Trying to understand the concept of divergence

[Taturana]

$$\operatorname{div}\,\mathbf{F}(p) = <br /> \lim_{V \rightarrow \{p\}}<br /> \iint_{S(V)} {\mathbf{F}\cdot\mathbf{n} \over |V| } \; dS$$

This is the definition of divergence from wikipedia...

The divergence is property of a point in space. Is that right?

If the divergence is zero at a point, that means that such point does not contribute with the field as source nor a sink. Is that right?

So, the divergence of a point measures how that point contributes as a source or a sink with the field?

The surface integral in the equation above means a certain area, right? Is that area the area of the entire surface (like a gaussian surface in the gauss's law) or the area of the micro-surface that is "around" the point I'm measuring the divergence on?

Usually I like to think in the dimensions of the conceps (units). I noticed that the unit of divergence will always be area/volume (m^-1). Does that have any meaning?

If someone can help me with some of these questions I would be grateful...

Thank you,
Rafael Andreatta

 

[Taturana]

Nobody can help me?

Either I asked some very noob questions or very hard questions haha

 

[mathman]

There is an alternate way of expressing divergence (which is the one I am used to).

In Cartesian coordinates:
divF=∂F/∂x + ∂F/∂y + ∂F/∂z

This definition is for any point in space where the partials are defined.

 

[arildno]

$$\operatorname{div}\,\mathbf{F}(p) = <br /> \lim_{V \rightarrow \{p\}}<br /> \iint_{S(V)} {\mathbf{F}\cdot\mathbf{n} \over |V| } \; dS$$

This is the definition of divergence from wikipedia..

It is, indeed.

The divergence is property of a point in space. Is that right?

No.
The divergence is a property of your vector field F.

If the divergence is zero at a point, that means that such point does not contribute with the field as source nor a sink. Is that right?

Indeed.
Then there is no net flux of F per unit volume centred about that point.

So, the divergence of a point measures how that point contributes as a source or a sink with the field?

Yes.

The surface integral in the equation above means a certain area, right? Is that area the area of the entire surface (like a gaussian surface in the gauss's law)

or the area of the micro-surface that is "around" the point I'm measuring the divergence on?

Your limiting process consists computing the net flux of F across the surrounding surfaces of ever-shrinking volumes V, giving you, in the limit of V to 0, the divergence of F at that point.

Usually I like to think in the dimensions of the conceps (units). I noticed that the unit of divergence will always be area/volume (m^-1). Does that have any meaning?

Units are: (area)*(unit of F)/volume.

If, for example, F is (velocity field of some fluid), then (area)*F gives the net amount of fluid flowing out of V; dividing with V gives you the volume flux per unit volume.

If someone can help me with some of these questions I would be grateful...

Thank you,
Rafael Andreatta

Ask more if you feel to.