In vector calculus a solenoidal vector field is a vector fieldv with divergence zero: Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in 2 or more dimensions. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ...
This condition is clearly satisfied whenever v has a vector potential, because if In vector calculus, a vector potential is a vector field whose curl is a given vector field. ...
then
The converse holds: for any solenoidal v there exists a vector potential A such that . (Strictly, this holds only subject to certain technical conditions on v.)
In vector calculus, a vector potential is a vectorfield whose curl is a given vectorfield.
This is analogous to a scalar potential, which is a scalar field whose negative gradient is a given vectorfield.
A generalization of this theorem is the Helmholtz decomposition which states that any vectorfield can be decomposed as a sum of a solenoidalvectorfield and an irrotational vectorfield.
In the case of fluid dynamics, the movement of a fluid is solenoidal when the fluid moves in a circle, or any kind of closed loop.
As the solenoidal movement decays with distance, it is as if the central gear were connected to several smaller surrounding gears moving counterclockwise, each with a positive curl, but none of them with a curl as strong as the central curl inside the solenoidal movement.
Anyway, the point is that not only does solenoidal (or vortical) velocity have non-zero curl, but it can itself be the curl of a potential (and the potential might not be too different from the curl of the velocity).
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