically:

$\nabla \cdot \vec{F}=\frac{1}{r^2}\frac{\partial}{\partial r} \left( r^2 \vec{F} \right) = 0 $

Result of the calculations give a value equal to zero for every point in the field. What's wrong here?

The answer is not trivial at all. If we observe again the definition of divergence, it can be appreciated that the answer is on the surface flow integral. When we choose a volume as the picture, it can be seen that field value is greater than in surface S than in surface S+dS (it can be seen that field value decreases with increasing r). But it must be mentioned that surface through which field flow passes is also greater in S+dS than in S.

Globally we have that a DECREASE OF THE FIELD VALUE WHEN RADIUS IS INCREASED IS BALANCED WITH SURFACE INCREMENT WHEN WE INCREASE THE RADIUS. It can be concluded that, although F has more intensity in S than in S+dS, the surface through which field passes is smaller than in S+dS, balancing the integrals in both points. In this example, this balance is exactly equal, and then flux passing through both surfaces is exactly the same.

So, resuming:

- Divergence of a vectorial field is a scalar
- The divergence implies a net flux balance in a certain volume through a surface
- When calculating the divergence in a given volume by visual analysis, be aware about value of the field and considered surfaces.


In following posts I will do a similar example, but with curl operator.