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

$\mathbf {F} \times (\nabla \times \mathbf {F} )=0.$ Thus $\mathbf {F}$ and $\nabla \times \mathbf {F}$ are parallel vectors in other words, $\nabla \times \mathbf {F} =\lambda \mathbf {F}$ .

If $\mathbf {F}$ is solenoidal - that is, if $\nabla \cdot \mathbf {F} =0$ such as for an incompressible fluid or a magnetic field, the identity $\nabla \times (\nabla \times \mathbf {F} )\equiv -\nabla ^{2}\mathbf {F} +\nabla (\nabla \cdot \mathbf {F} )$ becomes $\nabla \times (\nabla \times \mathbf {F} )\equiv -\nabla ^{2}\mathbf {F}$ and this leads to

$-\nabla ^{2}\mathbf {F} =\nabla \times (\lambda \mathbf {F} )$ and if we further assume that $\lambda$ is a constant, we arrive at the simple form

$\nabla ^{2}\mathbf {F} =-\lambda ^{2}\mathbf {F} .$ Beltrami vector fields with nonzero curl correspond to Euclidean contact forms in three dimensions.

The vector field

$\mathbf {F} =-{\frac {z}{\sqrt {1+z^{2))))\mathbf {i} +{\frac {1}{\sqrt {1+z^{2))))\mathbf {j}$ is a multiple of the standard contact structure −zi + j, and furnishes an example of a Beltrami vector field.