Solenoidal Fields: Understanding Curl and Divergence

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SUMMARY

The discussion centers on solenoidal fields, specifically the concepts of curl and divergence in vector fields. A vector field with zero divergence indicates the absence of sources or sinks, and all solenoidal fields are divergenceless. The conversation highlights that while a field can exhibit curl at every point, it does not necessarily have to form closed lines, as demonstrated by the parabolic velocity profile in fluid dynamics. The relationship between closed lines and divergenceless fields is clarified, emphasizing that flow lines must either form loops or extend to infinity.

PREREQUISITES
  • Understanding of vector calculus concepts, particularly curl and divergence.
  • Familiarity with solenoidal fields and their properties.
  • Knowledge of fluid dynamics, especially velocity profiles.
  • Basic grasp of vector fields and their graphical representations.
NEXT STEPS
  • Study the mathematical definitions of curl and divergence in vector calculus.
  • Explore the properties of solenoidal fields in greater detail.
  • Investigate fluid dynamics and the implications of velocity profiles on vector fields.
  • Learn about the relationship between flow lines and divergenceless fields in various contexts.
USEFUL FOR

Students and professionals in mathematics, physics, and engineering, particularly those focusing on fluid dynamics and vector field analysis.

fisico30
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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!
 
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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...
 


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...
 

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