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 and are parallel vectors in other words, .
If is solenoidal - that is, if such as for an incompressible fluid or a magnetic field, the identity becomes and this leads to
and if we further assume that 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.