Vector identities

${\vec {a}}\times {\vec {b}}=-{\vec {b}}\times {\vec {a}}$

${\vec {a}}\cdot \left({{\vec {b}}\times {\vec {c}}}\right)={\vec {b}}\cdot \left({{\vec {c}}\times {\vec {a}}}\right)={\vec {c}}\cdot \left({{\vec {a}}\times {\vec {b}}}\right)$

${\vec {a}}\times \left({{\vec {b}}\times {\vec {c}}}\right)=({\vec {a}}\cdot {\vec {c}}){\vec {b}}-({\vec {a}}\cdot {\vec {b}}){\vec {c}}$

$({\vec {a}}\times {\vec {b}})\cdot ({\vec {c}}\times {\vec {d}})=({\vec {a}}\cdot {\vec {c}})({\vec {b}}\cdot {\vec {d}})-({\vec {a}}\cdot {\vec {d}})({\vec {b}}\cdot {\vec {c}})$

$\nabla \times \nabla \psi =0$

$\nabla \cdot \left({\nabla \times {\vec {a}}}\right)=0$

$\nabla \times \left({\nabla \times {\vec {a}}}\right)=\nabla \left({\nabla \cdot {\vec {a}}}\right)-{\nabla }^{2}{\vec {a}}$

$\nabla \cdot \left({\psi {\vec {a}}}\right)={\vec {a}}\cdot \nabla \psi +\psi \nabla \cdot \left.{\vec {a}}\right.$

$\nabla \times \left({\psi {\vec {a}}}\right)=\nabla \psi \times {\vec {a}}+\psi \nabla \times \left.{\vec {a}}\right.$

$\nabla \left({{\vec {a}}\cdot {\vec {b}}}\right)=\left({{\vec {a}}\cdot \nabla }\right){\vec {b}}+\left({{\vec {b}}\cdot \nabla }\right){\vec {a}}+{\vec {a}}\times \left({\nabla \times {\vec {b}}}\right)+{\vec {b}}\times \left({\nabla \times {\vec {a}}}\right)$

$\nabla \cdot \left({{\vec {a}}\times {\vec {b}}}\right)={\vec {b}}\cdot \left({\nabla \times {\vec {a}}}\right)-{\vec {a}}\cdot \left({\nabla \times {\vec {b}}}\right)$

$\nabla \times \left({{\vec {a}}\times {\vec {b}}}\right)={\vec {a}}\left({\nabla \cdot {\vec {b}}}\right)-{\vec {b}}\left({\nabla \cdot {\vec {a}}}\right)+\left({{\vec {b}}\cdot \nabla }\right){\vec {a}}-\left({{\vec {a}}\cdot \nabla }\right){\vec {b}}$