Useful Vector Identities

Notation: ${\bf a}$, ${\bf b}$, ${\bf c}$, ${\bf d}$ are general vectors; $\phi$, $\psi$ are general scalar fields; ${\bf A}$, ${\bf B}$ are general vector fields; $({\bf A}\cdot\nabla)\,{\bf B}\equiv ({\bf A}\cdot\nabla B_x,\, {\bf A}\cdot\nabla B_y,\,{\bf A}\cdot\nabla B_z)$ and $\nabla^{\,2}{\bf A} = (\nabla^{\,2} A_x,\, \nabla^{\,2} A_y,\,\nabla^{\,2} A_z)$ (but, only in Cartesian coordinates)

$${\bf a}\times ({\bf b}\times {\bf c}) = ({\bf a}\cdot{\bf c})\,{\bf b} - ({\bf a}\cdot{\bf b})\,{\bf c},$$ (1.176)

$$({\bf a}\times {\bf b})\times {\bf c} = ({\bf c}\cdot{\bf a})\,{\bf b} - ({\bf c}\cdot{\bf b})\,{\bf a},$$ (1.177)

$$({\bf a}\times {\bf b})\cdot({\bf c}\times {\bf d}) = ({\bf a}\cdot{\bf c})\,({\bf b}\cdot{\bf d}) - ({\bf a}\cdot{\bf d})\,({\bf b}\cdot{\bf c}),$$ (1.178)

$$({\bf a}\times {\bf b})\times ({\bf c}\times {\bf d}) = ({\bf a}\times {\bf b}\cdot{\bf d})\,{\bf c} - ({\bf a}\times {\bf b}\cdot{\bf c})\,{\bf d},$$ (1.179)

$$\nabla(\phi\,\psi) =\phi\,\nabla\psi+\psi\,\nabla\phi,$$ (1.180)

$$\nabla({\bf A}\cdot{\bf B}) = {\bf A}\times(\nabla\times{\bf B}) + {\bf B}\times(\nabla\times{\bf A}) +({\bf A}\cdot \nabla)\,{\bf B} + ({\bf B} \cdot \nabla)\,{\bf A},$$ (1.181)

$$\nabla\cdot\nabla \phi =\nabla^{\,2}\phi,$$ (1.182)

$$\nabla\cdot\nabla \times {\bf A} =0,$$ (1.183)

$$\nabla\cdot (\phi\,{\bf A}) =\phi \,\nabla\cdot {\bf A} + {\bf A}\cdot \nabla\phi,$$ (1.184)

$$\nabla\cdot({\bf A}\times{\bf B}) = {\bf B}\cdot \nabla\times{\bf A} - {\bf A}\cdot \nabla\times{\bf B},$$ (1.185)

$$\nabla\times\nabla\phi =0,$$ (1.186)

$$\nabla\times(\nabla\times{\bf A}) = \nabla\,(\nabla\cdot{\bf A})- \nabla^{\,2}{\bf A},$$ (1.187)

$$\nabla\times(\phi\,{\bf A}) =\phi\, \nabla\times{\bf A} +\nabla\phi\times{\bf A},$$ (1.188)

$$\nabla\times({\bf A}\times{\bf B}) = (\nabla\cdot {\bf B})\,{\bf A}- (\nabla \cdot{\bf A}) \,{\bf B}+({\bf B}\cdot\nabla)\,{\bf A}- ({\bf A}\cdot\nabla)\,{\bf B}.$$ (1.189)