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
Thus formula_2 and formula_3 are parallel vectors in other words, formula_4.
If formula_2 is solenoidal - that is, if formula_6 such as for an incompressible fluid or a magnetic field, the identity formula_7 becomes formula_8 and this leads to
and if we further assume that formula_10 is a constant, we arrive at the simple form
Beltrami vector fields with nonzero curl correspond to Euclidean contact forms in three dimensions.
The vector field
is a multiple of the standard contact structure −"z" i + j, and furnishes an example of a Beltrami vector field.