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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 F {\displaystyle \mathbf {F} } and ∇ × F {\displaystyle \nabla \times \mathbf {F} } are parallel vectors in other words, ∇ × F = λ F {\displaystyle \nabla \times \mathbf {F} =\lambda \mathbf {F} }.
If F {\displaystyle \mathbf {F} } is solenoidal - that is, if ∇ ⋅ F = 0 {\displaystyle \nabla \cdot \mathbf {F} =0} such as for an incompressible fluid or a magnetic field, the identity ∇ × ≡ − ∇ 2 F + ∇ {\displaystyle \nabla \times \equiv -\nabla ^{2}\mathbf {F} +\nabla } becomes ∇ × ≡ − ∇ 2 F {\displaystyle \nabla \times \equiv -\nabla ^{2}\mathbf {F} } and this leads to
and if we further assume that λ {\displaystyle \lambda } is a constant, we arrive at the simple form