Vector identities


 * $$ \vec{a} \times \vec{b} = - \vec{b} \times  \vec{a} $$
 * $$ \vec{a}\cdot \left({\vec{b} \times  \vec{c}}\right) = \vec{b}\cdot  \left({\vec{c} \times  \vec{a}}\right) = \vec{c}\cdot  \left({\vec{a} \times  \vec{b}}\right) $$
 * $$ \vec{a} \times \left({ \vec{b} \times  \vec{c} }\right) = (\vec{a}\cdot  \vec{c}) \vec{b} - (\vec{a}\cdot  \vec{b}) \vec{c} $$
 * $$ (\vec{a} \times \vec{b})\cdot  (\vec{c} \times  \vec{d}) = (\vec{a}\cdot  \vec{c})(\vec{b}\cdot  \vec{d})-(\vec{a}\cdot  \vec{d})(\vec{b}\cdot  \vec{c}) $$
 * $$ \nabla \times  \nabla \psi  = 0 $$
 * $$ \nabla \cdot  \left({\nabla  \times  \vec{a}}\right) = 0 $$
 * $$ \nabla \times  \left({\nabla  \times  \vec{a}}\right) = \nabla \left({\nabla \cdot  \vec{a}}\right) - {\nabla }^{2}\vec{a} $$
 * $$ \nabla \cdot \left({\psi  \vec{a}}\right) = \vec{a}\cdot \nabla \psi  + \psi  \nabla \cdot \left.{\vec{a}}\right.  $$
 * $$ \nabla \times  \left({\psi \vec{a}}\right) = \nabla \psi  \times  \vec{a} + \psi \nabla  \times  \left.{\vec{a}}\right.  $$
 * $$ \nabla \left({\vec{a}\cdot \vec{b}}\right) = \left({\vec{a}\cdot \nabla }\right) \vec{b}+ \left({\vec{b}\cdot \nabla }\right) \vec{a} + \vec{a} \times  \left({\nabla  \times  \vec{b}}\right) + \vec{b} \times  \left({\nabla  \times  \vec{a}}\right) $$
 * $$ \nabla \cdot \left({\vec{a} \times  \vec{b}}\right) = \vec{b}\cdot  \left({\nabla  \times  \vec{a}}\right) - \vec{a}\cdot  \left({\nabla  \times  \vec{b}}\right) $$
 * $$ \nabla \times  \left({\vec{a} \times  \vec{b}}\right) = \vec{a}\left({\nabla \cdot  \vec{b}}\right) - \vec{b} \left({\nabla \cdot  \vec{a}}\right) + \left({\vec{b}\cdot \nabla }\right)\vec{a} - \left({\vec{a}\cdot \nabla }\right)\vec{b} $$