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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--see Appendix C).

$\displaystyle {\bf a}\times ({\bf b}\times {\bf c})$ $\displaystyle = ({\bf a}\cdot{\bf c})\,{\bf b} - ({\bf a}\cdot{\bf b})\,{\bf c},$ (A.166)
$\displaystyle ({\bf a}\times {\bf b})\times {\bf c}$ $\displaystyle = ({\bf c}\cdot{\bf a})\,{\bf b} - ({\bf c}\cdot{\bf b})\,{\bf a},$ (A.167)
$\displaystyle ({\bf a}\times {\bf b})\cdot({\bf c}\times {\bf d})$ $\displaystyle = ({\bf a}\cdot{\bf c})\,({\bf b}\cdot{\bf d}) - ({\bf a}\cdot{\bf d})\,({\bf b}\cdot{\bf c}),$ (A.168)
$\displaystyle ({\bf a}\times {\bf b})\times ({\bf c}\times {\bf d})$ $\displaystyle = ({\bf a}\times {\bf b}\cdot{\bf d})\,{\bf c} - ({\bf a}\times {\bf b}\cdot{\bf c})\,{\bf d},$ (A.169)
$\displaystyle \nabla(\phi\,\psi)$ $\displaystyle =\phi\,\nabla\psi+\psi\,\nabla\phi,$ (A.170)
$\displaystyle \nabla({\bf A}\cdot{\bf B})$ $\displaystyle = {\bf A}\times(\nabla\times{\bf B}) + {\bf B}\times(\nabla\times{\bf A}) +({\bf A}\cdot \nabla)\,{\bf B} + ({\bf B} \cdot \nabla)\,{\bf A},$ (A.171)
$\displaystyle \nabla\cdot\nabla \phi$ $\displaystyle =\nabla^{\,2}\phi,$ (A.172)
$\displaystyle \nabla\cdot\nabla \times {\bf A}$ $\displaystyle = 0,$ (A.173)
$\displaystyle \nabla\cdot (\phi\,{\bf A})$ $\displaystyle =\phi \,\nabla\cdot {\bf A} + {\bf A}\cdot \nabla\phi,$ (A.174)
$\displaystyle \nabla\cdot({\bf A}\times{\bf B})$ $\displaystyle = {\bf B}\cdot \nabla\times{\bf A} - {\bf A}\cdot \nabla\times{\bf B},$ (A.175)
$\displaystyle \nabla\times\nabla\phi$ $\displaystyle = 0,$ (A.176)
$\displaystyle \nabla\times(\nabla\times{\bf A})$ $\displaystyle = \nabla\,(\nabla\cdot{\bf A})- \nabla^{\,2}{\bf A},$ (A.177)
$\displaystyle \nabla\times(\phi\,{\bf A})$ $\displaystyle =\phi\, \nabla\times{\bf A} +\nabla\phi\times{\bf A},$ (A.178)
$\displaystyle \nabla\times({\bf A}\times{\bf B})$ $\displaystyle = (\nabla\cdot {\bf B})\,{\bf A}- (\nabla \cdot{\bf A}) \,{\bf B}+({\bf B}\cdot\nabla)\,{\bf A}- ({\bf A}\cdot\nabla)\,{\bf B}.$ (A.179)


next up previous
Next: Exercises Up: Vectors and Vector Fields Previous: Curl
Richard Fitzpatrick 2016-03-31