In vector calculus, a Beltrami vector field, named after Eugenio Beltrami

Eugenio Beltrami

Eugenio Beltrami was an Italian mathematician notable for his work concerning differential geometry and mathematical physics...

, is a vector field

Vector field

In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

 in three dimensions that is parallel to its own curl. That is, F is a Beltrami vector field provided that


If is solenoidal - that is, if such as for an incompressible fluid or a magnetic field, we may examine and use apply this identity twice to find that

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.