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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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Divergence and curl of a cross product

Curl of a cross product ( $\nabla \times (U \times v) = U \nabla \cdot v - v \nabla \cdot U + (v \cdot \nabla) U - (U \cdot \nabla) v$ )

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Divergence and curl of a cross product

Curl of a cross product (∇×(U×v)=U∇⋅v−v∇⋅U+(v⋅∇)U−(U⋅∇)v)

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Curl of a cross product ( $\nabla \times (U \times v) = U \nabla \cdot v - v \nabla \cdot U + (v \cdot \nabla) U - (U \cdot \nabla) v$ )

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