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

If F is solenoidal - that is, if ∇ ⋅ F = 0 such as for an incompressible fluid or a magnetic field, we may examine ∇ × ( ∇ × F ) ≡ − ∇ 2 F + ∇ ( ∇ ⋅ F ) and apply this identity twice to find that

and if we further assume that λ 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.