I just want to be sure if I'm wrong or not, I want to know what happend for a vector fiel if his divergence is zero ?
Are the vectors have all the same lengh ? Or maybe are they all time parallel ?
PS : I read some questions about this on this site but none of the answers help me :/
Thank you
On
We are talking about a flow field $v$ which does not change in time; but the field vectors change from point to point. Consider a fixed point $p$ within this flow field and a cube $C$ of side length $s\ll1$ centered at $p$. The net flux $\Phi$ of $v$ through the surface $\partial C$ of this cube represents the amount of fluid produced within $C$ per second: This flux is the amount of outgoing fluid minus the amount of incoming fluid per second. When the flow field $v$ is homogeneous (i.e. constant in space) then this difference is zero. In order to compute $\Phi$ we compute approximatively the flux through the six faces of $C$ and obtain $$\Phi\doteq\sum_{i=1}^3 \biggl(v_i\bigl(p_i+{s\over2}\bigr)-v_i\bigl(p_i-{s\over2}\bigr)\biggr)s^2\doteq\sum_{i=1}^3{\partial v_i\over\partial x_i}(p)s\ s^2={\rm vol}(C)\>\sum_{i=1}^3{\partial v_i\over\partial x_i}(p)\qquad(s\ll1)\ .$$ This can be interpreted as follows: There is some "production intensity" at work within $C$ that produces the amount $${\rm div}\>v(p):=\sum_{i=1}^3{\partial v_i\over\partial x_i}(p)$$ of fluid per unit volume and second. This "production intensity" is called the divergence of $v$ at $p$. If this divergence is $\equiv0$ throughout the domain $\Omega$ of $v$ this means that no new fluid is produced anywhere, and this entails that for any body $B\subset\Omega$ the total flux through its surface $\partial B$ is zero.
Imagine that the vector field in question describes the velocity of fluid at a given point in a giant tank of fluid. In this instance, a net positive divergence over a solid region means that there is fluid flowing out of that region or, equivalently, that fluid is being produced within the region, a 'source' if you like. A net negative divergence, on the other hand, would mean that fluid is being sucked into that region, a 'sink' or 'drain', if you like.
If you think about what the formula for divergence is talking about, you'll see it's effectively asking "how much fluid is being 'produced' or 'sucked out' at this point?" similar to how the curl is asking "how much is the fluid circulating around this point?". If we sum over all the 'production' and 'suction' of a region (integrate the divergence over the region), then we get the the amount of fluid that's going through the boundary of that region (integrate the flux over the boundary), which exactly what the Divergence Theorem says. Hope that helps!