Beltrami vector field

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

${\displaystyle \mathbf {F} \times (\nabla \times \mathbf {F} )=0.}$

Thus ${\displaystyle \mathbf {F} }$ and ${\displaystyle \nabla \times \mathbf {F} }$ are parallel vectors in other words, ${\displaystyle \nabla \times \mathbf {F} =\lambda \mathbf {F} }$.

If ${\displaystyle \mathbf {F} }$ is solenoidal - that is, if ${\displaystyle \nabla \cdot \mathbf {F} =0}$ such as for an incompressible fluid or a magnetic field, the identity ${\displaystyle \nabla \times (\nabla \times \mathbf {F} )\equiv -\nabla ^{2}\mathbf {F} +\nabla (\nabla \cdot \mathbf {F} )}$ becomes ${\displaystyle \nabla \times (\nabla \times \mathbf {F} )\equiv -\nabla ^{2}\mathbf {F} }$ and this leads to

${\displaystyle -\nabla ^{2}\mathbf {F} =\nabla \times (\lambda \mathbf {F} )}$

and if we further assume that ${\displaystyle \lambda }$ is a constant, we arrive at the simple form

${\displaystyle \nabla ^{2}\mathbf {F} =-\lambda ^{2}\mathbf {F} .}$

Beltrami vector fields with nonzero curl correspond to Euclidean contact forms in three dimensions.

The vector field

${\displaystyle \mathbf {F} =-{\frac {z}{\sqrt {1+z^{2}}}}\mathbf {i} +{\frac {1}{\sqrt {1+z^{2}}}}\mathbf {j} }$

is a multiple of the standard contact structure −z i + j, and furnishes an example of a Beltrami vector field.