Concept# Beltrami vector field

Summary

In vector calculus, a Beltrami vector field, named after Eugenio Beltrami, is a vector field in three dimensions that is parallel to its own curl. That is, F is a Beltrami vector field provided that
\mathbf{F}\times (\nabla\times\mathbf{F})=0.
Thus \mathbf{F} and \nabla\times\mathbf{F} are parallel vectors in other words, \nabla\times\mathbf{F} = \lambda \mathbf{F}.
If \mathbf{F} is solenoidal - that is, if \nabla \cdot \mathbf{F} = 0 such as for an incompressible fluid or a magnetic field, the identity \nabla \times (\nabla \times \mathbf{F}) \equiv -\nabla^2 \mathbf{F} + \nabla (\nabla \cdot \mathbf{F}) becomes \nabla \times (\nabla \times \mathbf{F}) \equiv -\nabla^2 \mathbf{F} and this leads to
-\nabla^2 \mathbf{F} = \nabla \times(\lambda \mathbf{F})
and if we further assume that \lambda is a constant, we arrive at the

Official source

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