Solenoidal Fields: Understanding Curl and Divergence

[fisico30]

solenoidal fields...

hello forum,

curl and divergence are "local" concepts.
If a vector field has zero divergence it means that there is no source (or sink) at that point.
It could be divergenceless everywhere.

If the field is solenoidal it automatically is divergenceless.
I do not understand why a solenoidal field needs to have closed lines however.
Is that true only if we consider a field line that encircles many points?
for example, a field could have a curl at every point but not have closed line, like in the case of velocity field of a fluid in a tube. The parabolic velocity profile is such that the field has curl, but the field lines are straight (no closed lines).

thanks!

 

[Ben Niehoff]

Flow lines may only start and end on sources and sinks, respectively. Therefore, in a divergenceless vector field, the flow lines must either form closed loops, or they must extend to infinity. For example, a constant vector field is divergenceless...

 

[fisico30]

thanks Ben,
I see. So the closed line idea has nothing to do with the fact that the vector field has curl or not, but only on the fact that it is divergenceless...