Consider a vector field v and an arbitrary constant vector c. In index notation, the cross product is given by v × × c = ε ε i j k v j c k e i where ε ε i j k is the permutation symbol, otherwise known as the Levi-Civita symbol. Then, ∇ ∇ ⋅ ⋅ ( v × × c ) = ε ε i j k v j , i c k = ( ε ε i j k v j , i e k ) ⋅ ⋅ c = ( ∇ ∇ × × v ) ⋅ ⋅ c Therefore, ∇ ∇ × × v = ε ε i j k v j , i e k