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Divergence and curl of a cross product

For smooth vector fields

F, G

,

\nabla \cdot (F \times G) = G \cdot (\nabla \times F) - F \cdot (\nabla \times G)

and

\nabla \times (F \times G) = F (\nabla \cdot G) - G (\nabla \cdot F) + (G \cdot \nabla) F - (F \cdot \nabla) G .

Both follow by contracting two Levi-Civita symbols and applying the product rule.

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