
Beltrami vector field
Encyclopedia
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

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.
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



and if we further assume that


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.